Theorem hbal 2205

Description: If 𝑥 is not free in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by NM, 12-Mar-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 1844	. 2 ⊢ (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
3		ax-11 2195	. 2 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
4	2, 3	syl 18	1 ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1828  ax-4 1842  ax-11 2195

This theorem is used by:  nfal  2358  cbv3v  2369  cbv3  2431  hbral  3311  wl-nfalv  38239  nfalh  43043