Theorem ax13v 2373

Description: A weaker version of ax-13 2372 with distinct variable restrictions on pairs 𝑥, 𝑧 and 𝑦, 𝑧. In order to show (with ax13 2375) that this weakening is still adequate, this should be the only theorem referencing ax-13 2372 directly.

Had we additionally required 𝑥 and 𝑦 be distinct, too, this theorem would have been a direct consequence of ax-5 1913. So essentially this theorem states, that a distinct variable condition can be replaced with an inequality between set variables. Preferably, use the version ax13w 2132 to avoid the propagation of ax-13 2372. (Contributed by NM, 30-Jun-2016.) (New usage is discouraged.)

Assertion

Ref	Expression
ax13v	⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem ax13v

Step	Hyp	Ref	Expression
1		ax-13 2372	1 ⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-13 2372

This theorem is referenced by:  ax13lem1  2374  wl-spae  35680