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	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		alim	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜑 ) )
2		ax-11	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 )
3	1 2	syl6	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) )