Theorem elimif 3001

Description: Elimination of a conditional operator contained in a wff ps.

Hypotheses

Ref	Expression
sbif.1	|- (if(ph, A, B) = A -> (ps <-> ch))
sbif.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 717	. . 3 |- (ph \/ -. ph)
2	1	biantrur 794	. 2 |- (ps <-> ((ph \/ -. ph) /\ ps))
3		andir 666	. 2 |- (((ph \/ -. ph) /\ ps) <-> ((ph /\ ps) \/ (-. ph /\ ps)))
4		iftrue 2989	. . . . 5 |- (ph -> if(ph, A, B) = A)
5		sbif.1	. . . . 5 |- (if(ph, A, B) = A -> (ps <-> ch))
6	4, 5	syl 12	. . . 4 |- (ph -> (ps <-> ch))
7	6	pm5.32i 707	. . 3 |- ((ph /\ ps) <-> (ph /\ ch))
8		iffalse 2991	. . . . 5 |- (-. ph -> if(ph, A, B) = B)
9		sbif.2	. . . . 5 |- (if(ph, A, B) = B -> (ps <-> th))
10	8, 9	syl 12	. . . 4 |- (-. ph -> (ps <-> th))
11	10	pm5.32i 707	. . 3 |- ((-. ph /\ ps) <-> (-. ph /\ th))
12	7, 11	orbi12i 277	. 2 |- (((ph /\ ps) \/ (-. ph /\ ps)) <-> ((ph /\ ch) \/ (-. ph /\ th)))
13	2, 3, 12	3bitri 194	1 |- (ps <-> ((ph /\ ch) \/ (-. ph /\ th)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298  ifcif 2982

This theorem is referenced by:  eqif 3004  elif 3005  ifel 3006

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-if 2983

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