Theorem elint 4288

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 2539	. . . 4 |- ( y = A -> ( y e. x <-> A e. x ) )
3	2	imbi2d 316	. . 3 |- ( y = A -> ( ( x e. B -> y e. x ) <-> ( x e. B -> A e. x ) ) )
4	3	albidv 1689	. 2 |- ( y = A -> ( A. x ( x e. B -> y e. x ) <-> A. x ( x e. B -> A e. x ) ) )
5		df-int 4283	. 2 |- |^| B = { y | A. x ( x e. B -> y e. x ) }
6	1, 4, 5	elab2 3253	1 |- ( A e. |^| B <-> A. x ( x e. B -> A e. x ) )

Syntax hints:   -> wi 4   <-> wb 184   A.wal 1377   = wceq 1379   e. wcel 1767   _Vcvv 3113   |^|cint 4282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-int 4283

This theorem is referenced by:  elint2  4289  elintab  4293  intss1  4297  intss  4303  intun  4314  intpr  4315  cssmre  18531  dfom5b  29415