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Theorem elequ2 1617
Description: An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
elequ2  |-  ( x  =  y  ->  (
z  e.  x  <->  z  e.  y ) )

Proof of Theorem elequ2
StepHypRef Expression
1 ax-14 1421 . 2  |-  ( x  =  y  ->  (
z  e.  x  -> 
z  e.  y ) )
2 ax-14 1421 . . 3  |-  ( y  =  x  ->  (
z  e.  y  -> 
z  e.  x ) )
32equcoms 1610 . 2  |-  ( x  =  y  ->  (
z  e.  y  -> 
z  e.  x ) )
41, 3impbid 124 1  |-  ( x  =  y  ->  (
z  e.  x  <->  z  e.  y ) )
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-14 1421  ax-17 1435  ax-i9 1439
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  elsb4  1869  dveel2  1913  axext3  2039  axext4  2040  bm1.1  2041  bm1.3ii  3906  nalset  3915  zfun  4199  fv3  5225  tfrlemisucaccv  5970  bdsepnft  10394  bdsepnfALT  10396  bdbm1.3ii  10398  bj-nalset  10402  bj-nnelirr  10465  strcollnft  10496  strcollnfALT  10498
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