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Theorem ifeq2d 3746
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 3736 . 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 set class
Syntax hints:    -> wi 4    = wceq 1652   ifcif 3731
This theorem is referenced by:  ifeq12d  3747  ifbieq2d  3751  ifeq2da  3757  rdgeq1  6661  cantnflem1d  7636  cantnflem1  7637  rexmul  10842  1arithlem4  13286  ramcl  13389  mplcoe1  16520  mplcoe2  16522  subrgascl  16550  itg2monolem1  19634  iblmulc2  19714  itgmulc2lem1  19715  bddmulibl  19722  dvtaylp  20278  dchrinvcl  21029  rpvmasum2  21198  padicfval  21302  gxval  21838  itg2addnclem  26246  itg2addnclem3  26248  itg2addnc  26249  itgmulc2nclem1  26261  hdmap1fval  32532
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-un 3317  df-if 3732
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