Theorem ifbieq2d 3559

Description: Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)

Hypotheses

Ref	Expression
ifbieq2d.1	|- ( ph -> ( ps <-> ch ) )
ifbieq2d.2	|- ( ph -> A = B )

Assertion

Ref	Expression
ifbieq2d	|- ( ph -> if ( ps , C , A ) = if ( ch , C , B ) )

Proof of Theorem ifbieq2d

Step	Hyp	Ref	Expression
1		ifbieq2d.1	. . 3 |- ( ph -> ( ps <-> ch ) )
2	1	ifbid 3556	. 2 |- ( ph -> if ( ps , C , A ) = if ( ch , C , A ) )
3		ifbieq2d.2	. . 3 |- ( ph -> A = B )
4	3	ifeq2d 3553	. 2 |- ( ph -> if ( ch , C , A ) = if ( ch , C , B ) )
5	2, 4	eqtrd 2210	1 |- ( ph -> if ( ps , C , A ) = if ( ch , C , B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   ifcif 3535

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2740  df-un 3134  df-if 3536

This theorem is referenced by:  difinfsnlem  7098  ctmlemr  7107  xnegeq  9827  xaddval  9845  iseqf1olemqval  10487  iseqf1olemqk  10494  seq3f1olemqsum  10500  exp3val  10522  gcdval  11960  gcdass  12016  lcmval  12063  lcmass  12085  pcval  12296  ennnfonelemj0  12402  ennnfonelemjn  12403  ennnfonelem0  12406  ennnfonelemp1  12407  ennnfonelemnn0  12423  mulgval  12986  lgsval  14408  lgsfvalg  14409  lgsval2lem  14414  nnsf  14757  peano4nninf  14758  peano3nninf  14759  exmidsbthr  14774