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 | merco1lem4 | ⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | merco1lem3 | ⊢ ( ( ( ( 𝜓 → ⊥ ) → ( 𝜑 → ⊥ ) ) → ( ( 𝜒 → 𝜑 ) → ⊥ ) ) → ( ( 𝜒 → 𝜑 ) → ( 𝜓 → ⊥ ) ) ) | |
| 2 | merco1 | ⊢ ( ( ( ( ( 𝜓 → ⊥ ) → ( 𝜑 → ⊥ ) ) → ( ( 𝜒 → 𝜑 ) → ⊥ ) ) → ( ( 𝜒 → 𝜑 ) → ( 𝜓 → ⊥ ) ) ) → ( ( ( ( 𝜒 → 𝜑 ) → ( 𝜓 → ⊥ ) ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) | |
| 3 | 1 2 | ax-mp | ⊢ ( ( ( ( 𝜒 → 𝜑 ) → ( 𝜓 → ⊥ ) ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) |
| 4 | merco1 | ⊢ ( ( ( ( ( 𝜒 → 𝜑 ) → ( 𝜓 → ⊥ ) ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) ) | |
| 5 | 3 4 | ax-mp | ⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) |