Theorem ssrexf

Description: Restricted existential quantification follows from a subclass relationship. (Contributed by Glauco Siliprandi, 20-Apr-2017)

	Ref	Expression
Hypotheses	ssrexf.1	|- F/_ x A
	ssrexf.2	|- F/_ x B
Assertion	ssrexf	|- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) )

Proof

Step	Hyp	Ref	Expression
1		ssrexf.1	|- F/_ x A
2		ssrexf.2	|- F/_ x B
3	1 2	nfss	|- F/ x A C_ B
4		ssel	|- ( A C_ B -> ( x e. A -> x e. B ) )
5	4	anim1d	|- ( A C_ B -> ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
6	3 5	eximd	|- ( A C_ B -> ( E. x ( x e. A /\ ph ) -> E. x ( x e. B /\ ph ) ) )
7		df-rex	|- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
8		df-rex	|- ( E. x e. B ph <-> E. x ( x e. B /\ ph ) )
9	6 7 8	3imtr4g	|- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) )