Theorem nfres 4961

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 4687	. 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 2348	. . . 4 |- F/_ x _V
5	3, 4	nfxp 4702	. . 3 |- F/_ x ( B X. _V )
6	2, 5	nfin 3379	. 2 |- F/_ x ( A i^i ( B X. _V ) )
7	1, 6	nfcxfr 2345	1 |- F/_ x ( A |` B )

Syntax hints:   F/_wnfc 2335   _Vcvv 2772   i^i cin 3165   X. cxp 4673   |` cres 4677

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rab 2493  df-in 3172  df-opab 4106  df-xp 4681  df-res 4687

This theorem is referenced by:  nfima  5030  nffrec  6482