Theorem nfres 4937

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 4789	. 2 ⊢ (A ↾ B) = (A ∩ (B × V))
2		nfres.1	. . 3 ⊢ ℲxA
3		nfres.2	. . . 4 ⊢ ℲxB
4		nfcv 2490	. . . 4 ⊢ ℲxV
5	3, 4	nfxp 4811	. . 3 ⊢ Ⅎx(B × V)
6	2, 5	nfin 3231	. 2 ⊢ Ⅎx(A ∩ (B × V))
7	1, 6	nfcxfr 2487	1 ⊢ Ⅎx(A ↾ B)

Syntax hints:  Ⅎwnfc 2477  Vcvv 2860   ∩ cin 3209   × cxp 4771   ↾ cres 4775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-nin 3212  df-compl 3213  df-in 3214  df-opab 4624  df-xp 4785  df-res 4789

This theorem is referenced by: (None)