Theorem nfrabxy 2671

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
nfrabxy.1	⊢ Ⅎ𝑥𝜑
nfrabxy.2	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfrabxy	⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem nfrabxy

Step	Hyp	Ref	Expression
1		df-rab 2477	. 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
2		nfrabxy.2	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2326	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfrabxy.1	. . . 4 ⊢ Ⅎ𝑥𝜑
5	3, 4	nfan 1576	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)
6	5	nfab 2337	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
7	1, 6	nfcxfr 2329	1 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}

Syntax hints:   ∧ wa 104  Ⅎwnf 1471   ∈ wcel 2160  {cab 2175  Ⅎwnfc 2319  {crab 2472

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477

This theorem is referenced by:  nfdif  3271  nfin  3356  nfse  4359  elfvmptrab1  5631  mpoxopoveq  6265  nfsup  7021  caucvgprprlemaddq  7737  ctiunct  12491