This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem merco1lem18

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 . (Contributed by Anthony Hart, 18-Sep-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion merco1lem18 ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) )

Proof

Step Hyp Ref Expression
1 merco1 ⊢ ( ( ( ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ⊥ ) ) → ( ( 𝜓 → 𝜒 ) → 𝜓 ) ) → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) )
2 merco1lem17 ⊢ ( ( ( ( ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ⊥ ) ) → ( ( 𝜓 → 𝜒 ) → 𝜓 ) ) → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) )
3 1 2 ax-mp ⊢ ( ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) )
4 merco1lem17 ⊢ ( ( ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) )
5 3 4 ax-mp ⊢ ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) )
6 merco1lem5 ⊢ ( ( ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ⊥ ) → ⊥ ) → ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) )
7 merco1lem3 ⊢ ( ( ( ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ⊥ ) → ⊥ ) → ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) ) → ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ⊥ ) ) )
8 6 7 ax-mp ⊢ ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ⊥ ) )
9 merco1lem5 ⊢ ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ⊥ ) ) → ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ⊥ ) ) )
10 8 9 ax-mp ⊢ ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ⊥ ) )
11 merco1lem4 ⊢ ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ⊥ ) ) → ( ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) → ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ⊥ ) ) )
12 10 11 ax-mp ⊢ ( ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) → ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ⊥ ) )
13 merco1 ⊢ ( ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ( 𝜓 → 𝜑 ) ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) ) )
14 merco1lem2 ⊢ ( ( ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ( 𝜓 → 𝜑 ) ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) ) ) → ( ( ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) → ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ⊥ ) ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) ) ) )
15 13 14 ax-mp ⊢ ( ( ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) → ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ⊥ ) ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) ) )
16 12 15 ax-mp ⊢ ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) )
17 merco1lem9 ⊢ ( ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) )
18 16 17 ax-mp ⊢ ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) )
19 5 18 ax-mp ⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) )