Theorem elimif 2374

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 655	. . 3 |- (ph \/ -. ph)
2	1	biantrur 725	. 2 |- (ps <-> ((ph \/ -. ph) /\ ps))
3		andir 605	. 2 |- (((ph \/ -. ph) /\ ps) <-> ((ph /\ ps) \/ (-. ph /\ ps)))
4		iftrue 2366	. . . . 5 |- (ph -> if(ph, A, B) = A)
5		sbif.1	. . . . 5 |- (if(ph, A, B) = A -> (ps <-> ch))
6	4, 5	syl 10	. . . 4 |- (ph -> (ps <-> ch))
7	6	pm5.32i 645	. . 3 |- ((ph /\ ps) <-> (ph /\ ch))
8		iffalse 2367	. . . . 5 |- (-. ph -> if(ph, A, B) = B)
9		sbif.2	. . . . 5 |- (if(ph, A, B) = B -> (ps <-> th))
10	8, 9	syl 10	. . . 4 |- (-. ph -> (ps <-> th))
11	10	pm5.32i 645	. . 3 |- ((-. ph /\ ps) <-> (-. ph /\ th))
12	7, 11	orbi12i 257	. 2 |- (((ph /\ ps) \/ (-. ph /\ ps)) <-> ((ph /\ ch) \/ (-. ph /\ th)))
13	2, 3, 12	3bitr 177	1 |- (ps <-> ((ph /\ ch) \/ (-. ph /\ th)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 956  ifcif 2361

This theorem is referenced by:  eqif 2377  elif 2378  ifel 2379

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-if 2362

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