Theorem nfrabxy 2644

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 2451	. 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
2		nfrabxy.2	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2300	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfrabxy.1	. . . 4 ⊢ Ⅎ𝑥𝜑
5	3, 4	nfan 1552	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)
6	5	nfab 2311	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
7	1, 6	nfcxfr 2303	1 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}

Syntax hints:   ∧ wa 103  Ⅎwnf 1447   ∈ wcel 2135  {cab 2150  Ⅎwnfc 2293  {crab 2446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rab 2451

This theorem is referenced by:  nfdif  3238  nfin  3323  nfse  4313  elfvmptrab1  5574  mpoxopoveq  6199  nfsup  6948  caucvgprprlemaddq  7640  ctiunct  12310