Theorem nfint 4281

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 4273	. 2 |- |^| A = { y | A. z e. A y e. z }
2		nfint.1	. . . 4 |- F/_ x A
3		nfv 1712	. . . 4 |- F/ x y e. z
4	2, 3	nfral 2840	. . 3 |- F/ x A. z e. A y e. z
5	4	nfab 2620	. 2 |- F/_ x { y | A. z e. A y e. z }
6	1, 5	nfcxfr 2614	1 |- F/_ x |^| A

Syntax hints:   {cab 2439   F/_wnfc 2602   A.wral 2804   |^|cint 4271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-int 4272

This theorem is referenced by:  onminsb  6607  oawordeulem  7195  nnawordex  7278  rankidb  8209  cardmin2  8370  cardaleph  8461  cardmin  8930  sltval2  29656  nobndlem5  29696  aomclem8  31246