Theorem nfres 4821

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

Step	Hyp	Ref	Expression
1		df-res 4551	. 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 2281	. . . 4 |- F/_ x _V
5	3, 4	nfxp 4566	. . 3 |- F/_ x ( B X. _V )
6	2, 5	nfin 3282	. 2 |- F/_ x ( A i^i ( B X. _V ) )
7	1, 6	nfcxfr 2278	1 |- F/_ x ( A |` B )

Syntax hints:   F/_wnfc 2268   _Vcvv 2686   i^i cin 3070   X. cxp 4537   |` cres 4541

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-in 3077  df-opab 3990  df-xp 4545  df-res 4551

This theorem is referenced by:  nfima  4889  nffrec  6293