Theorem nfreuw 3399

Description: Bound-variable hypothesis builder for restricted unique existence. Version of nfreu 3415 with a disjoint variable condition, which does not require ax-13 2404. (Contributed by NM, 30-Oct-2010.) Avoid ax-13 2404. (Revised by GG, 10-Jan-2024.) Avoid ax-9 2157, ax-ext 2735. (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 3370	. 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜑 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
2		nfrmow.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2918	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfrmow.2	. . . 4 ⊢ Ⅎ𝑥𝜑
5	3, 4	nfan 1934	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)
6	5	nfeuw 2621	. 2 ⊢ Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)
7	1, 6	nfxfr 1886	1 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑

This proof depends on syntax axioms:   ∧ wa 401  Ⅎwnf 1816   ∈ wcel 2147  ∃!weu 2596  Ⅎwnfc 2911  ∃!wreu 3367

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-10 2180  ax-11 2197  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-mo 2567  df-eu 2597  df-clel 2839  df-nfc 2913  df-reu 3370

This theorem is used by:  sbcreu  3827  reuccatpfxs1  14921  2reu7  48385  2reu8  48386  nfralseu  51209