Theorem ssralv

Description: Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006)

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

Proof

Step	Hyp	Ref	Expression
1		ssel	$$⊢A\subseteq B\to \left(x\in A\to x\in B\right)$$
2	1	imim1d	$$⊢A\subseteq B\to \left(\left(x\in B\to \phi \right)\to \left(x\in A\to \phi \right)\right)$$
3	2	ralimdv2	$$⊢A\subseteq B\to \left(\forall x\in B\phi \to \forall x\in A\phi \right)$$