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Theorem nfres 4869
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 4599 . 2 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
2 nfres.1 . . 3 𝑥𝐴
3 nfres.2 . . . 4 𝑥𝐵
4 nfcv 2299 . . . 4 𝑥V
53, 4nfxp 4614 . . 3 𝑥(𝐵 × V)
62, 5nfin 3313 . 2 𝑥(𝐴 ∩ (𝐵 × V))
71, 6nfcxfr 2296 1 𝑥(𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wnfc 2286  Vcvv 2712  cin 3101   × cxp 4585  cres 4589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rab 2444  df-in 3108  df-opab 4027  df-xp 4593  df-res 4599
This theorem is referenced by:  nfima  4937  nffrec  6344
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