Theorem nfres 4861

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 4591	. 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 2296	. . . 4 |- F/_ x _V
5	3, 4	nfxp 4606	. . 3 |- F/_ x ( B X. _V )
6	2, 5	nfin 3309	. 2 |- F/_ x ( A i^i ( B X. _V ) )
7	1, 6	nfcxfr 2293	1 |- F/_ x ( A |` B )

Syntax hints:   F/_wnfc 2283   _Vcvv 2709   i^i cin 3097   X. cxp 4577   |` cres 4581

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-rab 2441  df-in 3104  df-opab 4022  df-xp 4585  df-res 4591

This theorem is referenced by:  nfima  4929  nffrec  6333