Theorem pm11.53v

Description: Version of pm11.53 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020)

	Ref	Expression
Assertion	pm11.53v	$$⊢\forall x\forall y\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\phi \to \forall y\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.21v	$$⊢\forall y\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall y\psi \right)$$
2	1	albii	$$⊢\forall x\forall y\left(\phi \to \psi \right)\leftrightarrow \forall x\left(\phi \to \forall y\psi \right)$$
3		19.23v	$$⊢\forall x\left(\phi \to \forall y\psi \right)\leftrightarrow \left(\exists x\phi \to \forall y\psi \right)$$
4	2 3	bitri	$$⊢\forall x\forall y\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\phi \to \forall y\psi \right)$$