Theorem nfrab 3471

Description: A variable not free in a wff remains so in a restricted class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2366. Use the weaker nfrabw 3467 when possible. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 9-Oct-2016.) (New usage is discouraged.)

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 3431	. 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
2		nftru 1798	. . . 4 ⊢ Ⅎ𝑦⊤
3		nfrab.2	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
4	3	nfcri 2886	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
5		eleq1w 2812	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
6	4, 5	dvelimnf 2447	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 ∈ 𝐴)
7		nfrab.1	. . . . . . 7 ⊢ Ⅎ𝑥𝜑
8	7	a1i 11	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)
9	6, 8	nfand 1892	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
10	9	adantl 480	. . . 4 ⊢ ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
11	2, 10	nfabd2 2926	. . 3 ⊢ (⊤ → Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)})
12	11	mptru 1540	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
13	1, 12	nfcxfr 2897	1 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}

Syntax hints:  ¬ wn 3   ∧ wa 394  ∀wal 1531  ⊤wtru 1534  Ⅎwnf 1777   ∈ wcel 2098  {cab 2705  Ⅎwnfc 2879  {crab 3430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-13 2366  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-rab 3431

This theorem is referenced by:  elfvmptrab1  7038  elovmporab1  7675