Theorem nfrabxy 2561

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 2379	. 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
2		nfrabxy.2	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2229	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfrabxy.1	. . . 4 ⊢ Ⅎ𝑥𝜑
5	3, 4	nfan 1509	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)
6	5	nfab 2240	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
7	1, 6	nfcxfr 2232	1 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}

Syntax hints:   ∧ wa 103  Ⅎwnf 1401   ∈ wcel 1445  {cab 2081  Ⅎwnfc 2222  {crab 2374

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rab 2379

This theorem is referenced by:  nfdif  3136  nfin  3221  nfse  4192  elfvmptrab1  5433  mpt2xopoveq  6043  nfsup  6767  caucvgprprlemaddq  7364