Theorem nfrabxy 2650

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 2457	. 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
2		nfrabxy.2	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2306	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfrabxy.1	. . . 4 ⊢ Ⅎ𝑥𝜑
5	3, 4	nfan 1558	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)
6	5	nfab 2317	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
7	1, 6	nfcxfr 2309	1 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}

Syntax hints:   ∧ wa 103  Ⅎwnf 1453   ∈ wcel 2141  {cab 2156  Ⅎwnfc 2299  {crab 2452

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457

This theorem is referenced by:  nfdif  3248  nfin  3333  nfse  4326  elfvmptrab1  5590  mpoxopoveq  6219  nfsup  6969  caucvgprprlemaddq  7670  ctiunct  12395