Theorem ssrexv

Description: Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007)

	Ref	Expression
Assertion	ssrexv	$$⊢A\subseteq B\to \left(\exists x\in A\phi \to \exists x\in B\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ssel	$$⊢A\subseteq B\to \left(x\in A\to x\in B\right)$$
2	1	anim1d	$$⊢A\subseteq B\to \left(\left(x\in A\wedge \phi \right)\to \left(x\in B\wedge \phi \right)\right)$$
3	2	reximdv2	$$⊢A\subseteq B\to \left(\exists x\in A\phi \to \exists x\in B\phi \right)$$