Theorem hbal 1412

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 1390	. 2 ⊢ (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
3		ax-7 1383	. 2 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
4	2, 3	syl 14	1 ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)

Syntax hints:   → wi 4  ∀wal 1288

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-5 1382  ax-7 1383  ax-gen 1384

This theorem is referenced by:  hba2  1489  nfal  1514  aaanh  1524  hbex  1573  pm11.53  1824  euf  1954  hbral  2408