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Theorem elequ1 1647
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 1449 . 2  |-  ( x  =  y  ->  (
x  e.  z  -> 
y  e.  z ) )
2 ax-13 1449 . . 3  |-  ( y  =  x  ->  (
y  e.  z  ->  x  e.  z )
)
32equcoms 1641 . 2  |-  ( x  =  y  ->  (
y  e.  z  ->  x  e.  z )
)
41, 3impbid 127 1  |-  ( x  =  y  ->  (
x  e.  z  <->  y  e.  z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-gen 1383  ax-ie2 1428  ax-8 1440  ax-13 1449  ax-17 1464  ax-i9 1468
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  cleljust  1861  elsb3  1900  dveel1  1944  nalset  3961  zfpow  4002  mss  4044  zfun  4252  bj-nalset  11443  bj-nnelirr  11505  nninfsellemqall  11564  nninfomni  11568
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