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Metamath Proof Explorer

Theorem merco1lem17

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 merco1lem17 ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜏 ) → ( ( 𝜑 → 𝜒 ) → 𝜏 ) )

Proof

Step Hyp Ref Expression
1 merco1lem11 ⊢ ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) )
2 merco1lem7 ⊢ ( ( ( ( ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) → 𝜑 ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) → ⊥ ) ) → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) )
3 merco1 ⊢ ( ( ( ( ( ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) → 𝜑 ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) → ⊥ ) ) → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) ) )
4 2 3 ax-mp ⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) )
5 merco1lem9 ⊢ ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) )
6 4 5 ax-mp ⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) )
7 1 6 ax-mp ⊢ ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 )
8 merco1 ⊢ ( ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) → ( ( 𝜑 → 𝜒 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) ) )
9 7 8 ax-mp ⊢ ( ( 𝜑 → 𝜒 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) )
10 merco1lem11 ⊢ ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) )
11 merco1lem7 ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → 𝜑 ) → ( ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) → ⊥ ) ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) )
12 merco1 ⊢ ( ( ( ( ( ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → 𝜑 ) → ( ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) → ⊥ ) ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) ) )
13 11 12 ax-mp ⊢ ( ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) )
14 merco1lem9 ⊢ ( ( ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) ) → ( ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) )
15 13 14 ax-mp ⊢ ( ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) )
16 10 15 ax-mp ⊢ ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) )
17 merco1 ⊢ ( ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜒 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) ) ) )
18 16 17 ax-mp ⊢ ( ( ( 𝜑 → 𝜒 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) ) )
19 9 18 ax-mp ⊢ ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) )
20 merco1lem16 ⊢ ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → 𝜒 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) ) )
21 19 20 ax-mp ⊢ ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → 𝜒 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) )
22 merco1lem4 ⊢ ( ( ( ( 𝜏 → 𝜑 ) → ( ( 𝜑 → 𝜒 ) → ⊥ ) ) → 𝜒 ) → ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → 𝜒 ) )
23 merco1lem11 ⊢ ( ( ( ( ( 𝜏 → 𝜑 ) → ( ( 𝜑 → 𝜒 ) → ⊥ ) ) → 𝜒 ) → ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → 𝜒 ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜏 → 𝜑 ) → ( ( 𝜑 → 𝜒 ) → ⊥ ) ) → 𝜒 ) → ⊥ ) ) → ⊥ ) → ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → 𝜒 ) ) )
24 22 23 ax-mp ⊢ ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜏 → 𝜑 ) → ( ( 𝜑 → 𝜒 ) → ⊥ ) ) → 𝜒 ) → ⊥ ) ) → ⊥ ) → ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → 𝜒 ) )
25 merco1 ⊢ ( ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜏 → 𝜑 ) → ( ( 𝜑 → 𝜒 ) → ⊥ ) ) → 𝜒 ) → ⊥ ) ) → ⊥ ) → ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → 𝜒 ) ) → ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → 𝜒 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) ) → ( ( ( ( 𝜏 → 𝜑 ) → ( ( 𝜑 → 𝜒 ) → ⊥ ) ) → 𝜒 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) ) ) )
26 24 25 ax-mp ⊢ ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → 𝜒 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) ) → ( ( ( ( 𝜏 → 𝜑 ) → ( ( 𝜑 → 𝜒 ) → ⊥ ) ) → 𝜒 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) ) )
27 21 26 ax-mp ⊢ ( ( ( ( 𝜏 → 𝜑 ) → ( ( 𝜑 → 𝜒 ) → ⊥ ) ) → 𝜒 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) )
28 merco1 ⊢ ( ( ( ( ( 𝜏 → 𝜑 ) → ( ( 𝜑 → 𝜒 ) → ⊥ ) ) → 𝜒 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜏 ) → ( ( 𝜑 → 𝜒 ) → 𝜏 ) ) )
29 27 28 ax-mp ⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜏 ) → ( ( 𝜑 → 𝜒 ) → 𝜏 ) )