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Metamath Proof Explorer

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 ) )