Theorem nfin 3369

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 3164	. 2 |- ( A i^i B ) = { y e. A | y e. B }
2		nfin.2	. . . 4 |- F/_ x B
3	2	nfcri 2333	. . 3 |- F/ x y e. B
4		nfin.1	. . 3 |- F/_ x A
5	3, 4	nfrabw 2678	. 2 |- F/_ x { y e. A | y e. B }
6	1, 5	nfcxfr 2336	1 |- F/_ x ( A i^i B )

Syntax hints:   e. wcel 2167   F/_wnfc 2326   {crab 2479   i^i cin 3156

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rab 2484  df-in 3163

This theorem is referenced by:  csbing  3370  nfres  4948