Theorem ifbieq2d 3630

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 3627	. 2 |- ( ph -> if ( ps , C , A ) = if ( ch , C , A ) )
3		ifbieq2d.2	. . 3 |- ( ph -> A = B )
4	3	ifeq2d 3624	. 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 1397   ifcif 3605

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-v 2804  df-un 3204  df-if 3606

This theorem is referenced by:  difinfsnlem  7297  ctmlemr  7306  xnegeq  10061  xaddval  10079  iseqf1olemqval  10761  iseqf1olemqk  10768  seq3f1olemqsum  10774  exp3val  10802  gcdval  12529  gcdass  12585  lcmval  12634  lcmass  12656  pcval  12868  ennnfonelemj0  13021  ennnfonelemjn  13022  ennnfonelem0  13025  ennnfonelemp1  13026  ennnfonelemnn0  13042  mulgval  13708  znval  14649  lgsval  15732  lgsfvalg  15733  lgsval2lem  15738  nnsf  16607  peano4nninf  16608  peano3nninf  16609  exmidsbthr  16627