Theorem nfrmo 3115

Description: Bound-variable hypothesis builder for restricted uniqueness. (Contributed by NM, 16-Jun-2017.)

Hypotheses

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

Assertion

Ref	Expression
nfrmo	⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑

Proof of Theorem nfrmo

Step	Hyp	Ref	Expression
1		df-rmo 2920	. 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜑 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
2		nftru 1730	. . . 4 ⊢ Ⅎ𝑦⊤
3		nfcvf 2788	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
4		nfreu.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
5	4	a1i 11	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐴)
6	3, 5	nfeld 2773	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 ∈ 𝐴)
7		nfreu.2	. . . . . . 7 ⊢ Ⅎ𝑥𝜑
8	7	a1i 11	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)
9	6, 8	nfand 1826	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
10	9	adantl 482	. . . 4 ⊢ ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
11	2, 10	nfmod2 2483	. . 3 ⊢ (⊤ → Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
12	11	trud 1493	. 2 ⊢ Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)
13	1, 12	nfxfr 1779	1 ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑

Syntax hints:  ¬ wn 3   ∧ wa 384  ∀wal 1481  ⊤wtru 1484  Ⅎwnf 1708   ∈ wcel 1990  ∃*wmo 2471  Ⅎwnfc 2751  ∃*wrmo 2915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rmo 2920

This theorem is referenced by:  2rmorex  3412  2reurex  41181