Theorem nfrabw 2727

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 2531	. 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
2		nfrabw.2	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2380	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfrabw.1	. . . 4 ⊢ Ⅎ𝑥𝜑
5	3, 4	nfan 1614	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)
6	5	nfab 2391	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
7	1, 6	nfcxfr 2383	1 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}

Syntax hints:   ∧ wa 104  Ⅎwnf 1509   ∈ wcel 2205  {cab 2220  Ⅎwnfc 2373  {crab 2526

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rab 2531

This theorem is referenced by:  nfdif  3342  nfin  3429  nfse  4464  elfvmptrab1  5774  elovmporab  6256  elovmporab1w  6257  mpoxopoveq  6473  nfsup  7285  caucvgprprlemaddq  8028  ctiunct  13212