Theorem nfrexw

Description: Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999) (Revised by Mario Carneiro, 7-Oct-2016) (Proof shortened by Wolf Lammen, 30-Dec-2019) Add disjoint variable condition to avoid ax-13 . See nfrex for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024)

	Ref	Expression
Hypotheses	nfralw.1	⊢ Ⅎ 𝑥 𝐴
	nfralw.2	⊢ Ⅎ 𝑥 𝜑
Assertion	nfrexw	⊢ Ⅎ 𝑥 ∃ 𝑦 ∈ 𝐴 𝜑

Proof

Step	Hyp	Ref	Expression
1		nfralw.1	⊢ Ⅎ 𝑥 𝐴
2		nfralw.2	⊢ Ⅎ 𝑥 𝜑
3		nftru	⊢ Ⅎ 𝑦 ⊤
4	1	a1i	⊢ ( ⊤ → Ⅎ 𝑥 𝐴 )
5	2	a1i	⊢ ( ⊤ → Ⅎ 𝑥 𝜑 )
6	3 4 5	nfrexdw	⊢ ( ⊤ → Ⅎ 𝑥 ∃ 𝑦 ∈ 𝐴 𝜑 )
7	6	mptru	⊢ Ⅎ 𝑥 ∃ 𝑦 ∈ 𝐴 𝜑