Theorem nfrabw 2692

Description: A variable not free in a wff remains so in a restricted class abstraction. (Contributed by Jim Kingdon, 19-Jul-2018.)

Hypotheses

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

Assertion

Ref	Expression
nfrabw	⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfrabw

Step	Hyp	Ref	Expression
1		df-rab 2497	. 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
2		nfrabw.2	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2346	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfrabw.1	. . . 4 ⊢ Ⅎ𝑥𝜑
5	3, 4	nfan 1591	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)
6	5	nfab 2357	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
7	1, 6	nfcxfr 2349	1 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}

Syntax hints:   ∧ wa 104  Ⅎwnf 1486   ∈ wcel 2180  {cab 2195  Ⅎwnfc 2339  {crab 2492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-rab 2497

This theorem is referenced by:  nfdif  3305  nfin  3390  nfse  4409  elfvmptrab1  5702  elovmporab  6176  elovmporab1w  6177  mpoxopoveq  6356  nfsup  7127  caucvgprprlemaddq  7863  ctiunct  12977