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Metamath Proof Explorer

Theorem mercolem6

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 . (Contributed by Anthony Hart, 16-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion mercolem6 ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) )

Proof

Step Hyp Ref Expression
1 merco2 ⊢ ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) )
2 mercolem1 ⊢ ( ( ( 𝜑 → ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) → ( 𝜑 → 𝜒 ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) )
3 mercolem1 ⊢ ( ( ( ( 𝜑 → ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) → ( 𝜑 → 𝜒 ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) → ( ( 𝜑 → 𝜒 ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) )
4 2 3 ax-mp ⊢ ( ( 𝜑 → 𝜒 ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) )
5 mercolem5 ⊢ ( 𝜑 → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) )
6 mercolem4 ⊢ ( ( 𝜑 → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) → ( ( ( 𝜑 → 𝜒 ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) )
7 5 6 ax-mp ⊢ ( ( ( 𝜑 → 𝜒 ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) )
8 4 7 ax-mp ⊢ ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) )
9 1 8 ax-mp ⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) )
10 mercolem1 ⊢ ( ( ( 𝜑 → ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) )
11 mercolem1 ⊢ ( ( ( ( 𝜑 → ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) → ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) )
12 10 11 ax-mp ⊢ ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) )
13 mercolem5 ⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) )
14 mercolem4 ⊢ ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) ) → ( ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) ) ) )
15 13 14 ax-mp ⊢ ( ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) ) )
16 12 15 ax-mp ⊢ ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) )
17 1 16 ax-mp ⊢ ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) )
18 9 17 ax-mp ⊢ ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) )
19 1 18 ax-mp ⊢ ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) )
20 1 19 ax-mp ⊢ ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) )
21 1 20 ax-mp ⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) )