Theorem ifeq1d 3620

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 3605	. 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 1395   ifcif 3602

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-v 2801  df-un 3201  df-if 3603

This theorem is referenced by:  ifeq12d  3622  ifbieq1d  3625  ifeq1dadc  3633  iseqf1olemjpcl  10742  iseqf1olemqpcl  10743  iseqf1olemfvp  10744  seq3f1olemqsum  10747  seq3f1olemp  10749  summodc  11910  fsum3  11914  fsum3ser  11924  isumlessdc  12023  prodeq2w  12083  prodmodc  12105  fprodseq  12110  prodssdc  12116  subgmulg  13741  lgsval  15699