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  7258  updjudhcoinrg  7259  omp1eom  7273  xaddval  10053  iseqf1olemqval  10734  iseqf1olemqk  10741  seq3f1olemqsum  10747  seqf1oglem2  10754  exp3val  10775  ccatfvalfi  11140  ccatval1  11145  ccatval2  11146  ccatalpha  11161  cvgratz  12058  eucalgval2  12590  ennnfonelemg  12989  ennnfonelem1  12993  mulgval  13674  lgsval  15698  gausslemma2dlem1a  15752  gausslemma2dlem1f1o  15754  gausslemma2dlem2  15756  gausslemma2dlem3  15757  gausslemma2dlem4  15758  vtxvalg  15832  iedgvalg  15833