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Theorem elequ1 1691
Description: An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
elequ1 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))

Proof of Theorem elequ1
StepHypRef Expression
1 ax-13 1492 . 2 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
2 ax-13 1492 . . 3 (𝑦 = 𝑥 → (𝑦𝑧𝑥𝑧))
32equcoms 1685 . 2 (𝑥 = 𝑦 → (𝑦𝑧𝑥𝑧))
41, 3impbid 128 1 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
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 1426  ax-ie2 1471  ax-8 1483  ax-13 1492  ax-17 1507  ax-i9 1511
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  cleljust  1911  elsb3  1952  dveel1  1996  nalset  4065  zfpow  4106  mss  4155  zfun  4363  ctssdc  7005  acfun  7079  ccfunen  7095  bj-nalset  13262  bj-nnelirr  13320  nninfsellemqall  13384  nninfomni  13388
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