Theorem elint 3971

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 2301	. . . 4 |- ( y = A -> ( y e. x <-> A e. x ) )
3	2	imbi2d 230	. . 3 |- ( y = A -> ( ( x e. B -> y e. x ) <-> ( x e. B -> A e. x ) ) )
4	3	albidv 1877	. 2 |- ( y = A -> ( A. x ( x e. B -> y e. x ) <-> A. x ( x e. B -> A e. x ) ) )
5		df-int 3966	. 2 |- |^| B = { y | A. x ( x e. B -> y e. x ) }
6	1, 4, 5	elab2 2974	1 |- ( A e. |^| B <-> A. x ( x e. B -> A e. x ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1400   = wceq 1402   e. wcel 2209   _Vcvv 2821   |^|cint 3965

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-int 3966

This theorem is referenced by:  elint2  3972  elintab  3976  intss1  3980  intss  3986  intun  3996  intpr  3997  peano1  4736