Theorem elint 3621

Description: Membership in class intersection. (Contributed by NM, 21-May-1994.)

Hypothesis

Ref	Expression
elint.1	|- A e. _V

Assertion

Ref	Expression
elint	|- ( A e. |^| B <-> A. x ( x e. B -> A e. x ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem elint

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elint.1	. 2 |- A e. _V
2		eleq1 2100	. . . 4 |- ( y = A -> ( y e. x <-> A e. x ) )
3	2	imbi2d 219	. . 3 |- ( y = A -> ( ( x e. B -> y e. x ) <-> ( x e. B -> A e. x ) ) )
4	3	albidv 1705	. 2 |- ( y = A -> ( A. x ( x e. B -> y e. x ) <-> A. x ( x e. B -> A e. x ) ) )
5		df-int 3616	. 2 |- |^| B = { y | A. x ( x e. B -> y e. x ) }
6	1, 4, 5	elab2 2690	1 |- ( A e. |^| B <-> A. x ( x e. B -> A e. x ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   = wceq 1243   e. wcel 1393   _Vcvv 2557   |^|cint 3615

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-int 3616

This theorem is referenced by:  elint2  3622  elintab  3626  intss1  3630  intss  3636  intun  3646  intpr  3647  peano1  4317