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Theorem ifval 3968
Description: Another expression of the value of the  if predicate, analogous to eqif 3967. See also the more specialized iftrue 3935 and iffalse 3938. (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 3967 . 2  |-  ( A  =  if ( ph ,  B ,  C )  <-> 
( ( ph  /\  A  =  B )  \/  ( -.  ph  /\  A  =  C )
) )
2 cases2 969 . 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 366    /\ wa 367    = wceq 1398   ifcif 3929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-if 3930
This theorem is referenced by:  bj-projval  34955
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