Theorem nfrab1 2685

Description: The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)

Assertion

Ref	Expression
nfrab1	⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑}

Proof of Theorem nfrab1

Step	Hyp	Ref	Expression
1		df-rab 2492	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2		nfab1 2349	. 2 ⊢ Ⅎ𝑥{𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
3	1, 2	nfcxfr 2344	1 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑}

Syntax hints:   ∧ wa 104   ∈ wcel 2175  {cab 2190  Ⅎwnfc 2334  {crab 2487

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-5 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rab 2492

This theorem is referenced by:  repizf2  4205  rabxfrd  4514  onintrab2im  4564  tfis  4629  fvmptssdm  5658  infssuzcldc  10359  nnwosdc  12279  imasnopn  14689