MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfrabw Structured version   Visualization version   GIF version

Theorem nfrabw 3376
Description: A variable not free in a wff remains so in a restricted class abstraction. Version of nfrab 3377 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by NM, 13-Oct-2003.) (Revised by Gino Giotto, 10-Jan-2024.)
Hypotheses
Ref Expression
nfrabw.1 𝑥𝜑
nfrabw.2 𝑥𝐴
Assertion
Ref Expression
nfrabw 𝑥{𝑦𝐴𝜑}
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfrabw
StepHypRef Expression
1 df-rab 3141 . 2 {𝑦𝐴𝜑} = {𝑦 ∣ (𝑦𝐴𝜑)}
2 nftru 1806 . . . 4 𝑦
3 nfrabw.2 . . . . . . 7 𝑥𝐴
43nfcri 2971 . . . . . 6 𝑥 𝑦𝐴
54a1i 11 . . . . 5 (⊤ → Ⅎ𝑥 𝑦𝐴)
6 nfrabw.1 . . . . . 6 𝑥𝜑
76a1i 11 . . . . 5 (⊤ → Ⅎ𝑥𝜑)
85, 7nfand 1899 . . . 4 (⊤ → Ⅎ𝑥(𝑦𝐴𝜑))
92, 8nfabdw 3001 . . 3 (⊤ → 𝑥{𝑦 ∣ (𝑦𝐴𝜑)})
109mptru 1545 . 2 𝑥{𝑦 ∣ (𝑦𝐴𝜑)}
111, 10nfcxfr 2978 1 𝑥{𝑦𝐴𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 399  wtru 1539  wnf 1785  wcel 2115  {cab 2802  wnfc 2962  {crab 3136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3141
This theorem is referenced by:  nfdif  4086  nfin  4176  nfse  5513  elfvmptrab1w  6777  elovmporab  7376  elovmporab1w  7377  ovmpt3rab1  7388  elovmpt3rab1  7390  mpoxopoveq  7870  nfoi  8964  scottex  9300  elmptrab  22423  iundisjf  30338  nnindf  30534  fedgmullem2  31049  bnj1398  32326  bnj1445  32336  bnj1449  32340  nfwlim  33129  finminlem  33686  poimirlem26  34988  poimirlem27  34989  indexa  35076  nfscott  40798  binomcxplemdvbinom  40908  binomcxplemdvsum  40910  binomcxplemnotnn0  40911  infnsuprnmpt  41744  allbutfiinf  41914  supminfrnmpt  41939  supminfxrrnmpt  41967  fnlimfvre  42173  fnlimabslt  42178  dvnprodlem1  42445  stoweidlem16  42515  stoweidlem31  42530  stoweidlem34  42533  stoweidlem35  42534  stoweidlem48  42547  stoweidlem51  42550  stoweidlem53  42552  stoweidlem54  42553  stoweidlem57  42556  stoweidlem59  42558  fourierdlem31  42637  fourierdlem48  42653  fourierdlem51  42656  etransclem32  42765  ovncvrrp  43060  smflim  43267  smflimmpt  43298  smfsupmpt  43303  smfsupxr  43304  smfinfmpt  43307  smflimsuplem7  43314
  Copyright terms: Public domain W3C validator