Theorem nfiinxy 3939

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

Hypotheses

Ref	Expression
nfiunxy.1	|- F/_ y A
nfiunxy.2	|- F/_ y B

Assertion

Ref	Expression
nfiinxy	|- F/_ y |^|_ x e. A B

Distinct variable group:   x, y

Allowed substitution hints:   A( x, y)   B( x, y)

Proof of Theorem nfiinxy

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iin 3915	. 2 |- |^|_ x e. A B = { z | A. x e. A z e. B }
2		nfiunxy.1	. . . 4 |- F/_ y A
3		nfiunxy.2	. . . . 5 |- F/_ y B
4	3	nfcri 2330	. . . 4 |- F/ y z e. B
5	2, 4	nfralxy 2532	. . 3 |- F/ y A. x e. A z e. B
6	5	nfab 2341	. 2 |- F/_ y { z | A. x e. A z e. B }
7	1, 6	nfcxfr 2333	1 |- F/_ y |^|_ x e. A B

Syntax hints:   e. wcel 2164   {cab 2179   F/_wnfc 2323   A.wral 2472   |^|_ciin 3913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-iin 3915

This theorem is referenced by:  iinab  3974