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 ) ) ) |