Theorem ifbieq12d 3685

Description: Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypotheses

Ref	Expression
ifbieq12d.1	|- (ph -> (ps <-> ch))
ifbieq12d.2	|- (ph -> A = C)
ifbieq12d.3	|- (ph -> B = D)

Assertion

Ref	Expression
ifbieq12d	|- (ph -> if(ps, A, B) = if(ch, C, D))

Proof of Theorem ifbieq12d

Step	Hyp	Ref	Expression
1		ifbieq12d.1	. . 3 |- (ph -> (ps <-> ch))
2	1	ifbid 3681	. 2 |- (ph -> if(ps, A, B) = if(ch, A, B))
3		ifbieq12d.2	. . 3 |- (ph -> A = C)
4		ifbieq12d.3	. . 3 |- (ph -> B = D)
5	3, 4	ifeq12d 3679	. 2 |- (ph -> if(ch, A, B) = if(ch, C, D))
6	2, 5	eqtrd 2385	1 |- (ph -> if(ps, A, B) = if(ch, C, D))

Syntax hints:   -> wi 4   <-> wb 176   = wceq 1642  ifcif 3663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-if 3664

This theorem is referenced by:  csbifg  3691  phi11lem1  4596