Theorem elintg 3644

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 2141	. 2 |- ( y = A -> ( y e. |^| B <-> A e. |^| B ) )
2		eleq1 2141	. . 3 |- ( y = A -> ( y e. x <-> A e. x ) )
3	2	ralbidv 2368	. 2 |- ( y = A -> ( A. x e. B y e. x <-> A. x e. B A e. x ) )
4		vex 2604	. . 3 |- y e. _V
5	4	elint2 3643	. 2 |- ( y e. |^| B <-> A. x e. B y e. x )
6	1, 3, 5	vtoclbg 2659	1 |- ( A e. V -> ( A e. |^| B <-> A. x e. B A e. x ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   e. wcel 1433   A.wral 2348   |^|cint 3636

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-int 3637

This theorem is referenced by:  elinti  3645  elrint  3676  peano2  4336  pitonn  7016  peano1nnnn  7020  peano2nnnn  7021  1nn  8050  peano2nn  8051