Theorem nfiinya 3727

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

Hypotheses

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

Assertion

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

Distinct variable group:   x, A

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

Proof of Theorem nfiinya

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iin 3701	. 2 |- |^|_ x e. A B = { z | A. x e. A z e. B }
2		nfiunya.1	. . . 4 |- F/_ y A
3		nfiunya.2	. . . . 5 |- F/_ y B
4	3	nfcri 2217	. . . 4 |- F/ y z e. B
5	2, 4	nfralya 2409	. . 3 |- F/ y A. x e. A z e. B
6	5	nfab 2227	. 2 |- F/_ y { z | A. x e. A z e. B }
7	1, 6	nfcxfr 2220	1 |- F/_ y |^|_ x e. A B

Syntax hints:   e. wcel 1434   {cab 2069   F/_wnfc 2210   A.wral 2353   |^|_ciin 3699

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-iin 3701

This theorem is referenced by: (None)