Theorem nfin 3277

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 3073	. 2 |- ( A i^i B ) = { y e. A | y e. B }
2		nfin.2	. . . 4 |- F/_ x B
3	2	nfcri 2273	. . 3 |- F/ x y e. B
4		nfin.1	. . 3 |- F/_ x A
5	3, 4	nfrabxy 2609	. 2 |- F/_ x { y e. A | y e. B }
6	1, 5	nfcxfr 2276	1 |- F/_ x ( A i^i B )

Syntax hints:   e. wcel 1480   F/_wnfc 2266   {crab 2418   i^i cin 3065

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rab 2423  df-in 3072

This theorem is referenced by:  csbing  3278  nfres  4816