Theorem ifbieq2d 3595

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 3592	. 2 |- ( ph -> if ( ps , C , A ) = if ( ch , C , A ) )
3		ifbieq2d.2	. . 3 |- ( ph -> A = B )
4	3	ifeq2d 3589	. 2 |- ( ph -> if ( ch , C , A ) = if ( ch , C , B ) )
5	2, 4	eqtrd 2238	1 |- ( ph -> if ( ps , C , A ) = if ( ch , C , B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   ifcif 3571

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rab 2493  df-v 2774  df-un 3170  df-if 3572

This theorem is referenced by:  difinfsnlem  7201  ctmlemr  7210  xnegeq  9949  xaddval  9967  iseqf1olemqval  10645  iseqf1olemqk  10652  seq3f1olemqsum  10658  exp3val  10686  gcdval  12280  gcdass  12336  lcmval  12385  lcmass  12407  pcval  12619  ennnfonelemj0  12772  ennnfonelemjn  12773  ennnfonelem0  12776  ennnfonelemp1  12777  ennnfonelemnn0  12793  mulgval  13458  znval  14398  lgsval  15481  lgsfvalg  15482  lgsval2lem  15487  nnsf  15942  peano4nninf  15943  peano3nninf  15944  exmidsbthr  15962