Theorem hbal 1457

Description: If 𝑥 is not free in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
hbal.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
hbal	⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)

Proof of Theorem hbal

Step	Hyp	Ref	Expression
1		hbal.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2	1	alimi 1435	. 2 ⊢ (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
3		ax-7 1428	. 2 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
4	2, 3	syl 14	1 ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)

Syntax hints:   → wi 4  ∀wal 1333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-5 1427  ax-7 1428  ax-gen 1429

This theorem is referenced by:  hba2  1531  aaanh  1566  hbex  1616  pm11.53  1875  euf  2011  hbral  2486