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Metamath Proof Explorer

Theorem mercolem3

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 mercolem3 ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) )

Proof

Step Hyp Ref Expression
1 merco2 ⊢ ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) )
2 merco2 ⊢ ( ( ( 𝜒 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜓 ) ) → ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) )
3 mercolem2 ⊢ ( ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → 𝜓 ) → ( ( ⊥ → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜓 ) ) )
4 merco2 ⊢ ( ( ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → 𝜓 ) → ( ( ⊥ → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜓 ) ) ) → ( ( ( ( ⊥ → 𝜑 ) → 𝜓 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ⊥ → 𝜑 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) )
5 3 4 ax-mp ⊢ ( ( ( ( ⊥ → 𝜑 ) → 𝜓 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ⊥ → 𝜑 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) )
6 merco2 ⊢ ( ( ( ( ( ⊥ → 𝜑 ) → 𝜓 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ⊥ → 𝜑 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ⊥ → 𝜑 ) → 𝜓 ) ) → ( ( ⊥ → 𝜑 ) → ( ( 𝜒 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜓 ) ) ) ) )
7 5 6 ax-mp ⊢ ( ( ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ⊥ → 𝜑 ) → 𝜓 ) ) → ( ( ⊥ → 𝜑 ) → ( ( 𝜒 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜓 ) ) ) )
8 merco2 ⊢ ( ( ( ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ⊥ → 𝜑 ) → 𝜓 ) ) → ( ( ⊥ → 𝜑 ) → ( ( 𝜒 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜓 ) ) ) ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜓 ) ) → ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) )
9 7 8 ax-mp ⊢ ( ( ( ( 𝜒 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜓 ) ) → ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) )
10 2 9 ax-mp ⊢ ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) )
11 1 10 ax-mp ⊢ ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) )
12 1 11 ax-mp ⊢ ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) )