Theorem nfiin 3996

Description: Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)

Hypotheses

Ref	Expression
nfiun.1	|- F/_yA
nfiun.2	|- F/_yB

Assertion

Ref	Expression
nfiin	|- F/_y|^|_x e. A B

Proof of Theorem nfiin

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iin 3972	. 2 |- |^|_x e. A B = {z | A.x e. A z e. B}
2		nfiun.1	. . . 4 |- F/_yA
3		nfiun.2	. . . . 5 |- F/_yB
4	3	nfcri 2483	. . . 4 |- F/y z e. B
5	2, 4	nfral 2667	. . 3 |- F/yA.x e. A z e. B
6	5	nfab 2493	. 2 |- F/_y{z | A.x e. A z e. B}
7	1, 6	nfcxfr 2486	1 |- F/_y|^|_x e. A B

Syntax hints:   e. wcel 1710  {cab 2339   F/_wnfc 2476  A.wral 2614  |^|_ciin 3970

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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-iin 3972

This theorem is referenced by:  iinab  4027