Theorem nfalh 43024

Description: Version of nfal 2359 with an 'h' hypothesis, avoiding ax-12 2216. (Contributed by SN, 11-Feb-2026.)

Hypothesis

Ref	Expression
nfalh.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
nfalh	⊢ Ⅎ𝑥∀𝑦𝜑

Proof of Theorem nfalh

Step	Hyp	Ref	Expression
1		nfalh.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2	1	hbal 2205	. 2 ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)
3	2	nf5i 2184	1 ⊢ Ⅎ𝑥∀𝑦𝜑

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-10 2179  ax-11 2195

This proof depends on definitions:  df-bi 210  df-ex 1813  df-nf 1817

This theorem is used by:  nfale2  43026