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Theorem eqif 3774
Description: Expansion of an equality with a conditional operator. (Contributed by NM, 14-Feb-2005.)
Assertion
Ref Expression
eqif  |-  ( A  =  if ( ph ,  B ,  C )  <-> 
( ( ph  /\  A  =  B )  \/  ( -.  ph  /\  A  =  C )
) )

Proof of Theorem eqif
StepHypRef Expression
1 eqeq2 2447 . 2  |-  ( if ( ph ,  B ,  C )  =  B  ->  ( A  =  if ( ph ,  B ,  C )  <->  A  =  B ) )
2 eqeq2 2447 . 2  |-  ( if ( ph ,  B ,  C )  =  C  ->  ( A  =  if ( ph ,  B ,  C )  <->  A  =  C ) )
31, 2elimif 3770 1  |-  ( A  =  if ( ph ,  B ,  C )  <-> 
( ( ph  /\  A  =  B )  \/  ( -.  ph  /\  A  =  C )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653   ifcif 3741
This theorem is referenced by:  xpima  5315  fin23lem19  8218  fin23lem28  8222  fin23lem29  8223  fin23lem30  8224  aalioulem3  20253  ifbieq12d2  24004  iocinif  24146  ind1a  24420  esumsn  24458  itg2addnclem2  26259  afvpcfv0  27988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-if 3742
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