Theorem ifeq1d 3484

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

Step	Hyp	Ref	Expression
1		ifeq1d.1	. 2 |- ( ph -> A = B )
2		ifeq1 3472	. 2 |- ( A = B -> if ( ps , A , C ) = if ( ps , B , C ) )
3	1, 2	syl 14	1 |- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) )

Syntax hints:   -> wi 4   = wceq 1331   ifcif 3469

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 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rab 2423  df-v 2683  df-un 3070  df-if 3470

This theorem is referenced by:  ifeq12d  3486  ifbieq1d  3489  ifeq1dadc  3497  iseqf1olemjpcl  10261  iseqf1olemqpcl  10262  iseqf1olemfvp  10263  seq3f1olemqsum  10266  seq3f1olemp  10268  summodc  11145  fsum3  11149  fsum3ser  11159  isumlessdc  11258  prodeq2w  11318