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  10247  iseqf1olemqval  10937  iseqf1olemqk  10944  seq3f1olemqsum  10950  seqf1oglem2  10957  exp3val  10978  ccatfvalfi  11360  ccatval1  11365  ccatval2  11366  ccatalpha  11381  cvgratz  12299  eucalgval2  12831  ballotfilemsv  13253  ballotfilemsf1o  13257  ballotfi  13282  ennnfonelemg  13294  ennnfonelem1  13298  mulgval  13925  lgsval  16123  gausslemma2dlem1a  16177  gausslemma2dlem1f1o  16179  gausslemma2dlem2  16181  gausslemma2dlem3  16182  gausslemma2dlem4  16183  vtxvalg  16257  iedgvalg  16258