Theorem nfres 4711

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 4448	. 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 2228	. . . 4 |- F/_ x _V
5	3, 4	nfxp 4462	. . 3 |- F/_ x ( B X. _V )
6	2, 5	nfin 3206	. 2 |- F/_ x ( A i^i ( B X. _V ) )
7	1, 6	nfcxfr 2225	1 |- F/_ x ( A |` B )

Syntax hints:   F/_wnfc 2215   _Vcvv 2619   i^i cin 2998   X. cxp 4434   |` cres 4438

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-in 3005  df-opab 3898  df-xp 4442  df-res 4448

This theorem is referenced by:  nfima  4777  nffrec  6153