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Theorem elequ1 2139
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 2137 . 2  |-  ( x  =  y  ->  (
x  e.  z  -> 
y  e.  z ) )
2 ax-13 2137 . . 3  |-  ( y  =  x  ->  (
y  e.  z  ->  x  e.  z )
)
32equcoms 1695 . 2  |-  ( x  =  y  ->  (
y  e.  z  ->  x  e.  z )
)
41, 3impbid 128 1  |-  ( x  =  y  ->  (
x  e.  z  <->  y  e.  z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1436  ax-ie2 1481  ax-8 1491  ax-17 1513  ax-i9 1517  ax-13 2137
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  cleljust  2141  elsb1  2142  dveel1  2144  nalset  4109  zfpow  4151  mss  4201  zfun  4409  ctssdc  7072  acfun  7157  ccfunen  7199  bj-nalset  13670  bj-nnelirr  13728  nninfsellemqall  13788  nninfomni  13792
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