Theorem ifeq1d 3655

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 3640	. 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 1402   ifcif 3635

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-un 3224  df-if 3636

This theorem is referenced by:  ifeq12d  3657  ifbieq1d  3660  ifeq1dadc  3668  iseqf1olemjpcl  10923  iseqf1olemqpcl  10924  iseqf1olemfvp  10925  seq3f1olemqsum  10928  seq3f1olemp  10930  summodc  12128  fsum3  12132  fsum3ser  12142  isumlessdc  12241  prodeq2w  12301  prodmodc  12323  fprodseq  12328  prodssdc  12334  subgmulg  13968  lgsval  16037