Theorem nfrmow 3301

Description: Bound-variable hypothesis builder for restricted uniqueness. Version of nfrmo 3303 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 16-Jun-2017.) (Revised by Gino Giotto, 10-Jan-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 3071	. 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜑 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
2		nftru 1808	. . . 4 ⊢ Ⅎ𝑦⊤
3		nfcvd 2907	. . . . . 6 ⊢ (⊤ → Ⅎ𝑥𝑦)
4		nfreuw.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
5	4	a1i 11	. . . . . 6 ⊢ (⊤ → Ⅎ𝑥𝐴)
6	3, 5	nfeld 2917	. . . . 5 ⊢ (⊤ → Ⅎ𝑥 𝑦 ∈ 𝐴)
7		nfreuw.2	. . . . . 6 ⊢ Ⅎ𝑥𝜑
8	7	a1i 11	. . . . 5 ⊢ (⊤ → Ⅎ𝑥𝜑)
9	6, 8	nfand 1901	. . . 4 ⊢ (⊤ → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
10	2, 9	nfmodv 2559	. . 3 ⊢ (⊤ → Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
11	10	mptru 1546	. 2 ⊢ Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)
12	1, 11	nfxfr 1856	1 ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 395  ⊤wtru 1540  Ⅎwnf 1787   ∈ wcel 2108  ∃*wmo 2538  Ⅎwnfc 2886  ∃*wrmo 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-mo 2540  df-cleq 2730  df-clel 2817  df-nfc 2888  df-rmo 3071

This theorem is referenced by:  2rmorex  3684  2reurex  3690