Theorem moanim 2378

Description: Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 24-Dec-2018.)

Hypothesis

Ref	Expression
moanim.1	|- F/ x ph

Assertion

Ref	Expression
moanim	|- ( E* x ( ph /\ ps ) <-> ( ph -> E* x ps ) )

Proof of Theorem moanim

Step	Hyp	Ref	Expression
1		moanim.1	. . . 4 |- F/ x ph
2		ibar 512	. . . 4 |- ( ph -> ( ps <-> ( ph /\ ps ) ) )
3	1, 2	mobid 2338	. . 3 |- ( ph -> ( E* x ps <-> E* x ( ph /\ ps ) ) )
4	3	biimprcd 233	. 2 |- ( E* x ( ph /\ ps ) -> ( ph -> E* x ps ) )
5		simpl 464	. . . . . 6 |- ( ( ph /\ ps ) -> ph )
6	1, 5	exlimi 2015	. . . . 5 |- ( E. x ( ph /\ ps ) -> ph )
7		exmo 2344	. . . . . 6 |- ( E. x ( ph /\ ps ) \/ E* x ( ph /\ ps ) )
8	7	ori 382	. . . . 5 |- ( -. E. x ( ph /\ ps ) -> E* x ( ph /\ ps ) )
9	6, 8	nsyl4 149	. . . 4 |- ( -. E* x ( ph /\ ps ) -> ph )
10	9	con1i 134	. . 3 |- ( -. ph -> E* x ( ph /\ ps ) )
11		moan 2373	. . 3 |- ( E* x ps -> E* x ( ph /\ ps ) )
12	10, 11	ja 166	. 2 |- ( ( ph -> E* x ps ) -> E* x ( ph /\ ps ) )
13	4, 12	impbii 192	1 |- ( E* x ( ph /\ ps ) <-> ( ph -> E* x ps ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   E.wex 1671   F/wnf 1675   E*wmo 2320

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-or 377  df-an 378  df-ex 1672  df-nf 1676  df-eu 2323  df-mo 2324

This theorem is referenced by:  moanimv  2380  moanmo  2381