Theorem nfrab 3312

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 2372. Use the weaker nfrabw 3311 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 3072	. 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
2		nftru 1808	. . . 4 ⊢ Ⅎ𝑦⊤
3		nfrab.2	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
4	3	nfcri 2893	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
5		eleq1w 2821	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
6	4, 5	dvelimnf 2453	. . . . . 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 1901	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
10	9	adantl 481	. . . 4 ⊢ ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
11	2, 10	nfabd2 2932	. . 3 ⊢ (⊤ → Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)})
12	11	mptru 1546	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
13	1, 12	nfcxfr 2904	1 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}

Syntax hints:  ¬ wn 3   ∧ wa 395  ∀wal 1537  ⊤wtru 1540  Ⅎwnf 1787   ∈ wcel 2108  {cab 2715  Ⅎwnfc 2886  {crab 3067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-13 2372  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-rab 3072

This theorem is referenced by:  elfvmptrab1  6884  elovmporab1  7495