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)

	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		nftru	⊢ Ⅎ 𝑦 ⊤
5		nfcvd	⊢ ( ⊤ → Ⅎ 𝑥 𝑦 )
6	1	a1i	⊢ ( ⊤ → Ⅎ 𝑥 𝐴 )
7	5 6	nfeld	⊢ ( ⊤ → Ⅎ 𝑥 𝑦 ∈ 𝐴 )
8	2	a1i	⊢ ( ⊤ → Ⅎ 𝑥 𝜑 )
9	7 8	nfand	⊢ ( ⊤ → Ⅎ 𝑥 ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) )
10	4 9	nfeudw	⊢ ( ⊤ → Ⅎ 𝑥 ∃! 𝑦 ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) )
11	3 10	nfxfrd	⊢ ( ⊤ → Ⅎ 𝑥 ∃! 𝑦 ∈ 𝐴 𝜑 )
12	11	mptru	⊢ Ⅎ 𝑥 ∃! 𝑦 ∈ 𝐴 𝜑