ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfres GIF version

Theorem nfres 4927
Description: Bound-variable hypothesis builder for restriction. (Contributed by NM, 15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.)
Hypotheses
Ref Expression
nfres.1 𝑥𝐴
nfres.2 𝑥𝐵
Assertion
Ref Expression
nfres 𝑥(𝐴𝐵)

Proof of Theorem nfres
StepHypRef Expression
1 df-res 4656 . 2 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
2 nfres.1 . . 3 𝑥𝐴
3 nfres.2 . . . 4 𝑥𝐵
4 nfcv 2332 . . . 4 𝑥V
53, 4nfxp 4671 . . 3 𝑥(𝐵 × V)
62, 5nfin 3356 . 2 𝑥(𝐴 ∩ (𝐵 × V))
71, 6nfcxfr 2329 1 𝑥(𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wnfc 2319  Vcvv 2752  cin 3143   × cxp 4642  cres 4646
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-in 3150  df-opab 4080  df-xp 4650  df-res 4656
This theorem is referenced by:  nfima  4996  nffrec  6421
  Copyright terms: Public domain W3C validator