Theorem nfrabw 3318

Description: A variable not free in a wff remains so in a restricted class abstraction. Version of nfrab 3320 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 13-Oct-2003.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 23-Nov-2024.)

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 3073	. 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
2		nftru 1807	. . . 4 ⊢ Ⅎ𝑦⊤
3		nfrabw.2	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
4	3	nfcri 2894	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
5		nfrabw.1	. . . . . 6 ⊢ Ⅎ𝑥𝜑
6	4, 5	nfan 1902	. . . . 5 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)
7	6	a1i 11	. . . 4 ⊢ (⊤ → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
8	2, 7	nfabdw 2930	. . 3 ⊢ (⊤ → Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)})
9	8	mptru 1546	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
10	1, 9	nfcxfr 2905	1 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}

Syntax hints:   ∧ wa 396  ⊤wtru 1540  Ⅎwnf 1786   ∈ wcel 2106  {cab 2715  Ⅎwnfc 2887  {crab 3068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rab 3073

This theorem is referenced by:  nfdif  4060  nfin  4150  nfse  5564  elfvmptrab1w  6901  elovmporab  7515  elovmporab1w  7516  ovmpt3rab1  7527  elovmpt3rab1  7529  mpoxopoveq  8035  nfoi  9273  scottex  9643  elmptrab  22978  iundisjf  30928  nnindf  31133  fedgmullem2  31711  bnj1398  33014  bnj1445  33024  bnj1449  33028  nfwlim  33816  finminlem  34507  poimirlem26  35803  poimirlem27  35804  indexa  35891  nfscott  41857  binomcxplemdvbinom  41971  binomcxplemdvsum  41973  binomcxplemnotnn0  41974  infnsuprnmpt  42796  allbutfiinf  42960  supminfrnmpt  42985  supminfxrrnmpt  43011  fnlimfvre  43215  fnlimabslt  43220  dvnprodlem1  43487  stoweidlem16  43557  stoweidlem31  43572  stoweidlem34  43575  stoweidlem35  43576  stoweidlem48  43589  stoweidlem51  43592  stoweidlem53  43594  stoweidlem54  43595  stoweidlem57  43598  stoweidlem59  43600  fourierdlem31  43679  fourierdlem48  43695  fourierdlem51  43698  etransclem32  43807  ovncvrrp  44102  smflim  44312  smflimmpt  44343  smfsupmpt  44348  smfsupxr  44349  smfinfmpt  44352  smflimsuplem7  44359