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Theorem nfres 4921
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 4650 . 2 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
2 nfres.1 . . 3 𝑥𝐴
3 nfres.2 . . . 4 𝑥𝐵
4 nfcv 2329 . . . 4 𝑥V
53, 4nfxp 4665 . . 3 𝑥(𝐵 × V)
62, 5nfin 3353 . 2 𝑥(𝐴 ∩ (𝐵 × V))
71, 6nfcxfr 2326 1 𝑥(𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wnfc 2316  Vcvv 2749  cin 3140   × cxp 4636  cres 4640
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rab 2474  df-in 3147  df-opab 4077  df-xp 4644  df-res 4650
This theorem is referenced by:  nfima  4990  nffrec  6411
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