Theorem nfrabxy 2490

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 2315	. 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
2		nfrabxy.2	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2172	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfrabxy.1	. . . 4 ⊢ Ⅎ𝑥𝜑
5	3, 4	nfan 1457	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)
6	5	nfab 2182	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
7	1, 6	nfcxfr 2175	1 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}

Syntax hints:   ∧ wa 97  Ⅎwnf 1349   ∈ wcel 1393  {cab 2026  Ⅎwnfc 2165  {crab 2310

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315

This theorem is referenced by:  nfdif  3065  nfin  3143  nfse  4078  mpt2xopoveq  5855  caucvgprprlemaddq  6806