Theorem nfin 3343

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 3138	. 2 |- ( A i^i B ) = { y e. A | y e. B }
2		nfin.2	. . . 4 |- F/_ x B
3	2	nfcri 2313	. . 3 |- F/ x y e. B
4		nfin.1	. . 3 |- F/_ x A
5	3, 4	nfrabxy 2658	. 2 |- F/_ x { y e. A | y e. B }
6	1, 5	nfcxfr 2316	1 |- F/_ x ( A i^i B )

Syntax hints:   e. wcel 2148   F/_wnfc 2306   {crab 2459   i^i cin 3130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-in 3137

This theorem is referenced by:  csbing  3344  nfres  4911