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Theorem wl-speqv 33617
 Description: Under the assumption ¬ 𝑥 = 𝑦 a specialized version of sp 2217 is provable from Tarski's FOL and ax13v 2420 only. Note that this reverts the implication in ax13lem1 2421, so in fact (¬ 𝑥 = 𝑦 → (∀𝑥𝑧 = 𝑦 ↔ 𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.)
Assertion
Ref Expression
wl-speqv 𝑥 = 𝑦 → (∀𝑥 𝑧 = 𝑦𝑧 = 𝑦))
Distinct variable group:   𝑥,𝑧

Proof of Theorem wl-speqv
StepHypRef Expression
1 19.2 2071 . 2 (∀𝑥 𝑧 = 𝑦 → ∃𝑥 𝑧 = 𝑦)
2 ax13lem2 2462 . 2 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦𝑧 = 𝑦))
31, 2syl5 34 1 𝑥 = 𝑦 → (∀𝑥 𝑧 = 𝑦𝑧 = 𝑦))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1635  ∃wex 1859 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-13 2419 This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1860 This theorem is referenced by: (None)
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