Theorem nfrmow 3375

Description: Bound-variable hypothesis builder for restricted uniqueness. Version of nfrmo 3393 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 16-Jun-2017.) Avoid ax-13 2372. (Revised by GG, 10-Jan-2024.) Avoid ax-9 2121, ax-ext 2703. (Revised by Wolf Lammen, 21-Nov-2024.)

Hypotheses

Ref	Expression
nfrmow.1	⊢ Ⅎ𝑥𝐴
nfrmow.2	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfrmow	⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfrmow

Step	Hyp	Ref	Expression
1		df-rmo 3346	. 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜑 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
2		nfrmow.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2886	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfrmow.2	. . . 4 ⊢ Ⅎ𝑥𝜑
5	3, 4	nfan 1900	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)
6	5	nfmov 2555	. 2 ⊢ Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)
7	1, 6	nfxfr 1854	1 ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 395  Ⅎwnf 1784   ∈ wcel 2111  ∃*wmo 2533  Ⅎwnfc 2879  ∃*wrmo 3345

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 2113  ax-10 2144  ax-11 2160  ax-12 2180

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 2535  df-clel 2806  df-nfc 2881  df-rmo 3346

This theorem is referenced by:  2rmorex  3713  2reurex  3719