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Theorem elequ1 2164
Description: An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
elequ1  |-  ( x  =  y  ->  (
x  e.  z  <->  y  e.  z ) )

Proof of Theorem elequ1
StepHypRef Expression
1 ax-13 2162 . 2  |-  ( x  =  y  ->  (
x  e.  z  -> 
y  e.  z ) )
2 ax-13 2162 . . 3  |-  ( y  =  x  ->  (
y  e.  z  ->  x  e.  z )
)
32equcoms 1719 . 2  |-  ( x  =  y  ->  (
y  e.  z  ->  x  e.  z )
)
41, 3impbid 129 1  |-  ( x  =  y  ->  (
x  e.  z  <->  y  e.  z ) )
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 1460  ax-ie2 1505  ax-8 1515  ax-17 1537  ax-i9 1541  ax-13 2162
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  cleljust  2166  elsb1  2167  dveel1  2169  nalset  4148  zfpow  4193  mss  4244  zfun  4452  pw2f1odclem  6862  ctssdc  7142  acfun  7236  ccfunen  7293  bj-nalset  15108  bj-nnelirr  15166  nninfsellemqall  15226  nninfomni  15230
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