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Theorem nfres 4642
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 4383 . 2 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
2 nfres.1 . . 3 𝑥𝐴
3 nfres.2 . . . 4 𝑥𝐵
4 nfcv 2220 . . . 4 𝑥V
53, 4nfxp 4397 . . 3 𝑥(𝐵 × V)
62, 5nfin 3179 . 2 𝑥(𝐴 ∩ (𝐵 × V))
71, 6nfcxfr 2217 1 𝑥(𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wnfc 2207  Vcvv 2602  cin 2973   × cxp 4369  cres 4373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rab 2358  df-in 2980  df-opab 3848  df-xp 4377  df-res 4383
This theorem is referenced by:  nfima  4706  nffrec  6045
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