Theorem euan 2379

Description: Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 24-Dec-2018.)

Hypothesis

Ref	Expression
moanim.1	|- F/ x ph

Assertion

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

Proof of Theorem euan

Step	Hyp	Ref	Expression
1		euex 2343	. . . 4 |- ( E! x ( ph /\ ps ) -> E. x ( ph /\ ps ) )
2		moanim.1	. . . . 5 |- F/ x ph
3		simpl 464	. . . . 5 |- ( ( ph /\ ps ) -> ph )
4	2, 3	exlimi 2015	. . . 4 |- ( E. x ( ph /\ ps ) -> ph )
5	1, 4	syl 17	. . 3 |- ( E! x ( ph /\ ps ) -> ph )
6		ibar 512	. . . . 5 |- ( ph -> ( ps <-> ( ph /\ ps ) ) )
7	2, 6	eubid 2337	. . . 4 |- ( ph -> ( E! x ps <-> E! x ( ph /\ ps ) ) )
8	7	biimprcd 233	. . 3 |- ( E! x ( ph /\ ps ) -> ( ph -> E! x ps ) )
9	5, 8	jcai 545	. 2 |- ( E! x ( ph /\ ps ) -> ( ph /\ E! x ps ) )
10	7	biimpa 492	. 2 |- ( ( ph /\ E! x ps ) -> E! x ( ph /\ ps ) )
11	9, 10	impbii 192	1 |- ( E! x ( ph /\ ps ) <-> ( ph /\ E! x ps ) )

Syntax hints:   <-> wb 189   /\ wa 376   E.wex 1671   F/wnf 1675   E!weu 2319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950

This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676  df-eu 2323

This theorem is referenced by:  euanv  2383  2eu7  2408  2eu8  2409