Theorem 19.31v

Description: Version of 19.31 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020)

	Ref	Expression
Assertion	19.31v	$$⊢\forall x\left(\phi \vee \psi \right)\leftrightarrow \left(\forall x\phi \vee \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.32v	$$⊢\forall x\left(\psi \vee \phi \right)\leftrightarrow \left(\psi \vee \forall x\phi \right)$$
2		orcom	$$⊢\left(\phi \vee \psi \right)\leftrightarrow \left(\psi \vee \phi \right)$$
3	2	albii	$$⊢\forall x\left(\phi \vee \psi \right)\leftrightarrow \forall x\left(\psi \vee \phi \right)$$
4		orcom	$$⊢\left(\forall x\phi \vee \psi \right)\leftrightarrow \left(\psi \vee \forall x\phi \right)$$
5	1 3 4	3bitr4i	$$⊢\forall x\left(\phi \vee \psi \right)\leftrightarrow \left(\forall x\phi \vee \psi \right)$$