Description: Show that ax-c11 can be derived from ax-c11n and ax-12 . An open problem is whether this theorem can be derived from ax-c11n and the others when ax-12 is replaced with ax-c15 or ax12v . See Theorems axc11nfromc11 for the rederivation of ax-c11n from axc11 .
Normally, axc11 should be used rather than ax-c11 or axc11-o , except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | axc11-o | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜑 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-c11n | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑦 𝑦 = 𝑥 ) | |
| 2 | ax12 | ⊢ ( 𝑦 = 𝑥 → ( ∀ 𝑥 𝜑 → ∀ 𝑦 ( 𝑦 = 𝑥 → 𝜑 ) ) ) | |
| 3 | 2 | equcoms | ⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 → ∀ 𝑦 ( 𝑦 = 𝑥 → 𝜑 ) ) ) |
| 4 | 3 | sps-o | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 → ∀ 𝑦 ( 𝑦 = 𝑥 → 𝜑 ) ) ) |
| 5 | pm2.27 | ⊢ ( 𝑦 = 𝑥 → ( ( 𝑦 = 𝑥 → 𝜑 ) → 𝜑 ) ) | |
| 6 | 5 | al2imi | ⊢ ( ∀ 𝑦 𝑦 = 𝑥 → ( ∀ 𝑦 ( 𝑦 = 𝑥 → 𝜑 ) → ∀ 𝑦 𝜑 ) ) |
| 7 | 1 4 6 | sylsyld | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜑 ) ) |