Theorem moanim

Description: Introduction of a conjunct into "at most one" quantifier. For a version requiring disjoint variables, but fewer axioms, see moanimv . (Contributed by NM, 3-Dec-2001) (Proof shortened by Wolf Lammen, 24-Dec-2018)

	Ref	Expression
Hypothesis	moanim.1	|- F/ x ph
Assertion	moanim	|- ( E* x ( ph /\ ps ) <-> ( ph -> E* x ps ) )

Proof

Step	Hyp	Ref	Expression
1		moanim.1	|- F/ x ph
2		ibar	|- ( ph -> ( ps <-> ( ph /\ ps ) ) )
3	1 2	mobid	|- ( ph -> ( E* x ps <-> E* x ( ph /\ ps ) ) )
4		simpl	|- ( ( ph /\ ps ) -> ph )
5	1 4	exlimi	|- ( E. x ( ph /\ ps ) -> ph )
6	3 5	moanimlem	|- ( E* x ( ph /\ ps ) <-> ( ph -> E* x ps ) )