Theorem ralss

Description: Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015)

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

Proof

Step	Hyp	Ref	Expression
1		ssel	$$⊢A\subseteq B\to \left(x\in A\to x\in B\right)$$
2	1	pm4.71rd	$$⊢A\subseteq B\to \left(x\in A\leftrightarrow \left(x\in B\wedge x\in A\right)\right)$$
3	2	imbi1d	$$⊢A\subseteq B\to \left(\left(x\in A\to \phi \right)\leftrightarrow \left(\left(x\in B\wedge x\in A\right)\to \phi \right)\right)$$
4		impexp	$$⊢\left(\left(x\in B\wedge x\in A\right)\to \phi \right)\leftrightarrow \left(x\in B\to \left(x\in A\to \phi \right)\right)$$
5	3 4	bitrdi	$$⊢A\subseteq B\to \left(\left(x\in A\to \phi \right)\leftrightarrow \left(x\in B\to \left(x\in A\to \phi \right)\right)\right)$$
6	5	ralbidv2	$$⊢A\subseteq B\to \left(\forall x\in A\phi \leftrightarrow \forall x\in B\left(x\in A\to \phi \right)\right)$$