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Theorem eqeqan12rd 2299
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 2298 . 2  |-  ( (
ph  /\  ps )  ->  ( A  =  C  <-> 
B  =  D ) )
43ancoms 439 1  |-  ( ( ps  /\  ph )  ->  ( A  =  C  <-> 
B  =  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623
This theorem is referenced by:  tfrlem5  6396  eigorthi  22417  axcontlem4  24595  expdiophlem2  27115  pwssplit4  27191
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-cleq 2276
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