Theorem ifbieq12d 3649

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 3644	. 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 3642	. 2 |- ( ph -> if ( ch , A , B ) = if ( ch , C , D ) )
6	2, 5	eqtrd 2265	1 |- ( ph -> if ( ps , A , B ) = if ( ch , C , D ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   ifcif 3620

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rab 2529  df-v 2815  df-un 3215  df-if 3621

This theorem is referenced by:  updjudhcoinlf  7371  updjudhcoinrg  7372  omp1eom  7386  xaddval  10178  iseqf1olemqval  10862  iseqf1olemqk  10869  seq3f1olemqsum  10875  seqf1oglem2  10882  exp3val  10903  ccatfvalfi  11280  ccatval1  11285  ccatval2  11286  ccatalpha  11301  cvgratz  12218  eucalgval2  12750  ennnfonelemg  13154  ennnfonelem1  13158  mulgval  13839  lgsval  15877  gausslemma2dlem1a  15931  gausslemma2dlem1f1o  15933  gausslemma2dlem2  15935  gausslemma2dlem3  15936  gausslemma2dlem4  15937  vtxvalg  16011  iedgvalg  16012