Theorem nfrmo 3380

Description: Bound-variable hypothesis builder for restricted uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2389. Use the weaker nfrmow 3378 when possible. (Contributed by NM, 16-Jun-2017.) (New usage is discouraged.)

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 3149	. 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜑 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
2		nftru 1804	. . . 4 ⊢ Ⅎ𝑦⊤
3		nfcvf 3010	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
4		nfreu.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
5	4	a1i 11	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐴)
6	3, 5	nfeld 2992	. . . . . 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 1897	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
10	9	adantl 484	. . . 4 ⊢ ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
11	2, 10	nfmod2 2641	. . 3 ⊢ (⊤ → Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
12	11	mptru 1543	. 2 ⊢ Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)
13	1, 12	nfxfr 1852	1 ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑

Syntax hints:  ¬ wn 3   ∧ wa 398  ∀wal 1534  ⊤wtru 1537  Ⅎwnf 1783   ∈ wcel 2113  ∃*wmo 2619  Ⅎwnfc 2964  ∃*wrmo 3144

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-13 2389  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-mo 2621  df-cleq 2817  df-clel 2896  df-nfc 2966  df-rmo 3149

This theorem is referenced by: (None)