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

Metamath Proof Explorer

Theorem merco1lem1

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 . (Contributed by Anthony Hart, 17-Sep-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion merco1lem1 ( 𝜑 → ( ⊥ → 𝜒 ) )

Proof

Step Hyp Ref Expression
1 merco1 ⊢ ( ( ( ( ( ⊥ → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜑 ) ) → ( ( ( ⊥ → 𝜑 ) → ⊥ ) → ( 𝜑 → ⊥ ) ) )
2 merco1 ⊢ ( ( ( ( ( ( ⊥ → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜑 ) ) → ( ( ( ⊥ → 𝜑 ) → ⊥ ) → ( 𝜑 → ⊥ ) ) ) → ( ( ( ( ( ⊥ → 𝜑 ) → ⊥ ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜑 ) ) → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) )
3 1 2 ax-mp ⊢ ( ( ( ( ( ⊥ → 𝜑 ) → ⊥ ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜑 ) ) → ( 𝜑 → ( ⊥ → 𝜑 ) ) )
4 merco1 ⊢ ( ( ( ( ( ( ⊥ → 𝜑 ) → ⊥ ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜑 ) ) → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜑 ) ) → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) )
5 3 4 ax-mp ⊢ ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜑 ) ) → ( 𝜑 → ( ⊥ → 𝜑 ) ) )
6 merco1 ⊢ ( ( ( ( ( ⊥ → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ⊥ ) ) → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ⊥ ) → ( 𝜑 → ⊥ ) ) )
7 merco1 ⊢ ( ( ( ( ( ( ⊥ → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ⊥ ) ) → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ⊥ ) → ( 𝜑 → ⊥ ) ) ) → ( ( ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ⊥ ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜑 ) ) → ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜑 ) ) ) )
8 6 7 ax-mp ⊢ ( ( ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ⊥ ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜑 ) ) → ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜑 ) ) )
9 merco1 ⊢ ( ( ( ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ⊥ ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜑 ) ) → ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜑 ) ) ) → ( ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜑 ) ) → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) ) )
10 8 9 ax-mp ⊢ ( ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜑 ) ) → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) )
11 5 10 ax-mp ⊢ ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) )
12 merco1 ⊢ ( ( ( ( ( ⊥ → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜒 ) ) → ( ( ( ⊥ → 𝜒 ) → ⊥ ) → ( 𝜑 → ⊥ ) ) )
13 merco1 ⊢ ( ( ( ( ( ( ⊥ → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜒 ) ) → ( ( ( ⊥ → 𝜒 ) → ⊥ ) → ( 𝜑 → ⊥ ) ) ) → ( ( ( ( ( ⊥ → 𝜒 ) → ⊥ ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜑 ) ) → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) )
14 12 13 ax-mp ⊢ ( ( ( ( ( ⊥ → 𝜒 ) → ⊥ ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜑 ) ) → ( 𝜑 → ( ⊥ → 𝜑 ) ) )
15 merco1 ⊢ ( ( ( ( ( ( ⊥ → 𝜒 ) → ⊥ ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜑 ) ) → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜒 ) ) → ( 𝜑 → ( ⊥ → 𝜒 ) ) ) )
16 14 15 ax-mp ⊢ ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜒 ) ) → ( 𝜑 → ( ⊥ → 𝜒 ) ) )
17 merco1 ⊢ ( ( ( ( ( ⊥ → 𝜒 ) → ( ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ⊥ ) ) → ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ⊥ ) ) → ( 𝜑 → ( ⊥ → 𝜒 ) ) ) → ( ( ( 𝜑 → ( ⊥ → 𝜒 ) ) → ⊥ ) → ( ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ⊥ ) ) )
18 merco1 ⊢ ( ( ( ( ( ( ⊥ → 𝜒 ) → ( ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ⊥ ) ) → ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ⊥ ) ) → ( 𝜑 → ( ⊥ → 𝜒 ) ) ) → ( ( ( 𝜑 → ( ⊥ → 𝜒 ) ) → ⊥ ) → ( ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ⊥ ) ) ) → ( ( ( ( ( 𝜑 → ( ⊥ → 𝜒 ) ) → ⊥ ) → ( ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ⊥ ) ) → ( ⊥ → 𝜒 ) ) → ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜒 ) ) ) )
19 17 18 ax-mp ⊢ ( ( ( ( ( 𝜑 → ( ⊥ → 𝜒 ) ) → ⊥ ) → ( ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ⊥ ) ) → ( ⊥ → 𝜒 ) ) → ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜒 ) ) )
20 merco1 ⊢ ( ( ( ( ( ( 𝜑 → ( ⊥ → 𝜒 ) ) → ⊥ ) → ( ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ⊥ ) ) → ( ⊥ → 𝜒 ) ) → ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜒 ) ) ) → ( ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜒 ) ) → ( 𝜑 → ( ⊥ → 𝜒 ) ) ) → ( ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ( 𝜑 → ( ⊥ → 𝜒 ) ) ) ) )
21 19 20 ax-mp ⊢ ( ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜒 ) ) → ( 𝜑 → ( ⊥ → 𝜒 ) ) ) → ( ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ( 𝜑 → ( ⊥ → 𝜒 ) ) ) )
22 16 21 ax-mp ⊢ ( ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ( 𝜑 → ( ⊥ → 𝜒 ) ) )
23 11 22 ax-mp ⊢ ( 𝜑 → ( ⊥ → 𝜒 ) )