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Theorem elequ1 1616
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 1420 . 2 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
2 ax-13 1420 . . 3 (𝑦 = 𝑥 → (𝑦𝑧𝑥𝑧))
32equcoms 1610 . 2 (𝑥 = 𝑦 → (𝑦𝑧𝑥𝑧))
41, 3impbid 124 1 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-gen 1354  ax-ie2 1399  ax-8 1411  ax-13 1420  ax-17 1435  ax-i9 1439
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  cleljust  1829  elsb3  1868  dveel1  1912  nalset  3915  zfpow  3956  mss  3990  zfun  4199  bj-nalset  10402  bj-nnelirr  10465
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