Theorem ifbieq12d 3653

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 3648	. 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 3646	. 2 |- ( ph -> if ( ch , A , B ) = if ( ch , C , D ) )
6	2, 5	eqtrd 2267	1 |- ( ph -> if ( ps , A , B ) = if ( ch , C , D ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   ifcif 3624

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rab 2531  df-v 2817  df-un 3218  df-if 3625

This theorem is referenced by:  updjudhcoinlf  7384  updjudhcoinrg  7385  omp1eom  7399  xaddval  10197  iseqf1olemqval  10886  iseqf1olemqk  10893  seq3f1olemqsum  10899  seqf1oglem2  10906  exp3val  10927  ccatfvalfi  11305  ccatval1  11310  ccatval2  11311  ccatalpha  11326  cvgratz  12243  eucalgval2  12775  ballotfilemsv  13197  ballotfilemsf1o  13201  ballotfi  13226  ennnfonelemg  13238  ennnfonelem1  13242  mulgval  13875  lgsval  16003  gausslemma2dlem1a  16057  gausslemma2dlem1f1o  16059  gausslemma2dlem2  16061  gausslemma2dlem3  16062  gausslemma2dlem4  16063  vtxvalg  16137  iedgvalg  16138