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Theorem ifeq1d 3551
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 3537 . 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 1353   ifcif 3534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2739  df-un 3133  df-if 3535
This theorem is referenced by:  ifeq12d  3553  ifbieq1d  3556  ifeq1dadc  3564  iseqf1olemjpcl  10478  iseqf1olemqpcl  10479  iseqf1olemfvp  10480  seq3f1olemqsum  10483  seq3f1olemp  10485  summodc  11372  fsum3  11376  fsum3ser  11386  isumlessdc  11485  prodeq2w  11545  prodmodc  11567  fprodseq  11572  prodssdc  11578  lgsval  14065
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