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Theorem elequ2 2153
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 2151 . 2 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
2 ax-14 2151 . . 3 (𝑦 = 𝑥 → (𝑧𝑦𝑧𝑥))
32equcoms 1708 . 2 (𝑥 = 𝑦 → (𝑧𝑦𝑧𝑥))
41, 3impbid 129 1 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
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-14 2151
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  elsb2  2156  dveel2  2158  axext3  2160  axext4  2161  bm1.1  2162  eleq2w  2239  bm1.3ii  4126  nalset  4135  zfun  4436  fv3  5540  tfrlemisucaccv  6328  tfr1onlemsucaccv  6344  tfrcllemsucaccv  6357  acfun  7208  ccfunen  7265  cc1  7266  bdsepnft  14678  bdsepnfALT  14680  bdbm1.3ii  14682  bj-nalset  14686  bj-nnelirr  14744  nninfalllem1  14796  nninfsellemeq  14802  nninfsellemqall  14803  nninfsellemeqinf  14804  nninfomni  14807
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