Theorem ifbieq2d 3662

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 3659	. 2 |- ( ph -> if ( ps , C , A ) = if ( ch , C , A ) )
3		ifbieq2d.2	. . 3 |- ( ph -> A = B )
4	3	ifeq2d 3656	. 2 |- ( ph -> if ( ch , C , A ) = if ( ch , C , B ) )
5	2, 4	eqtrd 2271	1 |- ( ph -> if ( ps , C , A ) = if ( ch , C , B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1402   ifcif 3635

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-un 3224  df-if 3636

This theorem is referenced by:  difinfsnlem  7429  ctmlemr  7438  xnegeq  10208  xaddval  10226  iseqf1olemqval  10915  iseqf1olemqk  10922  seq3f1olemqsum  10928  exp3val  10956  gcdval  12714  gcdass  12770  lcmval  12819  lcmass  12841  pcval  13053  ennnfonelemj0  13270  ennnfonelemjn  13271  ennnfonelem0  13274  ennnfonelemp1  13275  ennnfonelemnn0  13291  mulgval  13902  znval  14943  lgsval  16037  lgsfvalg  16038  lgsval2lem  16043  eupth2lem3lem3fi  16625  eupth2fi  16634  depindlem1  16661  nnsf  16953  peano4nninf  16954  peano3nninf  16955  exmidsbthr  16973