Theorem hbal 2203

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 1843	. 2 ⊢ (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
3		ax-11 2193	. 2 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
4	2, 3	syl 18	1 ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1827  ax-4 1841  ax-11 2193

This theorem is used by:  nfal  2353  cbv3v  2364  cbv3  2426  hbral  3306  wl-nfalv  38594  nfalh  43403