Theorem elint2 3933

Description: Membership in class intersection. (Contributed by NM, 14-Oct-1999.)

Hypothesis

Ref	Expression
elint2.1	|- A e. _V

Assertion

Ref	Expression
elint2	|- (A e. |^|B <-> A.x e. B A e. x)

Distinct variable groups:   x,A   x,B

Proof of Theorem elint2

Step	Hyp	Ref	Expression
1		elint2.1	. . 3 |- A e. _V
2	1	elint 3932	. 2 |- (A e. |^|B <-> A.x(x e. B -> A e. x))
3		df-ral 2619	. 2 |- (A.x e. B A e. x <-> A.x(x e. B -> A e. x))
4	2, 3	bitr4i 243	1 |- (A e. |^|B <-> A.x e. B A e. x)

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   e. wcel 1710  A.wral 2614  _Vcvv 2859  |^|cint 3926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-int 3927

This theorem is referenced by:  elintg  3934  ssint  3942  intssuni  3948  iinuni  4049  dfint3  4318