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Theorem sbequ12a 1731
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 1729 . 2  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
2 sbequ12 1729 . . 3  |-  ( y  =  x  ->  ( ph 
<->  [ x  /  y ] ph ) )
32equcoms 1669 . 2  |-  ( x  =  y  ->  ( ph 
<->  [ x  /  y ] ph ) )
41, 3bitr3d 189 1  |-  ( x  =  y  ->  ( [ y  /  x ] ph  <->  [ x  /  y ] ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   [wsb 1720
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 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-4 1472  ax-17 1491  ax-i9 1495
This theorem depends on definitions:  df-bi 116  df-sb 1721
This theorem is referenced by: (None)
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