Theorem elint 3864

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 2251	. . . 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 1834	. 2 |- ( y = A -> ( A. x ( x e. B -> y e. x ) <-> A. x ( x e. B -> A e. x ) ) )
5		df-int 3859	. 2 |- |^| B = { y | A. x ( x e. B -> y e. x ) }
6	1, 4, 5	elab2 2899	1 |- ( A e. |^| B <-> A. x ( x e. B -> A e. x ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1361   = wceq 1363   e. wcel 2159   _Vcvv 2751   |^|cint 3858

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-v 2753  df-int 3859

This theorem is referenced by:  elint2  3865  elintab  3869  intss1  3873  intss  3879  intun  3889  intpr  3890  peano1  4607