Theorem nfreuw 3422

Description: Bound-variable hypothesis builder for restricted unique existence. Version of nfreu 3442 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by NM, 30-Oct-2010.) Avoid ax-13 2380. (Revised by GG, 10-Jan-2024.) Avoid ax-9 2118, ax-ext 2711. (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 3389	. 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜑 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
2		nfrmow.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2900	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfrmow.2	. . . 4 ⊢ Ⅎ𝑥𝜑
5	3, 4	nfan 1898	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)
6	5	nfeuw 2596	. 2 ⊢ Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)
7	1, 6	nfxfr 1851	1 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 395  Ⅎwnf 1781   ∈ wcel 2108  ∃!weu 2571  Ⅎwnfc 2893  ∃!wreu 3386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-10 2141  ax-11 2158  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-mo 2543  df-eu 2572  df-clel 2819  df-nfc 2895  df-reu 3389

This theorem is referenced by:  sbcreu  3898  reuccatpfxs1  14795  2reu7  47026  2reu8  47027