Theorem euan 2056

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

Hypothesis

Ref	Expression
euan.1	|- ( ph -> A. x ph )

Assertion

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

Proof of Theorem euan

Step	Hyp	Ref	Expression
1		euan.1	. . . . . 6 |- ( ph -> A. x ph )
2		simpl 108	. . . . . 6 |- ( ( ph /\ ps ) -> ph )
3	1, 2	exlimih 1573	. . . . 5 |- ( E. x ( ph /\ ps ) -> ph )
4	3	adantr 274	. . . 4 |- ( ( E. x ( ph /\ ps ) /\ E* x ( ph /\ ps ) ) -> ph )
5		simpr 109	. . . . . 6 |- ( ( ph /\ ps ) -> ps )
6	5	eximi 1580	. . . . 5 |- ( E. x ( ph /\ ps ) -> E. x ps )
7	6	adantr 274	. . . 4 |- ( ( E. x ( ph /\ ps ) /\ E* x ( ph /\ ps ) ) -> E. x ps )
8		hbe1 1472	. . . . . 6 |- ( E. x ( ph /\ ps ) -> A. x E. x ( ph /\ ps ) )
9	3	a1d 22	. . . . . . . 8 |- ( E. x ( ph /\ ps ) -> ( ps -> ph ) )
10	9	ancrd 324	. . . . . . 7 |- ( E. x ( ph /\ ps ) -> ( ps -> ( ph /\ ps ) ) )
11	5, 10	impbid2 142	. . . . . 6 |- ( E. x ( ph /\ ps ) -> ( ( ph /\ ps ) <-> ps ) )
12	8, 11	mobidh 2034	. . . . 5 |- ( E. x ( ph /\ ps ) -> ( E* x ( ph /\ ps ) <-> E* x ps ) )
13	12	biimpa 294	. . . 4 |- ( ( E. x ( ph /\ ps ) /\ E* x ( ph /\ ps ) ) -> E* x ps )
14	4, 7, 13	jca32 308	. . 3 |- ( ( E. x ( ph /\ ps ) /\ E* x ( ph /\ ps ) ) -> ( ph /\ ( E. x ps /\ E* x ps ) ) )
15		eu5 2047	. . 3 |- ( E! x ( ph /\ ps ) <-> ( E. x ( ph /\ ps ) /\ E* x ( ph /\ ps ) ) )
16		eu5 2047	. . . 4 |- ( E! x ps <-> ( E. x ps /\ E* x ps ) )
17	16	anbi2i 453	. . 3 |- ( ( ph /\ E! x ps ) <-> ( ph /\ ( E. x ps /\ E* x ps ) ) )
18	14, 15, 17	3imtr4i 200	. 2 |- ( E! x ( ph /\ ps ) -> ( ph /\ E! x ps ) )
19		ibar 299	. . . 4 |- ( ph -> ( ps <-> ( ph /\ ps ) ) )
20	1, 19	eubidh 2006	. . 3 |- ( ph -> ( E! x ps <-> E! x ( ph /\ ps ) ) )
21	20	biimpa 294	. 2 |- ( ( ph /\ E! x ps ) -> E! x ( ph /\ ps ) )
22	18, 21	impbii 125	1 |- ( E! x ( ph /\ ps ) <-> ( ph /\ E! x ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1330   E.wex 1469   E!weu 2000   E*wmo 2001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004

This theorem is referenced by:  euanv  2057  2eu7  2094