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Theorem ifeqda 3920
Description: Separation of the values of the conditional operator. (Contributed by Alexander van der Vekens, 13-Apr-2018.)
Hypotheses
Ref Expression
ifeqda.1  |-  ( (
ph  /\  ps )  ->  A  =  C )
ifeqda.2  |-  ( (
ph  /\  -.  ps )  ->  B  =  C )
Assertion
Ref Expression
ifeqda  |-  ( ph  ->  if ( ps ,  A ,  B )  =  C )

Proof of Theorem ifeqda
StepHypRef Expression
1 iftrue 3895 . . . 4  |-  ( ps 
->  if ( ps ,  A ,  B )  =  A )
21adantl 466 . . 3  |-  ( (
ph  /\  ps )  ->  if ( ps ,  A ,  B )  =  A )
3 ifeqda.1 . . 3  |-  ( (
ph  /\  ps )  ->  A  =  C )
42, 3eqtrd 2492 . 2  |-  ( (
ph  /\  ps )  ->  if ( ps ,  A ,  B )  =  C )
5 iffalse 3897 . . . 4  |-  ( -. 
ps  ->  if ( ps ,  A ,  B
)  =  B )
65adantl 466 . . 3  |-  ( (
ph  /\  -.  ps )  ->  if ( ps ,  A ,  B )  =  B )
7 ifeqda.2 . . 3  |-  ( (
ph  /\  -.  ps )  ->  B  =  C )
86, 7eqtrd 2492 . 2  |-  ( (
ph  /\  -.  ps )  ->  if ( ps ,  A ,  B )  =  C )
94, 8pm2.61dan 789 1  |-  ( ph  ->  if ( ps ,  A ,  B )  =  C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370   ifcif 3889
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 3890
This theorem is referenced by:  cantnfp1  7990  ccatsymb  12383  swrdccat3blem  12488  repswccat  12525  linc0scn0  31064  fvmptnn04if  31303  chfacfscmulgsum  31314  chfacfpmmulgsum  31318
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