Theorem nfexhe 2209

Description: Version of nfex 2355 with the existential dual to the 'h' hypothesis, avoiding ax-12 2211. (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 2176	. . 3 ⊢ (∃𝑥∃𝑦𝜑 → ∀𝑥∃𝑥∃𝑦𝜑)
2		excomim 2196	. . . 4 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)
3		nfexhe.1	. . . . 5 ⊢ (∃𝑥𝜑 → 𝜑)
4	3	eximi 1854	. . . 4 ⊢ (∃𝑦∃𝑥𝜑 → ∃𝑦𝜑)
5	2, 4	syl 17	. . 3 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦𝜑)
6	1, 5	alrimih 1843	. 2 ⊢ (∃𝑥∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)
7	6	nfi 1807	1 ⊢ Ⅎ𝑥∃𝑦𝜑

Syntax hints:   → wi 4  ∃wex 1798  Ⅎwnf 1802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-10 2174  ax-11 2190

This theorem depends on definitions:  df-bi 209  df-ex 1799  df-nf 1803

This theorem is referenced by:  nfexa2  2210