Theorem nfint 4486

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 4477	. 2 |- |^| A = { y | A. z e. A y e. z }
2		nfint.1	. . . 4 |- F/_ x A
3		nfv 1843	. . . 4 |- F/ x y e. z
4	2, 3	nfral 2945	. . 3 |- F/ x A. z e. A y e. z
5	4	nfab 2769	. 2 |- F/_ x { y | A. z e. A y e. z }
6	1, 5	nfcxfr 2762	1 |- F/_ x |^| A

Syntax hints:   {cab 2608   F/_wnfc 2751   A.wral 2912   |^|cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-int 4476

This theorem is referenced by:  onminsb  6999  oawordeulem  7634  nnawordex  7717  rankidb  8663  cardmin2  8824  cardaleph  8912  cardmin  9386  ldsysgenld  30223  sltval2  31809  aomclem8  37631