Theorem nfalh 42706

Description: Version of nfal 2332 with an 'h' hypothesis, avoiding ax-12 2189. (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 2178	. 2 ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)
3	2	nf5i 2157	1 ⊢ Ⅎ𝑥∀𝑦𝜑

Syntax hints:   → wi 4  ∀wal 1545  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-10 2152  ax-11 2168

This theorem depends on definitions:  df-bi 208  df-ex 1787  df-nf 1791

This theorem is referenced by:  nfale2  42708