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Theorem ifeq1d 3489
Description: Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
Hypothesis
Ref Expression
ifeq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
ifeq1d  |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C )
)

Proof of Theorem ifeq1d
StepHypRef Expression
1 ifeq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 ifeq1 3477 . 2  |-  ( A  =  B  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C ) )
31, 2syl 14 1  |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331   ifcif 3474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-v 2688  df-un 3075  df-if 3475
This theorem is referenced by:  ifeq12d  3491  ifbieq1d  3494  ifeq1dadc  3502  iseqf1olemjpcl  10280  iseqf1olemqpcl  10281  iseqf1olemfvp  10282  seq3f1olemqsum  10285  seq3f1olemp  10287  summodc  11164  fsum3  11168  fsum3ser  11178  isumlessdc  11277  prodeq2w  11337  prodmodc  11359
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