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Theorem ifval 3929
Description: Another expression of the value of the  if predicate, analogous to eqif 3928. See also the more specialized iftrue 3898 and iffalse 3900. (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
ifval  |-  ( A  =  if ( ph ,  B ,  C )  <-> 
( ( ph  ->  A  =  B )  /\  ( -.  ph  ->  A  =  C ) ) )

Proof of Theorem ifval
StepHypRef Expression
1 eqif 3928 . 2  |-  ( A  =  if ( ph ,  B ,  C )  <-> 
( ( ph  /\  A  =  B )  \/  ( -.  ph  /\  A  =  C )
) )
2 cases2 963 . 2  |-  ( ( ( ph  /\  A  =  B )  \/  ( -.  ph  /\  A  =  C ) )  <->  ( ( ph  ->  A  =  B )  /\  ( -. 
ph  ->  A  =  C ) ) )
31, 2bitri 249 1  |-  ( A  =  if ( ph ,  B ,  C )  <-> 
( ( ph  ->  A  =  B )  /\  ( -.  ph  ->  A  =  C ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370   ifcif 3892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-if 3893
This theorem is referenced by:  bj-projval  32792
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