Theorem nfreuw 3378

Description: Bound-variable hypothesis builder for restricted unique existence. Version of nfreu 3396 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by NM, 30-Oct-2010.) Avoid ax-13 2374. (Revised by GG, 10-Jan-2024.) Avoid ax-9 2123, ax-ext 2706. (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 3349	. 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜑 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
2		nfrmow.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2888	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfrmow.2	. . . 4 ⊢ Ⅎ𝑥𝜑
5	3, 4	nfan 1900	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)
6	5	nfeuw 2591	. 2 ⊢ Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)
7	1, 6	nfxfr 1854	1 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 395  Ⅎwnf 1784   ∈ wcel 2113  ∃!weu 2566  Ⅎwnfc 2881  ∃!wreu 3346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-10 2146  ax-11 2162  ax-12 2182

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-mo 2537  df-eu 2567  df-clel 2809  df-nfc 2883  df-reu 3349

This theorem is referenced by:  sbcreu  3824  reuccatpfxs1  14668  2reu7  47299  2reu8  47300