Theorem nfiinxy 3848

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 3824	. 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 2276	. . . 4 |- F/ y z e. B
5	2, 4	nfralxy 2474	. . 3 |- F/ y A. x e. A z e. B
6	5	nfab 2287	. 2 |- F/_ y { z | A. x e. A z e. B }
7	1, 6	nfcxfr 2279	1 |- F/_ y |^|_ x e. A B

Syntax hints:   e. wcel 1481   {cab 2126   F/_wnfc 2269   A.wral 2417   |^|_ciin 3822

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-iin 3824

This theorem is referenced by:  iinab  3882