Theorem r19.21

Description: Restricted quantifier version of 19.21 . (Contributed by Scott Fenton, 30-Mar-2011)

	Ref	Expression
Hypothesis	r19.21.1	$$⊢Ⅎx\phi$$
Assertion	r19.21	$$⊢\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\in A\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		r19.21.1	$$⊢Ⅎx\phi$$
2		r19.21t	$$⊢Ⅎx\phi \to \left(\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\in A\psi \right)\right)$$
3	1 2	ax-mp	$$⊢\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\in A\psi \right)$$