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Theorem nfres 4711
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  |-  F/_ x A
nfres.2  |-  F/_ x B
Assertion
Ref Expression
nfres  |-  F/_ x
( A  |`  B )

Proof of Theorem nfres
StepHypRef Expression
1 df-res 4448 . 2  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
2 nfres.1 . . 3  |-  F/_ x A
3 nfres.2 . . . 4  |-  F/_ x B
4 nfcv 2228 . . . 4  |-  F/_ x _V
53, 4nfxp 4462 . . 3  |-  F/_ x
( B  X.  _V )
62, 5nfin 3206 . 2  |-  F/_ x
( A  i^i  ( B  X.  _V ) )
71, 6nfcxfr 2225 1  |-  F/_ x
( A  |`  B )
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2215   _Vcvv 2619    i^i cin 2998    X. cxp 4434    |` cres 4438
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-in 3005  df-opab 3898  df-xp 4442  df-res 4448
This theorem is referenced by:  nfima  4777  nffrec  6153
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