Theorem bj-nnflemaa

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

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

Proof

Step	Hyp	Ref	Expression
1		alim	$$⊢\forall x\left(\phi \to \forall y\psi \right)\to \left(\forall x\phi \to \forall x\forall y\psi \right)$$
2		ax-11	$$⊢\forall x\forall y\psi \to \forall y\forall x\psi$$
3	1 2	syl6	$$⊢\forall x\left(\phi \to \forall y\psi \right)\to \left(\forall x\phi \to \forall y\forall x\psi \right)$$