Theorem bj-wnf1

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-wnf1	$$⊢\left(\exists x\phi \to \forall x\psi \right)\to \forall x\left(\exists x\phi \to \forall x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-modal4e	$$⊢\exists x\exists x\phi \to \exists x\phi$$
2		hba1	$$⊢\forall x\psi \to \forall x\forall x\psi$$
3	1 2	imim12i	$$⊢\left(\exists x\phi \to \forall x\psi \right)\to \left(\exists x\exists x\phi \to \forall x\forall x\psi \right)$$
4		19.38	$$⊢\left(\exists x\exists x\phi \to \forall x\forall x\psi \right)\to \forall x\left(\exists x\phi \to \forall x\psi \right)$$
5	3 4	syl	$$⊢\left(\exists x\phi \to \forall x\psi \right)\to \forall x\left(\exists x\phi \to \forall x\psi \right)$$