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Theorem ifeq12d 3715
Description: Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.)
Hypotheses
Ref Expression
ifeq1d.1  |-  ( ph  ->  A  =  B )
ifeq12d.2  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
ifeq12d  |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  D )
)

Proof of Theorem ifeq12d
StepHypRef Expression
1 ifeq1d.1 . . 3  |-  ( ph  ->  A  =  B )
21ifeq1d 3713 . 2  |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C )
)
3 ifeq12d.2 . . 3  |-  ( ph  ->  C  =  D )
43ifeq2d 3714 . 2  |-  ( ph  ->  if ( ps ,  B ,  C )  =  if ( ps ,  B ,  D )
)
52, 4eqtrd 2436 1  |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  D )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   ifcif 3699
This theorem is referenced by:  ifbieq12d  3721  oev  6717  dfac12r  7982  xaddpnf1  10768  ruclem1  12785  eucalgval  13028  gsumpropd  14731  gsumress  14732  mulgfval  14846  mulgpropd  14878  frgpup3lem  15364  subrgmvr  16479  isobs  16902  pcoval  18989  pcorevlem  19004  itg2const  19585  ditgeq3  19690  efrlim  20761  lgsval  21037  rpvmasum2  21159  gxfval  21798  gxval  21799  gsumpropd2lem  24173  xrhval  24337  itg2addnclem  26155  uvcfval  27101  dgrsub2  27207  hdmap1fval  32280
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rab 2675  df-v 2918  df-un 3285  df-if 3700
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