Theorem bj-nnflemea

Description: One of four lemmas for nonfreeness: antecedent expressed with existential quantifier and consequent expressed with universal quantifier. (Contributed by BJ, 12-Aug-2023) (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		bj-19.12	$$⊢\exists y\forall x\phi \to \forall x\exists y\phi$$
2		alim	$$⊢\forall x\left(\exists y\phi \to \phi \right)\to \left(\forall x\exists y\phi \to \forall x\phi \right)$$
3	1 2	syl5	$$⊢\forall x\left(\exists y\phi \to \phi \right)\to \left(\exists y\forall x\phi \to \forall x\phi \right)$$