Theorem ss2rexv

Description: Two existential quantifications restricted to a subclass. (Contributed by AV, 11-Mar-2023)

	Ref	Expression
Assertion	ss2rexv	$$⊢A\subseteq B\to \left(\exists x\in A\exists y\in A\phi \to \exists x\in B\exists y\in B\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ssrexv	$$⊢A\subseteq B\to \left(\exists y\in A\phi \to \exists y\in B\phi \right)$$
2	1	reximdv	$$⊢A\subseteq B\to \left(\exists x\in A\exists y\in A\phi \to \exists x\in A\exists y\in B\phi \right)$$
3		ssrexv	$$⊢A\subseteq B\to \left(\exists x\in A\exists y\in B\phi \to \exists x\in B\exists y\in B\phi \right)$$
4	2 3	syld	$$⊢A\subseteq B\to \left(\exists x\in A\exists y\in A\phi \to \exists x\in B\exists y\in B\phi \right)$$