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Theorem eqif 3982
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 2472 . 2  |-  ( if ( ph ,  B ,  C )  =  B  ->  ( A  =  if ( ph ,  B ,  C )  <->  A  =  B ) )
2 eqeq2 2472 . 2  |-  ( if ( ph ,  B ,  C )  =  C  ->  ( A  =  if ( ph ,  B ,  C )  <->  A  =  C ) )
31, 2elimif 3978 1  |-  ( A  =  if ( ph ,  B ,  C )  <-> 
( ( ph  /\  A  =  B )  \/  ( -.  ph  /\  A  =  C )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1395   ifcif 3944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-if 3945
This theorem is referenced by:  ifval  3983  xpima  5456  fin23lem19  8733  fin23lem28  8737  fin23lem29  8738  fin23lem30  8739  aalioulem3  22856  ifbieq12d2  27555  iocinif  27752  fsumcvg4  28093  ind1a  28195  esumsnf  28236  itg2addnclem2  30272  afvpcfv0  32434
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