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Theorem ax13v 2383
 Description: A weaker version of ax-13 2382 with distinct variable restrictions on pairs 𝑥, 𝑧 and 𝑦, 𝑧. In order to show (with ax13 2385) that this weakening is still adequate, this should be the only theorem referencing ax-13 2382 directly. Had we additionally required 𝑥 and 𝑦 be distinct, too, this theorem would have been a direct consequence of ax-5 1911. So essentially this theorem states, that a distinct variable condition can be replaced with an inequality between set variables. Preferably, use the version ax13w 2138 to avoid the propagation of ax-13 2382. (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
StepHypRef Expression
1 ax-13 2382 1 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1536 This theorem was proved from axioms:  ax-13 2382 This theorem is referenced by:  ax13lem1  2384  wl-spae  34925
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