Theorem hbalg

Description: Closed form of hbal . Derived from hbalgVD . (Contributed by Alan Sare, 8-Feb-2014) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	hbalg	⊢ ( ∀ 𝑦 ( 𝜑 → ∀ 𝑥 𝜑 ) → ∀ 𝑦 ( ∀ 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 ) )
4	3	axc4i	⊢ ( ∀ 𝑦 ( 𝜑 → ∀ 𝑥 𝜑 ) → ∀ 𝑦 ( ∀ 𝑦 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜑 ) )