Theorem nfrmow 3303

Description: Bound-variable hypothesis builder for restricted uniqueness. Version of nfrmo 3308 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by NM, 16-Jun-2017.) (Revised by Gino Giotto, 10-Jan-2024.) Avoid ax-9 2120, ax-ext 2711. (Revised by Wolf Lammen, 21-Nov-2024.)

Hypotheses

Ref	Expression
nfreuw.1	⊢ Ⅎ𝑥𝐴
nfreuw.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 3074	. 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜑 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
2		nfreuw.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2896	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfreuw.2	. . . 4 ⊢ Ⅎ𝑥𝜑
5	3, 4	nfan 1906	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)
6	5	nfmov 2562	. 2 ⊢ Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)
7	1, 6	nfxfr 1859	1 ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 396  Ⅎwnf 1790   ∈ wcel 2110  ∃*wmo 2540  Ⅎwnfc 2889  ∃*wrmo 3069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-10 2141  ax-11 2158  ax-12 2175

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1545  df-ex 1787  df-nf 1791  df-mo 2542  df-clel 2818  df-nfc 2891  df-rmo 3074

This theorem is referenced by:  2rmorex  3693  2reurex  3699