Theorem nfreuw

Description: Bound-variable hypothesis builder for restricted unique existence. Version of nfreu with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 30-Oct-2010) (Revised by Gino Giotto, 10-Jan-2024) Avoid ax-9 , ax-ext . (Revised by Wolf Lammen, 21-Nov-2024)

	Ref	Expression
Hypotheses	nfreuw.1	⊢ Ⅎ 𝑥 𝐴
	nfreuw.2	⊢ Ⅎ 𝑥 𝜑
Assertion	nfreuw	⊢ Ⅎ 𝑥 ∃! 𝑦 ∈ 𝐴 𝜑

Proof

Step	Hyp	Ref	Expression
1		nfreuw.1	⊢ Ⅎ 𝑥 𝐴
2		nfreuw.2	⊢ Ⅎ 𝑥 𝜑
3		df-reu	⊢ ( ∃! 𝑦 ∈ 𝐴 𝜑 ↔ ∃! 𝑦 ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) )
4	1	nfcri	⊢ Ⅎ 𝑥 𝑦 ∈ 𝐴
5	4 2	nfan	⊢ Ⅎ 𝑥 ( 𝑦 ∈ 𝐴 ∧ 𝜑 )
6	5	nfeuw	⊢ Ⅎ 𝑥 ∃! 𝑦 ( 𝑦 ∈ 𝐴 ∧ 𝜑 )
7	3 6	nfxfr	⊢ Ⅎ 𝑥 ∃! 𝑦 ∈ 𝐴 𝜑