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Theorem elequ1 2152
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 2150 . 2 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
2 ax-13 2150 . . 3 (𝑦 = 𝑥 → (𝑦𝑧𝑥𝑧))
32equcoms 1708 . 2 (𝑥 = 𝑦 → (𝑦𝑧𝑥𝑧))
41, 3impbid 129 1 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1449  ax-ie2 1494  ax-8 1504  ax-17 1526  ax-i9 1530  ax-13 2150
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  cleljust  2154  elsb1  2155  dveel1  2157  nalset  4132  zfpow  4174  mss  4225  zfun  4433  ctssdc  7109  acfun  7203  ccfunen  7260  bj-nalset  14507  bj-nnelirr  14565  nninfsellemqall  14624  nninfomni  14628
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