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Theorem eqeqan12rd 2098
Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.)
Hypotheses
Ref Expression
eqeqan12rd.1  |-  ( ph  ->  A  =  B )
eqeqan12rd.2  |-  ( ps 
->  C  =  D
)
Assertion
Ref Expression
eqeqan12rd  |-  ( ( ps  /\  ph )  ->  ( A  =  C  <-> 
B  =  D ) )

Proof of Theorem eqeqan12rd
StepHypRef Expression
1 eqeqan12rd.1 . . 3  |-  ( ph  ->  A  =  B )
2 eqeqan12rd.2 . . 3  |-  ( ps 
->  C  =  D
)
31, 2eqeqan12d 2097 . 2  |-  ( (
ph  /\  ps )  ->  ( A  =  C  <-> 
B  =  D ) )
43ancoms 264 1  |-  ( ( ps  /\  ph )  ->  ( A  =  C  <-> 
B  =  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285
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-5 1377  ax-gen 1379  ax-4 1441  ax-17 1460  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-cleq 2075
This theorem is referenced by: (None)
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