Theorem hbal

Description: If x is not free in ph , it is not free in A. y ph . (Contributed by NM, 12-Mar-1993)

	Ref	Expression
Hypothesis	hbal.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
Assertion	hbal	⊢ ( ∀ 𝑦 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		hbal.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2	1	alimi	⊢ ( ∀ 𝑦 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 )
3		ax-11	⊢ ( ∀ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜑 )
4	2 3	syl	⊢ ( ∀ 𝑦 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜑 )