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Theorem sbequ12a 1947
Description: An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbequ12a  |-  ( x  =  y  ->  ( [ y  /  x ] ph  <->  [ x  /  y ] ph ) )

Proof of Theorem sbequ12a
StepHypRef Expression
1 sbequ12 1945 . 2  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
2 sbequ12 1945 . . 3  |-  ( y  =  x  ->  ( ph 
<->  [ x  /  y ] ph ) )
32equcoms 1694 . 2  |-  ( x  =  y  ->  ( ph 
<->  [ x  /  y ] ph ) )
41, 3bitr3d 248 1  |-  ( x  =  y  ->  ( [ y  /  x ] ph  <->  [ x  /  y ] ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   [wsb 1659
This theorem is referenced by:  sbco3  2165  sbco3wAUX7  29649  sbco3OLD7  29816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-11 1762
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-sb 1660
  Copyright terms: Public domain W3C validator