Theorem nfres 4948

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	⊢ Ⅎ𝑥𝐴
nfres.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfres	⊢ Ⅎ𝑥(𝐴 ↾ 𝐵)

Proof of Theorem nfres

Step	Hyp	Ref	Expression
1		df-res 4675	. 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		nfres.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfres.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4		nfcv 2339	. . . 4 ⊢ Ⅎ𝑥V
5	3, 4	nfxp 4690	. . 3 ⊢ Ⅎ𝑥(𝐵 × V)
6	2, 5	nfin 3369	. 2 ⊢ Ⅎ𝑥(𝐴 ∩ (𝐵 × V))
7	1, 6	nfcxfr 2336	1 ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵)

Syntax hints:  Ⅎwnfc 2326  Vcvv 2763   ∩ cin 3156   × cxp 4661   ↾ cres 4665

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rab 2484  df-in 3163  df-opab 4095  df-xp 4669  df-res 4675

This theorem is referenced by:  nfima  5017  nffrec  6454