MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbequ12a Unicode version

Theorem sbequ12a 1874
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 1872 . 2  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
2 sbequ12 1872 . . 3  |-  ( y  =  x  ->  ( ph 
<->  [ x  /  y ] ph ) )
32equcoms 1666 . 2  |-  ( x  =  y  ->  ( ph 
<->  [ x  /  y ] ph ) )
41, 3bitr3d 246 1  |-  ( x  =  y  ->  ( [ y  /  x ] ph  <->  [ x  /  y ] ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   [wsb 1638
This theorem is referenced by:  sbco3  2041  sbco3wAUX7  29561  sbco3OLD7  29708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-sb 1639
  Copyright terms: Public domain W3C validator