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 | merco1lem7 | ⊢ ( 𝜑 → ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → 𝜓 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | merco1lem5 | ⊢ ( ( ( ( 𝜓 → ⊥ ) → ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → ⊥ ) ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) | |
| 2 | merco1 | ⊢ ( ( ( ( ( 𝜓 → ⊥ ) → ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → ⊥ ) ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) → ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → 𝜓 ) ) ) | |
| 3 | 1 2 | ax-mp | ⊢ ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → 𝜓 ) ) |
| 4 | merco1lem6 | ⊢ ( ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → 𝜓 ) ) → ( 𝜑 → ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → 𝜓 ) ) ) | |
| 5 | 3 4 | ax-mp | ⊢ ( 𝜑 → ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → 𝜓 ) ) |