Theorem ifbieq2d 3556

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 3553	. 2 |- ( ph -> if ( ps , C , A ) = if ( ch , C , A ) )
3		ifbieq2d.2	. . 3 |- ( ph -> A = B )
4	3	ifeq2d 3550	. 2 |- ( ph -> if ( ch , C , A ) = if ( ch , C , B ) )
5	2, 4	eqtrd 2208	1 |- ( ph -> if ( ps , C , A ) = if ( ch , C , B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   ifcif 3532

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rab 2462  df-v 2737  df-un 3131  df-if 3533

This theorem is referenced by:  difinfsnlem  7088  ctmlemr  7097  xnegeq  9796  xaddval  9814  iseqf1olemqval  10455  iseqf1olemqk  10462  seq3f1olemqsum  10468  exp3val  10490  gcdval  11925  gcdass  11981  lcmval  12028  lcmass  12050  pcval  12261  ennnfonelemj0  12367  ennnfonelemjn  12368  ennnfonelem0  12371  ennnfonelemp1  12372  ennnfonelemnn0  12388  mulgval  12845  lgsval  13974  lgsfvalg  13975  lgsval2lem  13980  nnsf  14313  peano4nninf  14314  peano3nninf  14315  exmidsbthr  14330