Theorem bj-pm11.53a

Description: A variant of pm11.53v . One can similarly prove a variant with DV ( y , ph ) and A. y F// x ps instead of DV ( x , ps ) and A. x F// y ph . (Contributed by BJ, 7-Oct-2024)

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

Proof

Step	Hyp	Ref	Expression
1		bj-nnfv	$$⊢Ⅎ\text{'}x\forall y\psi$$
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	mpan2	$$⊢\forall xℲ\text{'}y\phi \to \left(\forall x\forall y\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\phi \to \forall y\psi \right)\right)$$