Theorem elint 3837

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 2233	. . . 4 |- ( y = A -> ( y e. x <-> A e. x ) )
3	2	imbi2d 229	. . 3 |- ( y = A -> ( ( x e. B -> y e. x ) <-> ( x e. B -> A e. x ) ) )
4	3	albidv 1817	. 2 |- ( y = A -> ( A. x ( x e. B -> y e. x ) <-> A. x ( x e. B -> A e. x ) ) )
5		df-int 3832	. 2 |- |^| B = { y | A. x ( x e. B -> y e. x ) }
6	1, 4, 5	elab2 2878	1 |- ( A e. |^| B <-> A. x ( x e. B -> A e. x ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1346   = wceq 1348   e. wcel 2141   _Vcvv 2730   |^|cint 3831

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-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-int 3832

This theorem is referenced by:  elint2  3838  elintab  3842  intss1  3846  intss  3852  intun  3862  intpr  3863  peano1  4578