Theorem nfin 3188

Description: Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)

Hypotheses

Ref	Expression
nfin.1	|- F/_ x A
nfin.2	|- F/_ x B

Assertion

Ref	Expression
nfin	|- F/_ x ( A i^i B )

Proof of Theorem nfin

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfin5 2989	. 2 |- ( A i^i B ) = { y e. A | y e. B }
2		nfin.2	. . . 4 |- F/_ x B
3	2	nfcri 2217	. . 3 |- F/ x y e. B
4		nfin.1	. . 3 |- F/_ x A
5	3, 4	nfrabxy 2539	. 2 |- F/_ x { y e. A | y e. B }
6	1, 5	nfcxfr 2220	1 |- F/_ x ( A i^i B )

Syntax hints:   e. wcel 1434   F/_wnfc 2210   {crab 2357   i^i cin 2981

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-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rab 2362  df-in 2988

This theorem is referenced by:  csbing  3189  nfres  4662