Theorem euor2OLD 1840

Description: Introduce or eliminate a disjunct in a uniqueness quantifier.

Assertion

Ref	Expression
euor2OLD	|- (-. E.xph -> (E!x(ph \/ ps) <-> E!xps))

Proof of Theorem euor2OLD

Step	Hyp	Ref	Expression
1		euex 1788	. . . . . . 7 |- (E!x(ph \/ ps) -> E.x(ph \/ ps))
2		19.43 1440	. . . . . . 7 |- (E.x(ph \/ ps) <-> (E.xph \/ E.xps))
3	1, 2	sylib 215	. . . . . 6 |- (E!x(ph \/ ps) -> (E.xph \/ E.xps))
4	3	ord 249	. . . . 5 |- (E!x(ph \/ ps) -> (-. E.xph -> E.xps))
5	4	com12 14	. . . 4 |- (-. E.xph -> (E!x(ph \/ ps) -> E.xps))
6		eumo 1807	. . . . . 6 |- (E!x(ph \/ ps) -> E*x(ph \/ ps))
7		orcom 266	. . . . . . . 8 |- ((ph \/ ps) <-> (ps \/ ph))
8	7	mobii 1801	. . . . . . 7 |- (E*x(ph \/ ps) <-> E*x(ps \/ ph))
9		moor 1821	. . . . . . 7 |- (E*x(ps \/ ph) -> E*xps)
10	8, 9	sylbi 216	. . . . . 6 |- (E*x(ph \/ ps) -> E*xps)
11	6, 10	syl 12	. . . . 5 |- (E!x(ph \/ ps) -> E*xps)
12	11	a1i 8	. . . 4 |- (-. E.xph -> (E!x(ph \/ ps) -> E*xps))
13	5, 12	jcad 661	. . 3 |- (-. E.xph -> (E!x(ph \/ ps) -> (E.xps /\ E*xps)))
14		eu5 1805	. . 3 |- (E!xps <-> (E.xps /\ E*xps))
15	13, 14	syl6ibr 230	. 2 |- (-. E.xph -> (E!x(ph \/ ps) -> E!xps))
16		hbe1 1363	. . . . 5 |- (E.xph -> A.xE.xph)
17	16	euor 1793	. . . 4 |- ((-. E.xph /\ E!xps) -> E!x(E.xph \/ ps))
18		euex 1788	. . . . . 6 |- (E!xps -> E.xps)
19		olc 290	. . . . . . 7 |- (ps -> (ph \/ ps))
20	19	eximi 1387	. . . . . 6 |- (E.xps -> E.x(ph \/ ps))
21		19.8a 1376	. . . . . . . . 9 |- (ph -> E.xph)
22	21	orim1i 364	. . . . . . . 8 |- ((ph \/ ps) -> (E.xph \/ ps))
23	22	ax-gen 1305	. . . . . . 7 |- A.x((ph \/ ps) -> (E.xph \/ ps))
24		euim 1817	. . . . . . 7 |- ((E.x(ph \/ ps) /\ A.x((ph \/ ps) -> (E.xph \/ ps))) -> (E!x(E.xph \/ ps) -> E!x(ph \/ ps)))
25	23, 24	mpan2 760	. . . . . 6 |- (E.x(ph \/ ps) -> (E!x(E.xph \/ ps) -> E!x(ph \/ ps)))
26	18, 20, 25	3syl 24	. . . . 5 |- (E!xps -> (E!x(E.xph \/ ps) -> E!x(ph \/ ps)))
27	26	adantl 424	. . . 4 |- ((-. E.xph /\ E!xps) -> (E!x(E.xph \/ ps) -> E!x(ph \/ ps)))
28	17, 27	mpd 29	. . 3 |- ((-. E.xph /\ E!xps) -> E!x(ph \/ ps))
29	28	ex 402	. 2 |- (-. E.xph -> (E!xps -> E!x(ph \/ ps)))
30	15, 29	impbid 574	1 |- (-. E.xph -> (E!x(ph \/ ps) <-> E!xps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240  A.wal 1296  E.wex 1326  E!weu 1771  E*wmo 1772

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-9 1307  ax-10 1308  ax-11 1309  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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776

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