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Theorem wl-speqv 32487
 Description: Under the assumption ¬ 𝑥 = 𝑦 a specialized version of sp 2041 is provable from Tarski's FOL and ax13v 2235 only. Note that this reverts the implication in ax13lem1 2236, 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 1879 . 2 (∀𝑥 𝑧 = 𝑦 → ∃𝑥 𝑧 = 𝑦)
2 ax13lem2 2284 . 2 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦𝑧 = 𝑦))
31, 2syl5 33 1 𝑥 = 𝑦 → (∀𝑥 𝑧 = 𝑦𝑧 = 𝑦))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696 This theorem is referenced by: (None)
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