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  7421  updjudhcoinrg  7422  omp1eom  7436  xaddval  10258  iseqf1olemqval  10952  iseqf1olemqk  10959  seq3f1olemqsum  10965  seqf1oglem2  10972  exp3val  10993  ccatfvalfi  11376  ccatval1  11381  ccatval2  11382  ccatalpha  11397  cvgratz  12318  eucalgval2  12850  ballotfilemsv  13305  ballotfilemsf1o  13309  ballotfi  13334  ennnfonelemg  13346  ennnfonelem1  13350  mulgval  13978  lgsval  16289  gausslemma2dlem1a  16343  gausslemma2dlem1f1o  16345  gausslemma2dlem2  16347  gausslemma2dlem3  16348  gausslemma2dlem4  16349  vtxvalg  16423  iedgvalg  16424