Theorem nfrmo 2787

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 2623	. 2 ⊢ (∃*y ∈ A φ ↔ ∃*y(y ∈ A ∧ φ))
2		nftru 1554	. . . 4 ⊢ Ⅎy ⊤
3		nfcvf 2512	. . . . . . 7 ⊢ (¬ ∀x x = y → Ⅎxy)
4		nfreu.1	. . . . . . . 8 ⊢ ℲxA
5	4	a1i 10	. . . . . . 7 ⊢ (¬ ∀x x = y → ℲxA)
6	3, 5	nfeld 2505	. . . . . 6 ⊢ (¬ ∀x x = y → Ⅎx y ∈ A)
7		nfreu.2	. . . . . . 7 ⊢ Ⅎxφ
8	7	a1i 10	. . . . . 6 ⊢ (¬ ∀x x = y → Ⅎxφ)
9	6, 8	nfand 1822	. . . . 5 ⊢ (¬ ∀x x = y → Ⅎx(y ∈ A ∧ φ))
10	9	adantl 452	. . . 4 ⊢ (( ⊤ ∧ ¬ ∀x x = y) → Ⅎx(y ∈ A ∧ φ))
11	2, 10	nfmod2 2217	. . 3 ⊢ ( ⊤ → Ⅎx∃*y(y ∈ A ∧ φ))
12	11	trud 1323	. 2 ⊢ Ⅎx∃*y(y ∈ A ∧ φ)
13	1, 12	nfxfr 1570	1 ⊢ Ⅎx∃*y ∈ A φ

Syntax hints:  ¬ wn 3   ∧ wa 358   ⊤ wtru 1316  ∀wal 1540  Ⅎwnf 1544   ∈ wcel 1710  ∃*wmo 2205  Ⅎwnfc 2477  ∃*wrmo 2618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rmo 2623

This theorem is referenced by:  2rmorex  3041