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Theorem nfres 4614
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 4357 . 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 2178 . . . 4  |-  F/_ x _V
53, 4nfxp 4371 . . 3  |-  F/_ x
( B  X.  _V )
62, 5nfin 3143 . 2  |-  F/_ x
( A  i^i  ( B  X.  _V ) )
71, 6nfcxfr 2175 1  |-  F/_ x
( A  |`  B )
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2165   _Vcvv 2557    i^i cin 2916    X. cxp 4343    |` cres 4347
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-in 2924  df-opab 3819  df-xp 4351  df-res 4357
This theorem is referenced by:  nfima  4676  nffrec  5982
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