Theorem bj-nnflemee

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

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

Proof

Step	Hyp	Ref	Expression
1		excom	$$⊢\exists y\exists x\phi \leftrightarrow \exists x\exists y\phi$$
2		exim	$$⊢\forall x\left(\exists y\phi \to \phi \right)\to \left(\exists x\exists y\phi \to \exists x\phi \right)$$
3	1 2	syl5bi	$$⊢\forall x\left(\exists y\phi \to \phi \right)\to \left(\exists y\exists x\phi \to \exists x\phi \right)$$