Theorem bj-pm11.53v

Description: Version of pm11.53v with nonfreeness antecedents. (Contributed by BJ, 7-Oct-2024)

	Ref	Expression
Assertion	bj-pm11.53v	$$⊢\left(\forall xℲ\text{'}y\phi \wedge \forall yℲ\text{'}x\psi \right)\to \left(\forall x\forall y\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\phi \to \forall y\psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-nnfalt	$$⊢\forall yℲ\text{'}x\psi \to Ⅎ\text{'}x\forall y\psi$$
2		bj-pm11.53vw	$$⊢\left(\forall xℲ\text{'}y\phi \wedge Ⅎ\text{'}x\forall y\psi \right)\to \left(\forall x\forall y\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\phi \to \forall y\psi \right)\right)$$
3	1 2	sylan2	$$⊢\left(\forall xℲ\text{'}y\phi \wedge \forall yℲ\text{'}x\psi \right)\to \left(\forall x\forall y\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\phi \to \forall y\psi \right)\right)$$