Theorem euan 2261

Description: Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Hypothesis

Ref	Expression
moanim.1	|- F/xph

Assertion

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

Proof of Theorem euan

Step	Hyp	Ref	Expression
1		moanim.1	. . . . . 6 |- F/xph
2		simpl 443	. . . . . 6 |- ((ph /\ ps) -> ph)
3	1, 2	exlimi 1803	. . . . 5 |- (E.x(ph /\ ps) -> ph)
4	3	adantr 451	. . . 4 |- ((E.x(ph /\ ps) /\ E*x(ph /\ ps)) -> ph)
5		simpr 447	. . . . . 6 |- ((ph /\ ps) -> ps)
6	5	eximi 1576	. . . . 5 |- (E.x(ph /\ ps) -> E.xps)
7	6	adantr 451	. . . 4 |- ((E.x(ph /\ ps) /\ E*x(ph /\ ps)) -> E.xps)
8		nfe1 1732	. . . . . 6 |- F/xE.x(ph /\ ps)
9	3	a1d 22	. . . . . . . 8 |- (E.x(ph /\ ps) -> (ps -> ph))
10	9	ancrd 537	. . . . . . 7 |- (E.x(ph /\ ps) -> (ps -> (ph /\ ps)))
11	5, 10	impbid2 195	. . . . . 6 |- (E.x(ph /\ ps) -> ((ph /\ ps) <-> ps))
12	8, 11	mobid 2238	. . . . 5 |- (E.x(ph /\ ps) -> (E*x(ph /\ ps) <-> E*xps))
13	12	biimpa 470	. . . 4 |- ((E.x(ph /\ ps) /\ E*x(ph /\ ps)) -> E*xps)
14	4, 7, 13	jca32 521	. . 3 |- ((E.x(ph /\ ps) /\ E*x(ph /\ ps)) -> (ph /\ (E.xps /\ E*xps)))
15		eu5 2242	. . 3 |- (E!x(ph /\ ps) <-> (E.x(ph /\ ps) /\ E*x(ph /\ ps)))
16		eu5 2242	. . . 4 |- (E!xps <-> (E.xps /\ E*xps))
17	16	anbi2i 675	. . 3 |- ((ph /\ E!xps) <-> (ph /\ (E.xps /\ E*xps)))
18	14, 15, 17	3imtr4i 257	. 2 |- (E!x(ph /\ ps) -> (ph /\ E!xps))
19		ibar 490	. . . 4 |- (ph -> (ps <-> (ph /\ ps)))
20	1, 19	eubid 2211	. . 3 |- (ph -> (E!xps <-> E!x(ph /\ ps)))
21	20	biimpa 470	. 2 |- ((ph /\ E!xps) -> E!x(ph /\ ps))
22	18, 21	impbii 180	1 |- (E!x(ph /\ ps) <-> (ph /\ E!xps))

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   F/wnf 1544  E!weu 2204  E*wmo 2205

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

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-eu 2208  df-mo 2209

This theorem is referenced by:  euanv  2265  2eu7  2290  2eu8  2291