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Theorem elequ2 1676
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 1477 . 2 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
2 ax-14 1477 . . 3 (𝑦 = 𝑥 → (𝑧𝑦𝑧𝑥))
32equcoms 1669 . 2 (𝑥 = 𝑦 → (𝑧𝑦𝑧𝑥))
41, 3impbid 128 1 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1410  ax-ie2 1455  ax-8 1467  ax-14 1477  ax-17 1491  ax-i9 1495
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  elsb4  1930  dveel2  1974  axext3  2100  axext4  2101  bm1.1  2102  eleq2w  2179  bm1.3ii  4019  nalset  4028  zfun  4326  fv3  5412  tfrlemisucaccv  6190  tfr1onlemsucaccv  6206  tfrcllemsucaccv  6219  acfun  7031  ccfunen  7047  bdsepnft  13012  bdsepnfALT  13014  bdbm1.3ii  13016  bj-nalset  13020  bj-nnelirr  13078  strcollnft  13109  strcollnfALT  13111  nninfalllem1  13130  nninfsellemeq  13137  nninfsellemqall  13138  nninfsellemeqinf  13139  nninfomni  13142
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