Theorem nfiinya 3707

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 3681	. 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 2213	. . . 4 |- F/ y z e. B
5	2, 4	nfralya 2404	. . 3 |- F/ y A. x e. A z e. B
6	5	nfab 2223	. 2 |- F/_ y { z | A. x e. A z e. B }
7	1, 6	nfcxfr 2216	1 |- F/_ y |^|_ x e. A B

Syntax hints:   e. wcel 1433   {cab 2067   F/_wnfc 2206   A.wral 2348   |^|_ciin 3679

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-iin 3681

This theorem is referenced by: (None)