Description: Alternate proof of ax13 from FOL, sp , and axc9 . (Contributed by NM, 21-Dec-2015) (Proof shortened by Wolf Lammen, 31-Jan-2018) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ax13ALT | ⊢ ( ¬ 𝑥 = 𝑦 → ( 𝑦 = 𝑧 → ∀ 𝑥 𝑦 = 𝑧 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sp | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦 ) | |
| 2 | 1 | con3i | ⊢ ( ¬ 𝑥 = 𝑦 → ¬ ∀ 𝑥 𝑥 = 𝑦 ) |
| 3 | sp | ⊢ ( ∀ 𝑥 𝑥 = 𝑧 → 𝑥 = 𝑧 ) | |
| 4 | 3 | con3i | ⊢ ( ¬ 𝑥 = 𝑧 → ¬ ∀ 𝑥 𝑥 = 𝑧 ) |
| 5 | axc9 | ⊢ ( ¬ ∀ 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 ) ) |