Theorem nfreuw 3398

Description: Bound-variable hypothesis builder for restricted unique existence. Version of nfreu 3414 with a disjoint variable condition, which does not require ax-13 2403. (Contributed by NM, 30-Oct-2010.) Avoid ax-13 2403. (Revised by GG, 10-Jan-2024.) Avoid ax-9 2152, ax-ext 2734. (Revised by Wolf Lammen, 21-Nov-2024.)

Hypotheses

Ref	Expression
nfrmow.1	⊢ Ⅎ𝑥𝐴
nfrmow.2	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfreuw	⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐴(𝑥, 𝑦)

Proof of Theorem nfreuw

Step	Hyp	Ref	Expression
1		df-reu 3369	. 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜑 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
2		nfrmow.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2916	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfrmow.2	. . . 4 ⊢ Ⅎ𝑥𝜑
5	3, 4	nfan 1928	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)
6	5	nfeuw 2620	. 2 ⊢ Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)
7	1, 6	nfxfr 1882	1 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑

This proof depends on syntax axioms:   ∧ wa 400  Ⅎwnf 1812   ∈ wcel 2142  ∃!weu 2595  Ⅎwnfc 2909  ∃!wreu 3366

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-10 2175  ax-11 2191  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-ex 1809  df-nf 1813  df-mo 2566  df-eu 2596  df-clel 2837  df-nfc 2911  df-reu 3369

This theorem is used by:  sbcreu  3828  reuccatpfxs1  14791  2reu7  47876  2reu8  47877  nfralseu  50641