Theorem ifbieq12d 3632

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 3627	. 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 3625	. 2 |- ( ph -> if ( ch , A , B ) = if ( ch , C , D ) )
6	2, 5	eqtrd 2264	1 |- ( ph -> if ( ps , A , B ) = if ( ch , C , D ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   ifcif 3605

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-v 2804  df-un 3204  df-if 3606

This theorem is referenced by:  updjudhcoinlf  7278  updjudhcoinrg  7279  omp1eom  7293  xaddval  10079  iseqf1olemqval  10761  iseqf1olemqk  10768  seq3f1olemqsum  10774  seqf1oglem2  10781  exp3val  10802  ccatfvalfi  11168  ccatval1  11173  ccatval2  11174  ccatalpha  11189  cvgratz  12092  eucalgval2  12624  ennnfonelemg  13023  ennnfonelem1  13027  mulgval  13708  lgsval  15732  gausslemma2dlem1a  15786  gausslemma2dlem1f1o  15788  gausslemma2dlem2  15790  gausslemma2dlem3  15791  gausslemma2dlem4  15792  vtxvalg  15866  iedgvalg  15867