Theorem ifbieq2d 3543

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 3540	. 2 |- ( ph -> if ( ps , C , A ) = if ( ch , C , A ) )
3		ifbieq2d.2	. . 3 |- ( ph -> A = B )
4	3	ifeq2d 3537	. 2 |- ( ph -> if ( ch , C , A ) = if ( ch , C , B ) )
5	2, 4	eqtrd 2198	1 |- ( ph -> if ( ps , C , A ) = if ( ch , C , B ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   ifcif 3519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-rab 2452  df-v 2727  df-un 3119  df-if 3520

This theorem is referenced by:  difinfsnlem  7060  ctmlemr  7069  xnegeq  9759  xaddval  9777  iseqf1olemqval  10418  iseqf1olemqk  10425  seq3f1olemqsum  10431  exp3val  10453  gcdval  11888  gcdass  11944  lcmval  11991  lcmass  12013  pcval  12224  ennnfonelemj0  12330  ennnfonelemjn  12331  ennnfonelem0  12334  ennnfonelemp1  12335  ennnfonelemnn0  12351  lgsval  13505  lgsfvalg  13506  lgsval2lem  13511  nnsf  13845  peano4nninf  13846  peano3nninf  13847  exmidsbthr  13862