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Theorem nfres 5944
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 5650 . 2 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
2 nfres.1 . . 3 𝑥𝐴
3 nfres.2 . . . 4 𝑥𝐵
4 nfcv 2908 . . . 4 𝑥V
53, 4nfxp 5671 . . 3 𝑥(𝐵 × V)
62, 5nfin 4181 . 2 𝑥(𝐴 ∩ (𝐵 × V))
71, 6nfcxfr 2906 1 𝑥(𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2888  Vcvv 3448  cin 3914   × cxp 5636  cres 5640
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-rab 3411  df-in 3922  df-opab 5173  df-xp 5644  df-res 5650
This theorem is referenced by:  nfima  6026  nffrecs  8219  nfwrecsOLD  8253  frsucmpt  8389  frsucmptn  8390  nfoi  9457  prdsdsf  23736  prdsxmet  23738  limciun  25274  nosupbnd2  27080  noinfbnd2  27095  2ndresdju  31607  gsumpart  31939  bnj1446  33697  bnj1447  33698  bnj1448  33699  bnj1466  33705  bnj1467  33706  bnj1519  33717  bnj1520  33718  bnj1529  33722  feqresmptf  43528  limcperiod  43943  xlimconst2  44150  cncfiooicclem1  44208  stoweidlem28  44343  nfdfat  45433  setrec2lem2  47213  setrec2  47214
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