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Theorem elequ2 1601
 Description: An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
elequ2 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))

Proof of Theorem elequ2
StepHypRef Expression
1 ax-14 1405 . 2 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
2 ax-14 1405 . . 3 (𝑦 = 𝑥 → (𝑧𝑦𝑧𝑥))
32equcoms 1594 . 2 (𝑥 = 𝑦 → (𝑧𝑦𝑧𝑥))
41, 3impbid 120 1 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-gen 1338  ax-ie2 1383  ax-8 1395  ax-14 1405  ax-17 1419  ax-i9 1423 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  elsb4  1853  dveel2  1897  axext3  2023  axext4  2024  bm1.1  2025  bm1.3ii  3878  nalset  3887  zfun  4171  fv3  5197  tfrlemisucaccv  5939  bdsepnft  10007  bdsepnfALT  10009  bdbm1.3ii  10011  bj-nalset  10015  bj-nnelirr  10078  strcollnft  10109  strcollnfALT  10111
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