Theorem nfres 4927

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 4656	. 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 2332	. . . 4 |- F/_ x _V
5	3, 4	nfxp 4671	. . 3 |- F/_ x ( B X. _V )
6	2, 5	nfin 3356	. 2 |- F/_ x ( A i^i ( B X. _V ) )
7	1, 6	nfcxfr 2329	1 |- F/_ x ( A |` B )

Syntax hints:   F/_wnfc 2319   _Vcvv 2752   i^i cin 3143   X. cxp 4642   |` cres 4646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-in 3150  df-opab 4080  df-xp 4650  df-res 4656

This theorem is referenced by:  nfima  4996  nffrec  6421