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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  6513  dfac12r  7772  xaddpnf1  10553  ruclem1  12509  eucalgval  12752  gsumpropd  14453  gsumress  14454  mulgfval  14568  mulgpropd  14600  frgpup3lem  15086  subrgmvr  16205  isobs  16620  pcoval  18509  pcorevlem  18524  itg2const  19095  ditgeq3  19200  efrlim  20264  lgsval  20539  rpvmasum2  20661  gxfval  20924  gxval  20925  gsumpropd2lem  23379  prodeq2  25307  isconc1  26006  isconc2  26007  isconc3  26008  uvcfval  27233  dgrsub2  27339  hdmap1fval  31987
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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