Theorem nfreuw 3305

Description: Bound-variable hypothesis builder for restricted unique existence. Version of nfreu 3308 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 30-Oct-2010.) (Revised by Gino Giotto, 10-Jan-2024.) Avoid ax-9 2116, ax-ext 2709. (Revised by Wolf Lammen, 21-Nov-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 3072	. 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜑 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
2		nfreuw.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2894	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfreuw.2	. . . 4 ⊢ Ⅎ𝑥𝜑
5	3, 4	nfan 1902	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)
6	5	nfeuw 2593	. 2 ⊢ Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)
7	1, 6	nfxfr 1855	1 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 396  Ⅎwnf 1786   ∈ wcel 2106  ∃!weu 2568  Ⅎwnfc 2887  ∃!wreu 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-10 2137  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-mo 2540  df-eu 2569  df-clel 2816  df-nfc 2889  df-reu 3072

This theorem is referenced by:  sbcreu  3809  reuccatpfxs1  14460  2reu7  44603  2reu8  44604