Theorem nfres 5021

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 4743	. 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 2375	. . . 4 |- F/_ x _V
5	3, 4	nfxp 4758	. . 3 |- F/_ x ( B X. _V )
6	2, 5	nfin 3415	. 2 |- F/_ x ( A i^i ( B X. _V ) )
7	1, 6	nfcxfr 2372	1 |- F/_ x ( A |` B )

Syntax hints:   F/_wnfc 2362   _Vcvv 2803   i^i cin 3200   X. cxp 4729   |` cres 4733

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rab 2520  df-in 3207  df-opab 4156  df-xp 4737  df-res 4743

This theorem is referenced by:  nfima  5090  nffrec  6605