Theorem nfreuw 3327

Description: Bound-variable hypothesis builder for restricted unique existence. Version of nfreu 3329 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 30-Oct-2010.) (Revised by Gino Giotto, 10-Jan-2024.)

Hypotheses

Ref	Expression
nfreuw.1	⊢ Ⅎ𝑥𝐴
nfreuw.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 3113	. . 3 ⊢ (∃!𝑦 ∈ 𝐴 𝜑 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
2		nftru 1806	. . . 4 ⊢ Ⅎ𝑦⊤
3		nfcvd 2956	. . . . . 6 ⊢ (⊤ → Ⅎ𝑥𝑦)
4		nfreuw.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
5	4	a1i 11	. . . . . 6 ⊢ (⊤ → Ⅎ𝑥𝐴)
6	3, 5	nfeld 2966	. . . . 5 ⊢ (⊤ → Ⅎ𝑥 𝑦 ∈ 𝐴)
7		nfreuw.2	. . . . . 6 ⊢ Ⅎ𝑥𝜑
8	7	a1i 11	. . . . 5 ⊢ (⊤ → Ⅎ𝑥𝜑)
9	6, 8	nfand 1898	. . . 4 ⊢ (⊤ → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
10	2, 9	nfeudw 2652	. . 3 ⊢ (⊤ → Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
11	1, 10	nfxfrd 1855	. 2 ⊢ (⊤ → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑)
12	11	mptru 1545	1 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 399  ⊤wtru 1539  Ⅎwnf 1785   ∈ wcel 2111  ∃!weu 2628  Ⅎwnfc 2936  ∃!wreu 3108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-mo 2598  df-eu 2629  df-cleq 2791  df-clel 2870  df-nfc 2938  df-reu 3113

This theorem is referenced by:  sbcreu  3805  reuccatpfxs1  14100  2reu7  43667  2reu8  43668