Theorem nfexhe 2211

Description: Version of nfex 2355 with the existential dual to the 'h' hypothesis, avoiding ax-12 2213. (Contributed by SN, 11-Feb-2026.)

Hypothesis

Ref	Expression
nfexhe.1	⊢ (∃𝑥𝜑 → 𝜑)

Assertion

Ref	Expression
nfexhe	⊢ Ⅎ𝑥∃𝑦𝜑

Proof of Theorem nfexhe

Step	Hyp	Ref	Expression
1		hbe1 2180	. . 3 ⊢ (∃𝑥∃𝑦𝜑 → ∀𝑥∃𝑥∃𝑦𝜑)
2		excomim 2200	. . . 4 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)
3		nfexhe.1	. . . . 5 ⊢ (∃𝑥𝜑 → 𝜑)
4	3	eximi 1868	. . . 4 ⊢ (∃𝑦∃𝑥𝜑 → ∃𝑦𝜑)
5	2, 4	syl 18	. . 3 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦𝜑)
6	1, 5	alrimih 1857	. 2 ⊢ (∃𝑥∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)
7	6	nfi 1821	1 ⊢ Ⅎ𝑥∃𝑦𝜑

This proof depends on syntax axioms:   → wi 4  ∃wex 1812  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 2178  ax-11 2194

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

This theorem is used by:  nfexa2  2212