Theorem bj-wnfanf

Description: When ph is substituted for ps , this statement expresses that weak nonfreeness implies the "forall" form of nonfreeness. (Contributed by BJ, 9-Dec-2023)

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

Proof

Step	Hyp	Ref	Expression
1		bj-wnf1	$$⊢\left(\exists x\phi \to \forall x\psi \right)\to \forall x\left(\exists x\phi \to \forall x\psi \right)$$
2		bj-19.23bit	$$⊢\left(\exists x\phi \to \forall x\psi \right)\to \left(\phi \to \forall x\psi \right)$$
3	1 2	sylg	$$⊢\left(\exists x\phi \to \forall x\psi \right)\to \forall x\left(\phi \to \forall x\psi \right)$$