Description: tbw-ax1 rederived from merco1 .
This theorem, along with retbwax2 , retbwax3 , and retbwax4 , shows that merco1 with ax-mp can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 18-Sep-2011) (Proof modification is discouraged.) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Assertion | retbwax1 | ⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | merco1lem18 | ⊢ ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) | |
2 | merco1lem16 | ⊢ ( ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) ) | |
3 | 1 2 | ax-mp | ⊢ ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) |
4 | merco1lem15 | ⊢ ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) | |
5 | merco1lem15 | ⊢ ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) ) | |
6 | 4 5 | ax-mp | ⊢ ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) |
7 | merco1lem18 | ⊢ ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) ) | |
8 | 6 7 | ax-mp | ⊢ ( ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) |
9 | merco1lem14 | ⊢ ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) ) | |
10 | 8 9 | ax-mp | ⊢ ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) |
11 | merco1lem14 | ⊢ ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( 𝜑 → 𝜒 ) ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) | |
12 | merco1lem10 | ⊢ ( ( ( ( ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ⊥ ) → ⊥ ) → ( ( ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → 𝜑 ) → ⊥ ) ) → ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) ) → ( ( ( 𝜑 → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) ) ) | |
13 | merco1 | ⊢ ( ( ( ( ( ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ⊥ ) → ⊥ ) → ( ( ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → 𝜑 ) → ⊥ ) ) → ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) ) → ( ( ( 𝜑 → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) ) ) → ( ( ( ( ( 𝜑 → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) ) → ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ⊥ ) ) → ( ( ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → 𝜑 ) → ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ⊥ ) ) ) ) | |
14 | 12 13 | ax-mp | ⊢ ( ( ( ( ( 𝜑 → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) ) → ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ⊥ ) ) → ( ( ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → 𝜑 ) → ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ⊥ ) ) ) |
15 | merco1 | ⊢ ( ( ( ( ( ( 𝜑 → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) ) → ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ⊥ ) ) → ( ( ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → 𝜑 ) → ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ⊥ ) ) ) → ( ( ( ( ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → 𝜑 ) → ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ⊥ ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( 𝜑 → 𝜒 ) ) ) ) | |
16 | 14 15 | ax-mp | ⊢ ( ( ( ( ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → 𝜑 ) → ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ⊥ ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( 𝜑 → 𝜒 ) ) ) |
17 | merco1 | ⊢ ( ( ( ( ( ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → 𝜑 ) → ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ⊥ ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( 𝜑 → 𝜒 ) ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) | |
18 | 16 17 | ax-mp | ⊢ ( ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( 𝜑 → 𝜒 ) ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) |
19 | 11 18 | ax-mp | ⊢ ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) |
20 | merco1lem15 | ⊢ ( ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) | |
21 | 19 20 | ax-mp | ⊢ ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) |
22 | merco1lem10 | ⊢ ( ( ( ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) ) → ( ( ( ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) | |
23 | merco1lem9 | ⊢ ( ( ( ( ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) ) → ( ( ( ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) → ( ( ( ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) | |
24 | 22 23 | ax-mp | ⊢ ( ( ( ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) ) |
25 | merco1lem13 | ⊢ ( ( ( ( ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) ) → ( ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) | |
26 | 24 25 | ax-mp | ⊢ ( ( ( ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ⊥ ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) ) |
27 | 21 26 | ax-mp | ⊢ ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) |
28 | 10 27 | ax-mp | ⊢ ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) |
29 | 3 28 | ax-mp | ⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) |