Theorem 19.32v

Description: Version of 19.32 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020)

	Ref	Expression
Assertion	19.32v	$$⊢\forall x\left(\phi \vee \psi \right)\leftrightarrow \left(\phi \vee \forall x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.21v	$$⊢\forall x\left(\neg \phi \to \psi \right)\leftrightarrow \left(\neg \phi \to \forall x\psi \right)$$
2		df-or	$$⊢\left(\phi \vee \psi \right)\leftrightarrow \left(\neg \phi \to \psi \right)$$
3	2	albii	$$⊢\forall x\left(\phi \vee \psi \right)\leftrightarrow \forall x\left(\neg \phi \to \psi \right)$$
4		df-or	$$⊢\left(\phi \vee \forall x\psi \right)\leftrightarrow \left(\neg \phi \to \forall x\psi \right)$$
5	1 3 4	3bitr4i	$$⊢\forall x\left(\phi \vee \psi \right)\leftrightarrow \left(\phi \vee \forall x\psi \right)$$