Theorem nfreuw 3405

Description: Bound-variable hypothesis builder for restricted unique existence. Version of nfreu 3421 with a disjoint variable condition, which does not require ax-13 2410. (Contributed by NM, 30-Oct-2010.) Avoid ax-13 2410. (Revised by GG, 10-Jan-2024.) Avoid ax-9 2159, ax-ext 2741. (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 3376	. 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜑 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
2		nfrmow.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2923	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfrmow.2	. . . 4 ⊢ Ⅎ𝑥𝜑
5	3, 4	nfan 1926	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)
6	5	nfeuw 2627	. 2 ⊢ Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)
7	1, 6	nfxfr 1880	1 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 400  Ⅎwnf 1810   ∈ wcel 2149  ∃!weu 2602  Ⅎwnfc 2916  ∃!wreu 3373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-10 2182  ax-11 2198  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-mo 2573  df-eu 2603  df-clel 2844  df-nfc 2918  df-reu 3376

This theorem is referenced by:  sbcreu  3836  reuccatpfxs1  14784  2reu7  47772  2reu8  47773