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Theorem eqif 3977
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 2482 . 2  |-  ( if ( ph ,  B ,  C )  =  B  ->  ( A  =  if ( ph ,  B ,  C )  <->  A  =  B ) )
2 eqeq2 2482 . 2  |-  ( if ( ph ,  B ,  C )  =  C  ->  ( A  =  if ( ph ,  B ,  C )  <->  A  =  C ) )
31, 2elimif 3973 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 1379   ifcif 3939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-if 3940
This theorem is referenced by:  ifval  3978  xpima  5449  fin23lem19  8716  fin23lem28  8720  fin23lem29  8721  fin23lem30  8722  aalioulem3  22492  ifbieq12d2  27122  iocinif  27288  fsumcvg4  27596  ind1a  27702  esumsn  27740  itg2addnclem2  29672  afvpcfv0  31726
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