Theorem euan 2110

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 109	. . . . . 6 |- ( ( ph /\ ps ) -> ph )
3	1, 2	exlimih 1616	. . . . 5 |- ( E. x ( ph /\ ps ) -> ph )
4	3	adantr 276	. . . 4 |- ( ( E. x ( ph /\ ps ) /\ E* x ( ph /\ ps ) ) -> ph )
5		simpr 110	. . . . . 6 |- ( ( ph /\ ps ) -> ps )
6	5	eximi 1623	. . . . 5 |- ( E. x ( ph /\ ps ) -> E. x ps )
7	6	adantr 276	. . . 4 |- ( ( E. x ( ph /\ ps ) /\ E* x ( ph /\ ps ) ) -> E. x ps )
8		hbe1 1518	. . . . . 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 326	. . . . . . 7 |- ( E. x ( ph /\ ps ) -> ( ps -> ( ph /\ ps ) ) )
11	5, 10	impbid2 143	. . . . . 6 |- ( E. x ( ph /\ ps ) -> ( ( ph /\ ps ) <-> ps ) )
12	8, 11	mobidh 2088	. . . . 5 |- ( E. x ( ph /\ ps ) -> ( E* x ( ph /\ ps ) <-> E* x ps ) )
13	12	biimpa 296	. . . 4 |- ( ( E. x ( ph /\ ps ) /\ E* x ( ph /\ ps ) ) -> E* x ps )
14	4, 7, 13	jca32 310	. . 3 |- ( ( E. x ( ph /\ ps ) /\ E* x ( ph /\ ps ) ) -> ( ph /\ ( E. x ps /\ E* x ps ) ) )
15		eu5 2101	. . 3 |- ( E! x ( ph /\ ps ) <-> ( E. x ( ph /\ ps ) /\ E* x ( ph /\ ps ) ) )
16		eu5 2101	. . . 4 |- ( E! x ps <-> ( E. x ps /\ E* x ps ) )
17	16	anbi2i 457	. . 3 |- ( ( ph /\ E! x ps ) <-> ( ph /\ ( E. x ps /\ E* x ps ) ) )
18	14, 15, 17	3imtr4i 201	. 2 |- ( E! x ( ph /\ ps ) -> ( ph /\ E! x ps ) )
19		ibar 301	. . . 4 |- ( ph -> ( ps <-> ( ph /\ ps ) ) )
20	1, 19	eubidh 2060	. . 3 |- ( ph -> ( E! x ps <-> E! x ( ph /\ ps ) ) )
21	20	biimpa 296	. 2 |- ( ( ph /\ E! x ps ) -> E! x ( ph /\ ps ) )
22	18, 21	impbii 126	1 |- ( E! x ( ph /\ ps ) <-> ( ph /\ E! x ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1371   E.wex 1515   E!weu 2054   E*wmo 2055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058

This theorem is referenced by:  euanv  2111  2eu7  2148