Theorem elintg 3935

Description: Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.)

Assertion

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

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   V(x)

Proof of Theorem elintg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2413	. 2 |- (y = A -> (y e. |^|B <-> A e. |^|B))
2		eleq1 2413	. . 3 |- (y = A -> (y e. x <-> A e. x))
3	2	ralbidv 2635	. 2 |- (y = A -> (A.x e. B y e. x <-> A.x e. B A e. x))
4		vex 2863	. . 3 |- y e. _V
5	4	elint2 3934	. 2 |- (y e. |^|B <-> A.x e. B y e. x)
6	1, 3, 5	vtoclbg 2916	1 |- (A e. V -> (A e. |^|B <-> A.x e. B A e. x))

Syntax hints:   -> wi 4   <-> wb 176   = wceq 1642   e. wcel 1710  A.wral 2615  |^|cint 3927

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 2479  df-ral 2620  df-v 2862  df-int 3928

This theorem is referenced by:  elinti  3936  elrint  3968