Theorem bj-nnflemae

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

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

Proof

Step	Hyp	Ref	Expression
1		exim	$$⊢\forall x\left(\phi \to \forall y\phi \right)\to \left(\exists x\phi \to \exists x\forall y\phi \right)$$
2		bj-19.12	$$⊢\exists x\forall y\phi \to \forall y\exists x\phi$$
3	1 2	syl6	$$⊢\forall x\left(\phi \to \forall y\phi \right)\to \left(\exists x\phi \to \forall y\exists x\phi \right)$$