Theorem nfrmow 3401

Description: Bound-variable hypothesis builder for restricted uniqueness. Version of nfrmo 3417 with a disjoint variable condition, which does not require ax-13 2407. (Contributed by NM, 16-Jun-2017.) Avoid ax-13 2407. (Revised by GG, 10-Jan-2024.) Avoid ax-9 2156, ax-ext 2738. (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 3372	. 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜑 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
2		nfrmow.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2920	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfrmow.2	. . . 4 ⊢ Ⅎ𝑥𝜑
5	3, 4	nfan 1932	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)
6	5	nfmov 2591	. 2 ⊢ Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)
7	1, 6	nfxfr 1886	1 ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑

This proof depends on syntax axioms:   ∧ wa 401  Ⅎwnf 1816   ∈ wcel 2146  ∃*wmo 2568  Ⅎwnfc 2913  ∃*wrmo 3371

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-10 2179  ax-11 2195  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-mo 2570  df-clel 2841  df-nfc 2915  df-rmo 3372

This theorem is used by:  2rmorex  3720  2reurex  3726