Theorem nfrmow 3407

Description: Bound-variable hypothesis builder for restricted uniqueness. Version of nfrmo 3428 with a disjoint variable condition, which does not require ax-13 2366. (Contributed by NM, 16-Jun-2017.) Avoid ax-13 2366. (Revised by Gino Giotto, 10-Jan-2024.) Avoid ax-9 2108, ax-ext 2699. (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 3374	. 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 1894	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)
6	5	nfmov 2549	. 2 ⊢ Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)
7	1, 6	nfxfr 1847	1 ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 394  Ⅎwnf 1777   ∈ wcel 2098  ∃*wmo 2527  Ⅎwnfc 2879  ∃*wrmo 3373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-10 2129  ax-11 2146  ax-12 2166

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-mo 2529  df-clel 2806  df-nfc 2881  df-rmo 3374

This theorem is referenced by:  2rmorex  3751  2reurex  3757