Theorem ifbieq12d 3664

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 3659	. 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 3657	. 2 |- ( ph -> if ( ch , A , B ) = if ( ch , C , D ) )
6	2, 5	eqtrd 2271	1 |- ( ph -> if ( ps , A , B ) = if ( ch , C , D ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1402   ifcif 3635

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-un 3224  df-if 3636

This theorem is referenced by:  updjudhcoinlf  7410  updjudhcoinrg  7411  omp1eom  7425  xaddval  10226  iseqf1olemqval  10915  iseqf1olemqk  10922  seq3f1olemqsum  10928  seqf1oglem2  10935  exp3val  10956  ccatfvalfi  11338  ccatval1  11343  ccatval2  11344  ccatalpha  11359  cvgratz  12277  eucalgval2  12809  ballotfilemsv  13231  ballotfilemsf1o  13235  ballotfi  13260  ennnfonelemg  13272  ennnfonelem1  13276  mulgval  13902  lgsval  16037  gausslemma2dlem1a  16091  gausslemma2dlem1f1o  16093  gausslemma2dlem2  16095  gausslemma2dlem3  16096  gausslemma2dlem4  16097  vtxvalg  16171  iedgvalg  16172