Theorem nfint 4213

Description: Bound-variable hypothesis builder for intersection. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Hypothesis

Ref	Expression
nfint.1	|- F/_ x A

Assertion

Ref	Expression
nfint	|- F/_ x |^| A

Proof of Theorem nfint

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfint2 4205	. 2 |- |^| A = { y | A. z e. A y e. z }
2		nfint.1	. . . 4 |- F/_ x A
3		nfv 1764	. . . 4 |- F/ x y e. z
4	2, 3	nfral 2769	. . 3 |- F/ x A. z e. A y e. z
5	4	nfab 2596	. 2 |- F/_ x { y | A. z e. A y e. z }
6	1, 5	nfcxfr 2590	1 |- F/_ x |^| A

Syntax hints:   {cab 2437   F/_wnfc 2579   A.wral 2736   |^|cint 4203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2741  df-int 4204

This theorem is referenced by:  onminsb  6613  oawordeulem  7241  nnawordex  7324  rankidb  8257  cardmin2  8418  cardaleph  8506  cardmin  8975  ldsysgenld  28988  sltval2  30548  nobndlem5  30590  aomclem8  35920