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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  |-  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 4385 . 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 2194 . . . 4  |-  F/_ x _V
53, 4nfxp 4399 . . 3  |-  F/_ x
( B  X.  _V )
62, 5nfin 3171 . 2  |-  F/_ x
( A  i^i  ( B  X.  _V ) )
71, 6nfcxfr 2191 1  |-  F/_ x
( A  |`  B )
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2181   _Vcvv 2574    i^i cin 2944    X. cxp 4371    |` cres 4375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rab 2332  df-in 2952  df-opab 3847  df-xp 4379  df-res 4385
This theorem is referenced by:  nfima  4704  nffrec  6013
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