Theorem bj-wnf2

Description: When ph is substituted for ps , this is the first half of nonfreness ( . -> A. ) of the weak form of nonfreeness ( E. -> A. ) . (Contributed by BJ, 9-Dec-2023)

	Ref	Expression
Assertion	bj-wnf2	$$⊢\exists x\left(\exists x\phi \to \forall x\psi \right)\to \left(\exists x\phi \to \forall x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		hbe1	$$⊢\exists x\phi \to \forall x\exists x\phi$$
2		bj-eximcom	$$⊢\exists x\left(\exists x\phi \to \forall x\psi \right)\to \left(\forall x\exists x\phi \to \exists x\forall x\psi \right)$$
3		hbe1a	$$⊢\exists x\forall x\psi \to \forall x\psi$$
4	1 2 3	syl56	$$⊢\exists x\left(\exists x\phi \to \forall x\psi \right)\to \left(\exists x\phi \to \forall x\psi \right)$$