Theorem ifbieq2d 3634

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 3631	. 2 |- ( ph -> if ( ps , C , A ) = if ( ch , C , A ) )
3		ifbieq2d.2	. . 3 |- ( ph -> A = B )
4	3	ifeq2d 3628	. 2 |- ( ph -> if ( ch , C , A ) = if ( ch , C , B ) )
5	2, 4	eqtrd 2264	1 |- ( ph -> if ( ps , C , A ) = if ( ch , C , B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   ifcif 3607

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rab 2520  df-v 2805  df-un 3205  df-if 3608

This theorem is referenced by:  difinfsnlem  7358  ctmlemr  7367  xnegeq  10123  xaddval  10141  iseqf1olemqval  10825  iseqf1olemqk  10832  seq3f1olemqsum  10838  exp3val  10866  gcdval  12610  gcdass  12666  lcmval  12715  lcmass  12737  pcval  12949  ennnfonelemj0  13102  ennnfonelemjn  13103  ennnfonelem0  13106  ennnfonelemp1  13107  ennnfonelemnn0  13123  mulgval  13789  znval  14732  lgsval  15823  lgsfvalg  15824  lgsval2lem  15829  eupth2lem3lem3fi  16411  eupth2fi  16420  depindlem1  16447  nnsf  16731  peano4nninf  16732  peano3nninf  16733  exmidsbthr  16751