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  7426  updjudhcoinrg  7427  omp1eom  7441  xaddval  10263  iseqf1olemqval  10957  iseqf1olemqk  10964  seq3f1olemqsum  10970  seqf1oglem2  10977  exp3val  10998  ccatfvalfi  11381  ccatval1  11386  ccatval2  11387  ccatalpha  11402  cvgratz  12323  eucalgval2  12855  ballotfilemsv  13310  ballotfilemsf1o  13314  ballotfi  13339  ennnfonelemg  13351  ennnfonelem1  13355  mulgval  13983  mvrfval  15214  lgsval  16345  gausslemma2dlem1a  16399  gausslemma2dlem1f1o  16401  gausslemma2dlem2  16403  gausslemma2dlem3  16404  gausslemma2dlem4  16405  vtxvalg  16479  iedgvalg  16480