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Theorem ifeq12d 3581
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 3579 . 2  |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C )
)
3 ifeq12d.2 . . 3  |-  ( ph  ->  C  =  D )
43ifeq2d 3580 . 2  |-  ( ph  ->  if ( ps ,  B ,  C )  =  if ( ps ,  B ,  D )
)
52, 4eqtrd 2315 1  |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  D )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   ifcif 3565
This theorem is referenced by:  ifbieq12d  3587  oev  6508  dfac12r  7767  xaddpnf1  10548  ruclem1  12504  eucalgval  12747  gsumpropd  14448  gsumress  14449  mulgfval  14563  mulgpropd  14595  frgpup3lem  15081  subrgmvr  16200  isobs  16615  pcoval  18504  pcorevlem  18519  itg2const  19090  ditgeq3  19195  efrlim  20259  lgsval  20534  rpvmasum2  20656  gxfval  20917  gxval  20918  prodeq2  24718  isconc1  25417  isconc2  25418  isconc3  25419  uvcfval  26644  dgrsub2  26750  hdmap1fval  31290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-un 3157  df-if 3566
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