Theorem ifbieq2d 3647

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 3644	. 2 |- ( ph -> if ( ps , C , A ) = if ( ch , C , A ) )
3		ifbieq2d.2	. . 3 |- ( ph -> A = B )
4	3	ifeq2d 3641	. 2 |- ( ph -> if ( ch , C , A ) = if ( ch , C , B ) )
5	2, 4	eqtrd 2265	1 |- ( ph -> if ( ps , C , A ) = if ( ch , C , B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   ifcif 3620

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rab 2529  df-v 2815  df-un 3215  df-if 3621

This theorem is referenced by:  difinfsnlem  7390  ctmlemr  7399  xnegeq  10160  xaddval  10178  iseqf1olemqval  10862  iseqf1olemqk  10869  seq3f1olemqsum  10875  exp3val  10903  gcdval  12655  gcdass  12711  lcmval  12760  lcmass  12782  pcval  12994  ennnfonelemj0  13152  ennnfonelemjn  13153  ennnfonelem0  13156  ennnfonelemp1  13157  ennnfonelemnn0  13173  mulgval  13839  znval  14784  lgsval  15877  lgsfvalg  15878  lgsval2lem  15883  eupth2lem3lem3fi  16465  eupth2fi  16474  depindlem1  16501  nnsf  16783  peano4nninf  16784  peano3nninf  16785  exmidsbthr  16803