Theorem rmoimi2

Description: Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017)

	Ref	Expression
Hypothesis	rmoimi2.1	|- A. x ( ( x e. A /\ ph ) -> ( x e. B /\ ps ) )
Assertion	rmoimi2	|- ( E* x e. B ps -> E* x e. A ph )

Proof

Step	Hyp	Ref	Expression
1		rmoimi2.1	|- A. x ( ( x e. A /\ ph ) -> ( x e. B /\ ps ) )
2		moim	|- ( A. x ( ( x e. A /\ ph ) -> ( x e. B /\ ps ) ) -> ( E* x ( x e. B /\ ps ) -> E* x ( x e. A /\ ph ) ) )
3	1 2	ax-mp	|- ( E* x ( x e. B /\ ps ) -> E* x ( x e. A /\ ph ) )
4		df-rmo	|- ( E* x e. B ps <-> E* x ( x e. B /\ ps ) )
5		df-rmo	|- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
6	3 4 5	3imtr4i	|- ( E* x e. B ps -> E* x e. A ph )