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Theorem eqeqan12rd 2407
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 2405 . 2  |-  ( (
ph  /\  ps )  ->  ( A  =  C  <-> 
B  =  D ) )
43ancoms 451 1  |-  ( ( ps  /\  ph )  ->  ( A  =  C  <-> 
B  =  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-an 369  df-cleq 2374
This theorem is referenced by:  axcontlem4  24391  cusgrasize  24599  clwwlkf1  24917  eigorthi  26872  expdiophlem2  31130  pwssplit4  31201
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