Description: Alternate proof of equid and equid1 from older axioms ax-c7 , ax-c10 and ax-c9 . (Contributed by NM, 10-Jan-1993) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | equid1ALT | ⊢ 𝑥 = 𝑥 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-c9 | ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑥 → ( ¬ ∀ 𝑥 𝑥 = 𝑥 → ( 𝑥 = 𝑥 → ∀ 𝑥 𝑥 = 𝑥 ) ) ) | |
| 2 | 1 | pm2.43i | ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑥 → ( 𝑥 = 𝑥 → ∀ 𝑥 𝑥 = 𝑥 ) ) |
| 3 | 2 | alimi | ⊢ ( ∀ 𝑥 ¬ ∀ 𝑥 𝑥 = 𝑥 → ∀ 𝑥 ( 𝑥 = 𝑥 → ∀ 𝑥 𝑥 = 𝑥 ) ) |
| 4 | ax-c10 | ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑥 → ∀ 𝑥 𝑥 = 𝑥 ) → 𝑥 = 𝑥 ) | |
| 5 | 3 4 | syl | ⊢ ( ∀ 𝑥 ¬ ∀ 𝑥 𝑥 = 𝑥 → 𝑥 = 𝑥 ) |
| 6 | ax-c7 | ⊢ ( ¬ ∀ 𝑥 ¬ ∀ 𝑥 𝑥 = 𝑥 → 𝑥 = 𝑥 ) | |
| 7 | 5 6 | pm2.61i | ⊢ 𝑥 = 𝑥 |