Theorem nfint 4292

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 4284	. 2 |- |^| A = { y | A. z e. A y e. z }
2		nfint.1	. . . 4 |- F/_ x A
3		nfv 1683	. . . 4 |- F/ x y e. z
4	2, 3	nfral 2850	. . 3 |- F/ x A. z e. A y e. z
5	4	nfab 2633	. 2 |- F/_ x { y | A. z e. A y e. z }
6	1, 5	nfcxfr 2627	1 |- F/_ x |^| A

Syntax hints:   {cab 2452   F/_wnfc 2615   A.wral 2814   |^|cint 4282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-int 4283

This theorem is referenced by:  onminsb  6619  oawordeulem  7204  nnawordex  7287  rankidb  8219  cardmin2  8380  cardaleph  8471  cardmin  8940  sltval2  29269  nobndlem5  29309  aomclem8  30838