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Theorem ifval 3948
Description: Another expression of the value of the  if predicate, analogous to eqif 3947. See also the more specialized iftrue 3915 and iffalse 3918. (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 3947 . 2  |-  ( A  =  if ( ph ,  B ,  C )  <-> 
( ( ph  /\  A  =  B )  \/  ( -.  ph  /\  A  =  C )
) )
2 cases2 980 . 2  |-  ( ( ( ph  /\  A  =  B )  \/  ( -.  ph  /\  A  =  C ) )  <->  ( ( ph  ->  A  =  B )  /\  ( -. 
ph  ->  A  =  C ) ) )
31, 2bitri 252 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 187    \/ wo 369    /\ wa 370    = wceq 1437   ifcif 3909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-if 3910
This theorem is referenced by:  bj-projval  31545
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