Theorem r19.32v

Description: Restricted quantifier version of 19.32v . (Contributed by NM, 25-Nov-2003)

	Ref	Expression
Assertion	r19.32v	$$⊢\forall x\in A\left(\phi \vee \psi \right)\leftrightarrow \left(\phi \vee \forall x\in A\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		r19.21v	$$⊢\forall x\in A\left(\neg \phi \to \psi \right)\leftrightarrow \left(\neg \phi \to \forall x\in A\psi \right)$$
2		df-or	$$⊢\left(\phi \vee \psi \right)\leftrightarrow \left(\neg \phi \to \psi \right)$$
3	2	ralbii	$$⊢\forall x\in A\left(\phi \vee \psi \right)\leftrightarrow \forall x\in A\left(\neg \phi \to \psi \right)$$
4		df-or	$$⊢\left(\phi \vee \forall x\in A\psi \right)\leftrightarrow \left(\neg \phi \to \forall x\in A\psi \right)$$
5	1 3 4	3bitr4i	$$⊢\forall x\in A\left(\phi \vee \psi \right)\leftrightarrow \left(\phi \vee \forall x\in A\psi \right)$$