Theorem r19.23

Description: Restricted quantifier version of 19.23 . See r19.23v for a version requiring fewer axioms. (Contributed by NM, 22-Oct-2010) (Proof shortened by Mario Carneiro, 8-Oct-2016)

	Ref	Expression
Hypothesis	r19.23.1	$$⊢Ⅎx\psi$$
Assertion	r19.23	$$⊢\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\in A\phi \to \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		r19.23.1	$$⊢Ⅎx\psi$$
2		r19.23t	$$⊢Ⅎx\psi \to \left(\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\in A\phi \to \psi \right)\right)$$
3	1 2	ax-mp	$$⊢\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\in A\phi \to \psi \right)$$