Theorem nfin 3333

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 3128	. 2 |- ( A i^i B ) = { y e. A | y e. B }
2		nfin.2	. . . 4 |- F/_ x B
3	2	nfcri 2306	. . 3 |- F/ x y e. B
4		nfin.1	. . 3 |- F/_ x A
5	3, 4	nfrabxy 2650	. 2 |- F/_ x { y e. A | y e. B }
6	1, 5	nfcxfr 2309	1 |- F/_ x ( A i^i B )

Syntax hints:   e. wcel 2141   F/_wnfc 2299   {crab 2452   i^i cin 3120

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-in 3127

This theorem is referenced by:  csbing  3334  nfres  4893