Theorem moanim 2529

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 525	. . . 4 |- ( ph -> ( ps <-> ( ph /\ ps ) ) )
3	1, 2	mobid 2489	. . 3 |- ( ph -> ( E* x ps <-> E* x ( ph /\ ps ) ) )
4	3	biimprcd 240	. 2 |- ( E* x ( ph /\ ps ) -> ( ph -> E* x ps ) )
5		simpl 473	. . . . . 6 |- ( ( ph /\ ps ) -> ph )
6	1, 5	exlimi 2086	. . . . 5 |- ( E. x ( ph /\ ps ) -> ph )
7		exmo 2495	. . . . . 6 |- ( E. x ( ph /\ ps ) \/ E* x ( ph /\ ps ) )
8	7	ori 390	. . . . 5 |- ( -. E. x ( ph /\ ps ) -> E* x ( ph /\ ps ) )
9	6, 8	nsyl4 156	. . . 4 |- ( -. E* x ( ph /\ ps ) -> ph )
10	9	con1i 144	. . 3 |- ( -. ph -> E* x ( ph /\ ps ) )
11		moan 2524	. . 3 |- ( E* x ps -> E* x ( ph /\ ps ) )
12	10, 11	ja 173	. 2 |- ( ( ph -> E* x ps ) -> E* x ( ph /\ ps ) )
13	4, 12	impbii 199	1 |- ( E* x ( ph /\ ps ) <-> ( ph -> E* x ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   E.wex 1704   F/wnf 1708   E*wmo 2471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by:  moanimv  2531  moanmo  2532