Theorem nfexhe 2210

Description: Version of nfex 2354 with the existential dual to the 'h' hypothesis, avoiding ax-12 2212. (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 2179	. . 3 ⊢ (∃𝑥∃𝑦𝜑 → ∀𝑥∃𝑥∃𝑦𝜑)
2		excomim 2199	. . . 4 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)
3		nfexhe.1	. . . . 5 ⊢ (∃𝑥𝜑 → 𝜑)
4	3	eximi 1867	. . . 4 ⊢ (∃𝑦∃𝑥𝜑 → ∃𝑦𝜑)
5	2, 4	syl 18	. . 3 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦𝜑)
6	1, 5	alrimih 1856	. 2 ⊢ (∃𝑥∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)
7	6	nfi 1820	1 ⊢ Ⅎ𝑥∃𝑦𝜑

This proof depends on syntax axioms:   → wi 4  ∃wex 1811  Ⅎwnf 1815

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-10 2177  ax-11 2193

This proof depends on definitions:  df-bi 210  df-ex 1812  df-nf 1816

This theorem is used by:  nfexa2  2211