Theorem bj-hbalt

Description: Closed form of hbal . When in main part, prove hbal and hbald from it. (Contributed by BJ, 2-May-2019)

	Ref	Expression
Assertion	bj-hbalt	⊢ ( ∀ 𝑦 ( 𝜑 → ∀ 𝑥 𝜑 ) → ( ∀ 𝑦 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜑 ) )

Proof

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