Theorem euan 1953

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
euan.1	|- (ph -> A.xph)

Assertion

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

Proof of Theorem euan

Step	Hyp	Ref	Expression
1		euan.1	. . . . . 6 |- (ph -> A.xph)
2		simpl 102	. . . . . 6 |- ((ph /\ ps) -> ph)
3	1, 2	exlimih 1481	. . . . 5 |- (E.x(ph /\ ps) -> ph)
4	3	adantr 261	. . . 4 |- ((E.x(ph /\ ps) /\ E*x(ph /\ ps)) -> ph)
5		simpr 103	. . . . . 6 |- ((ph /\ ps) -> ps)
6	5	eximi 1488	. . . . 5 |- (E.x(ph /\ ps) -> E.xps)
7	6	adantr 261	. . . 4 |- ((E.x(ph /\ ps) /\ E*x(ph /\ ps)) -> E.xps)
8		hbe1 1381	. . . . . 6 |- (E.x(ph /\ ps) -> A.xE.x(ph /\ ps))
9	3	a1d 22	. . . . . . . 8 |- (E.x(ph /\ ps) -> (ps -> ph))
10	9	ancrd 309	. . . . . . 7 |- (E.x(ph /\ ps) -> (ps -> (ph /\ ps)))
11	5, 10	impbid2 131	. . . . . 6 |- (E.x(ph /\ ps) -> ((ph /\ ps) <-> ps))
12	8, 11	mobidh 1931	. . . . 5 |- (E.x(ph /\ ps) -> (E*x(ph /\ ps) <-> E*xps))
13	12	biimpa 280	. . . 4 |- ((E.x(ph /\ ps) /\ E*x(ph /\ ps)) -> E*xps)
14	4, 7, 13	jca32 293	. . 3 |- ((E.x(ph /\ ps) /\ E*x(ph /\ ps)) -> (ph /\ (E.xps /\ E*xps)))
15		eu5 1944	. . 3 |- (E!x(ph /\ ps) <-> (E.x(ph /\ ps) /\ E*x(ph /\ ps)))
16		eu5 1944	. . . 4 |- (E!xps <-> (E.xps /\ E*xps))
17	16	anbi2i 430	. . 3 |- ((ph /\ E!xps) <-> (ph /\ (E.xps /\ E*xps)))
18	14, 15, 17	3imtr4i 190	. 2 |- (E!x(ph /\ ps) -> (ph /\ E!xps))
19		ibar 285	. . . 4 |- (ph -> (ps <-> (ph /\ ps)))
20	1, 19	eubidh 1903	. . 3 |- (ph -> (E!xps <-> E!x(ph /\ ps)))
21	20	biimpa 280	. 2 |- ((ph /\ E!xps) -> E!x(ph /\ ps))
22	18, 21	impbii 117	1 |- (E!x(ph /\ ps) <-> (ph /\ E!xps))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  A.wal 1240  E.wex 1378  E!weu 1897  E*wmo 1898

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901

This theorem is referenced by:  euanv  1954  2eu7  1991