Theorem ifbieq12d 3667

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 3662	. 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 3660	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   ifcif 3638

This proof depends on 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 proof 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 3639

This theorem is used by:  updjudhcoinlf  7420  updjudhcoinrg  7421  omp1eom  7435  xaddval  10257  iseqf1olemqval  10950  iseqf1olemqk  10957  seq3f1olemqsum  10963  seqf1oglem2  10970  exp3val  10991  ccatfvalfi  11374  ccatval1  11379  ccatval2  11380  ccatalpha  11395  cvgratz  12315  eucalgval2  12847  ballotfilemsv  13302  ballotfilemsf1o  13306  ballotfi  13331  ennnfonelemg  13343  ennnfonelem1  13347  mulgval  13974  lgsval  16221  gausslemma2dlem1a  16275  gausslemma2dlem1f1o  16277  gausslemma2dlem2  16279  gausslemma2dlem3  16280  gausslemma2dlem4  16281  vtxvalg  16355  iedgvalg  16356