Theorem euor2 1430

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

Assertion

Ref	Expression
euor2	|- (-. E.xph -> (E!x(ph \/ ps) <-> E!xps))

Proof of Theorem euor2

Step	Hyp	Ref	Expression
1		euex 1387	. . . . . . 7 |- (E!x(ph \/ ps) -> E.x(ph \/ ps))
2		19.43 1084	. . . . . . 7 |- (E.x(ph \/ ps) <-> (E.xph \/ E.xps))
3	1, 2	sylib 198	. . . . . 6 |- (E!x(ph \/ ps) -> (E.xph \/ E.xps))
4	3	ord 232	. . . . 5 |- (E!x(ph \/ ps) -> (-. E.xph -> E.xps))
5	4	com12 11	. . . 4 |- (-. E.xph -> (E!x(ph \/ ps) -> E.xps))
6		eumo 1404	. . . . . 6 |- (E!x(ph \/ ps) -> E*x(ph \/ ps))
7		orcom 246	. . . . . . . 8 |- ((ph \/ ps) <-> (ps \/ ph))
8	7	mobii 1398	. . . . . . 7 |- (E*x(ph \/ ps) <-> E*x(ps \/ ph))
9		moor 1417	. . . . . . 7 |- (E*x(ps \/ ph) -> E*xps)
10	8, 9	sylbi 199	. . . . . 6 |- (E*x(ph \/ ps) -> E*xps)
11	6, 10	syl 10	. . . . 5 |- (E!x(ph \/ ps) -> E*xps)
12	11	a1i 8	. . . 4 |- (-. E.xph -> (E!x(ph \/ ps) -> E*xps))
13	5, 12	jcad 598	. . 3 |- (-. E.xph -> (E!x(ph \/ ps) -> (E.xps /\ E*xps)))
14		eu5 1402	. . 3 |- (E!xps <-> (E.xps /\ E*xps))
15	13, 14	syl6ibr 213	. 2 |- (-. E.xph -> (E!x(ph \/ ps) -> E!xps))
16		hbe1 1012	. . . . 5 |- (E.xph -> A.xE.xph)
17	16	euor 1391	. . . 4 |- ((-. E.xph /\ E!xps) -> E!x(E.xph \/ ps))
18		euex 1387	. . . . . 6 |- (E!xps -> E.xps)
19		olc 268	. . . . . . 7 |- (ps -> (ph \/ ps))
20	19	19.22i 1036	. . . . . 6 |- (E.xps -> E.x(ph \/ ps))
21		19.8a 1025	. . . . . . . . 9 |- (ph -> E.xph)
22	21	orim1i 337	. . . . . . . 8 |- ((ph \/ ps) -> (E.xph \/ ps))
23	22	ax-gen 960	. . . . . . 7 |- A.x((ph \/ ps) -> (E.xph \/ ps))
24		euim 1414	. . . . . . 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 694	. . . . . 6 |- (E.x(ph \/ ps) -> (E!x(E.xph \/ ps) -> E!x(ph \/ ps)))
26	18, 20, 25	3syl 20	. . . . 5 |- (E!xps -> (E!x(E.xph \/ ps) -> E!x(ph \/ ps)))
27	26	adantl 388	. . . 4 |- ((-. E.xph /\ E!xps) -> (E!x(E.xph \/ ps) -> E!x(ph \/ ps)))
28	17, 27	mpd 26	. . 3 |- ((-. E.xph /\ E!xps) -> E!x(ph \/ ps))
29	28	ex 373	. 2 |- (-. E.xph -> (E!xps -> E!x(ph \/ ps)))
30	15, 29	impbid 514	1 |- (-. E.xph -> (E!x(ph \/ ps) <-> E!xps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223  A.wal 951  E.wex 977  E!weu 1373  E*wmo 1374

This theorem is referenced by:  reuun2 2268

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376

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