Theorem moanim 2301

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 502	. . . 4 |- ( ph -> ( ps <-> ( ph /\ ps ) ) )
3	1, 2	mobid 2259	. . 3 |- ( ph -> ( E* x ps <-> E* x ( ph /\ ps ) ) )
4	3	biimprcd 225	. 2 |- ( E* x ( ph /\ ps ) -> ( ph -> E* x ps ) )
5		simpl 455	. . . . . 6 |- ( ( ph /\ ps ) -> ph )
6	1, 5	exlimi 1940	. . . . 5 |- ( E. x ( ph /\ ps ) -> ph )
7		exmo 2265	. . . . . 6 |- ( E. x ( ph /\ ps ) \/ E* x ( ph /\ ps ) )
8	7	ori 373	. . . . 5 |- ( -. E. x ( ph /\ ps ) -> E* x ( ph /\ ps ) )
9	6, 8	nsyl4 142	. . . 4 |- ( -. E* x ( ph /\ ps ) -> ph )
10	9	con1i 129	. . 3 |- ( -. ph -> E* x ( ph /\ ps ) )
11		moan 2296	. . 3 |- ( E* x ps -> E* x ( ph /\ ps ) )
12	10, 11	ja 161	. 2 |- ( ( ph -> E* x ps ) -> E* x ( ph /\ ps ) )
13	4, 12	impbii 188	1 |- ( E* x ( ph /\ ps ) <-> ( ph -> E* x ps ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   E.wex 1633   F/wnf 1637   E*wmo 2239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-12 1878

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-ex 1634  df-nf 1638  df-eu 2242  df-mo 2243

This theorem is referenced by:  moanimv  2303  moanmo  2304  2eu1OLD  2328