Theorem ifbieq2d 3649

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 3646	. 2 |- ( ph -> if ( ps , C , A ) = if ( ch , C , A ) )
3		ifbieq2d.2	. . 3 |- ( ph -> A = B )
4	3	ifeq2d 3643	. 2 |- ( ph -> if ( ch , C , A ) = if ( ch , C , B ) )
5	2, 4	eqtrd 2267	1 |- ( ph -> if ( ps , C , A ) = if ( ch , C , B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   ifcif 3622

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 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rab 2531  df-v 2817  df-un 3217  df-if 3623

This theorem is referenced by:  difinfsnlem  7392  ctmlemr  7401  xnegeq  10163  xaddval  10181  iseqf1olemqval  10866  iseqf1olemqk  10873  seq3f1olemqsum  10879  exp3val  10907  gcdval  12659  gcdass  12715  lcmval  12764  lcmass  12786  pcval  12998  ennnfonelemj0  13169  ennnfonelemjn  13170  ennnfonelem0  13173  ennnfonelemp1  13174  ennnfonelemnn0  13190  mulgval  13856  znval  14801  lgsval  15894  lgsfvalg  15895  lgsval2lem  15900  eupth2lem3lem3fi  16482  eupth2fi  16491  depindlem1  16518  nnsf  16800  peano4nninf  16801  peano3nninf  16802  exmidsbthr  16820