Theorem elimif 3691

Description: Elimination of a conditional operator contained in a wff ps. (Contributed by NM, 15-Feb-2005.)

Hypotheses

Ref	Expression
elimif.1	|- (if(ph, A, B) = A -> (ps <-> ch))
elimif.2	|- (if(ph, A, B) = B -> (ps <-> th))

Assertion

Ref	Expression
elimif	|- (ps <-> ((ph /\ ch) \/ (-. ph /\ th)))

Proof of Theorem elimif

Step	Hyp	Ref	Expression
1		exmid 404	. . 3 |- (ph \/ -. ph)
2	1	biantrur 492	. 2 |- (ps <-> ((ph \/ -. ph) /\ ps))
3		andir 838	. 2 |- (((ph \/ -. ph) /\ ps) <-> ((ph /\ ps) \/ (-. ph /\ ps)))
4		iftrue 3668	. . . . 5 |- (ph -> if(ph, A, B) = A)
5		elimif.1	. . . . 5 |- (if(ph, A, B) = A -> (ps <-> ch))
6	4, 5	syl 15	. . . 4 |- (ph -> (ps <-> ch))
7	6	pm5.32i 618	. . 3 |- ((ph /\ ps) <-> (ph /\ ch))
8		iffalse 3669	. . . . 5 |- (-. ph -> if(ph, A, B) = B)
9		elimif.2	. . . . 5 |- (if(ph, A, B) = B -> (ps <-> th))
10	8, 9	syl 15	. . . 4 |- (-. ph -> (ps <-> th))
11	10	pm5.32i 618	. . 3 |- ((-. ph /\ ps) <-> (-. ph /\ th))
12	7, 11	orbi12i 507	. 2 |- (((ph /\ ps) \/ (-. ph /\ ps)) <-> ((ph /\ ch) \/ (-. ph /\ th)))
13	2, 3, 12	3bitri 262	1 |- (ps <-> ((ph /\ ch) \/ (-. ph /\ th)))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358   = wceq 1642  ifcif 3662

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-if 3663

This theorem is referenced by:  eqif  3695  elif  3696  ifel  3697