Theorem nfres 4980

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 4705	. 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 2350	. . . 4 |- F/_ x _V
5	3, 4	nfxp 4720	. . 3 |- F/_ x ( B X. _V )
6	2, 5	nfin 3387	. 2 |- F/_ x ( A i^i ( B X. _V ) )
7	1, 6	nfcxfr 2347	1 |- F/_ x ( A |` B )

Syntax hints:   F/_wnfc 2337   _Vcvv 2776   i^i cin 3173   X. cxp 4691   |` cres 4695

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rab 2495  df-in 3180  df-opab 4122  df-xp 4699  df-res 4705

This theorem is referenced by:  nfima  5049  nffrec  6505