Description: Derive ax-13 from ax-c9 and other older axioms.
This proof uses newer axioms ax-4 and ax-6 , but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-c4 and ax-c10 . (Contributed by NM, 21-Dec-2015) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ax13fromc9 | ⊢ ( ¬ 𝑥 = 𝑦 → ( 𝑦 = 𝑧 → ∀ 𝑥 𝑦 = 𝑧 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-c5 | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦 ) | |
| 2 | 1 | con3i | ⊢ ( ¬ 𝑥 = 𝑦 → ¬ ∀ 𝑥 𝑥 = 𝑦 ) |
| 3 | ax-c5 | ⊢ ( ∀ 𝑥 𝑥 = 𝑧 → 𝑥 = 𝑧 ) | |
| 4 | 3 | con3i | ⊢ ( ¬ 𝑥 = 𝑧 → ¬ ∀ 𝑥 𝑥 = 𝑧 ) |
| 5 | ax-c9 | ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( ¬ ∀ 𝑥 𝑥 = 𝑧 → ( 𝑦 = 𝑧 → ∀ 𝑥 𝑦 = 𝑧 ) ) ) | |
| 6 | 2 4 5 | syl2im | ⊢ ( ¬ 𝑥 = 𝑦 → ( ¬ 𝑥 = 𝑧 → ( 𝑦 = 𝑧 → ∀ 𝑥 𝑦 = 𝑧 ) ) ) |
| 7 | ax13b | ⊢ ( ( ¬ 𝑥 = 𝑦 → ( 𝑦 = 𝑧 → ∀ 𝑥 𝑦 = 𝑧 ) ) ↔ ( ¬ 𝑥 = 𝑦 → ( ¬ 𝑥 = 𝑧 → ( 𝑦 = 𝑧 → ∀ 𝑥 𝑦 = 𝑧 ) ) ) ) | |
| 8 | 6 7 | mpbir | ⊢ ( ¬ 𝑥 = 𝑦 → ( 𝑦 = 𝑧 → ∀ 𝑥 𝑦 = 𝑧 ) ) |