Theorem ssralv

Description: Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006) Avoid axioms. (Revised by GG, 19-May-2025)

	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		df-ss	$$⊢A\subseteq B\leftrightarrow \forall x\left(x\in A\to x\in B\right)$$
2		imim1	$$⊢\left(x\in A\to x\in B\right)\to \left(\left(x\in B\to \phi \right)\to \left(x\in A\to \phi \right)\right)$$
3	2	al2imi	$$⊢\forall x\left(x\in A\to x\in B\right)\to \left(\forall x\left(x\in B\to \phi \right)\to \forall x\left(x\in A\to \phi \right)\right)$$
4		df-ral	$$⊢\forall x\in B\phi \leftrightarrow \forall x\left(x\in B\to \phi \right)$$
5		df-ral	$$⊢\forall x\in A\phi \leftrightarrow \forall x\left(x\in A\to \phi \right)$$
6	3 4 5	3imtr4g	$$⊢\forall x\left(x\in A\to x\in B\right)\to \left(\forall x\in B\phi \to \forall x\in A\phi \right)$$
7	1 6	sylbi	$$⊢A\subseteq B\to \left(\forall x\in B\phi \to \forall x\in A\phi \right)$$