Theorem ifbieq12d 3629

Description: Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypotheses

Ref	Expression
ifbieq12d.1	|- ( ph -> ( ps <-> ch ) )
ifbieq12d.2	|- ( ph -> A = C )
ifbieq12d.3	|- ( ph -> B = D )

Assertion

Ref	Expression
ifbieq12d	|- ( ph -> if ( ps , A , B ) = if ( ch , C , D ) )

Proof of Theorem ifbieq12d

Step	Hyp	Ref	Expression
1		ifbieq12d.1	. . 3 |- ( ph -> ( ps <-> ch ) )
2	1	ifbid 3624	. 2 |- ( ph -> if ( ps , A , B ) = if ( ch , A , B ) )
3		ifbieq12d.2	. . 3 |- ( ph -> A = C )
4		ifbieq12d.3	. . 3 |- ( ph -> B = D )
5	3, 4	ifeq12d 3622	. 2 |- ( ph -> if ( ch , A , B ) = if ( ch , C , D ) )
6	2, 5	eqtrd 2262	1 |- ( ph -> if ( ps , A , B ) = if ( ch , C , D ) )

Syntax hints:   -> wi 4   <-> wb 105   = 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-in1 617  ax-in2 618  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:  updjudhcoinlf  7243  updjudhcoinrg  7244  omp1eom  7258  xaddval  10037  iseqf1olemqval  10717  iseqf1olemqk  10724  seq3f1olemqsum  10730  seqf1oglem2  10737  exp3val  10758  ccatfvalfi  11122  ccatval1  11127  ccatval2  11128  cvgratz  12038  eucalgval2  12570  ennnfonelemg  12969  ennnfonelem1  12973  mulgval  13654  lgsval  15677  gausslemma2dlem1a  15731  gausslemma2dlem1f1o  15733  gausslemma2dlem2  15735  gausslemma2dlem3  15736  gausslemma2dlem4  15737  vtxvalg  15811  iedgvalg  15812