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Theorem ifeq2d 3807
Description: Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
Hypothesis
Ref Expression
ifeq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
ifeq2d  |-  ( ph  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B )
)

Proof of Theorem ifeq2d
StepHypRef Expression
1 ifeq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 ifeq2 3795 . 2  |-  ( A  =  B  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B ) )
31, 2syl 16 1  |-  ( ph  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369   ifcif 3790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-rab 2723  df-v 2973  df-un 3332  df-if 3791
This theorem is referenced by:  ifeq12d  3808  ifbieq2d  3813  ifeq2da  3819  ifcomnan  3837  rdgeq1  6866  cantnflem1d  7895  cantnflem1  7896  cantnflem1dOLD  7918  cantnflem1OLD  7919  rexmul  11233  1arithlem4  13986  ramcl  14089  mplcoe1  17543  mplcoe5  17547  mplcoe2OLD  17549  subrgascl  17579  marrepfval  18370  ma1repveval  18381  mulmarep1el  18382  mdetralt2  18414  mdetunilem8  18424  maduval  18443  maducoeval2  18445  madurid  18449  minmar1val0  18452  itg2monolem1  21227  iblmulc2  21307  itgmulc2lem1  21308  bddmulibl  21315  dvtaylp  21834  dchrinvcl  22591  rpvmasum2  22760  padicfval  22864  gxval  23744  plymulx  26948  itg2addnclem  28441  itg2addnclem3  28443  itg2addnc  28444  itgmulc2nclem1  28456  hdmap1fval  35440
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