Theorem nfiinxy 3893

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 3869	. 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 2302	. . . 4 |- F/ y z e. B
5	2, 4	nfralxy 2504	. . 3 |- F/ y A. x e. A z e. B
6	5	nfab 2313	. 2 |- F/_ y { z | A. x e. A z e. B }
7	1, 6	nfcxfr 2305	1 |- F/_ y |^|_ x e. A B

Syntax hints:   e. wcel 2136   {cab 2151   F/_wnfc 2295   A.wral 2444   |^|_ciin 3867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-iin 3869

This theorem is referenced by:  iinab  3927