Theorem nfrab 2793

Description: A variable not free in a wff remains so in a restricted class abstraction. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 9-Oct-2016.)

Hypotheses

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

Assertion

Ref	Expression
nfrab	⊢ Ⅎx{y ∈ A ∣ φ}

Proof of Theorem nfrab

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rab 2624	. 2 ⊢ {y ∈ A ∣ φ} = {y ∣ (y ∈ A ∧ φ)}
2		nftru 1554	. . . 4 ⊢ Ⅎy ⊤
3		nfrab.2	. . . . . . . 8 ⊢ ℲxA
4	3	nfcri 2484	. . . . . . 7 ⊢ Ⅎx z ∈ A
5		eleq1 2413	. . . . . . 7 ⊢ (z = y → (z ∈ A ↔ y ∈ A))
6	4, 5	dvelimnf 2017	. . . . . 6 ⊢ (¬ ∀x x = y → Ⅎx y ∈ A)
7		nfrab.1	. . . . . . 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	nfabd2 2508	. . 3 ⊢ ( ⊤ → Ⅎx{y ∣ (y ∈ A ∧ φ)})
12	11	trud 1323	. 2 ⊢ Ⅎx{y ∣ (y ∈ A ∧ φ)}
13	1, 12	nfcxfr 2487	1 ⊢ Ⅎx{y ∈ A ∣ φ}

Syntax hints:  ¬ wn 3   ∧ wa 358   ⊤ wtru 1316  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2477  {crab 2619

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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rab 2624

This theorem is referenced by: (None)