Theorem elintg 3787

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 2203	. 2 |- ( y = A -> ( y e. |^| B <-> A e. |^| B ) )
2		eleq1 2203	. . 3 |- ( y = A -> ( y e. x <-> A e. x ) )
3	2	ralbidv 2438	. 2 |- ( y = A -> ( A. x e. B y e. x <-> A. x e. B A e. x ) )
4		vex 2692	. . 3 |- y e. _V
5	4	elint2 3786	. 2 |- ( y e. |^| B <-> A. x e. B y e. x )
6	1, 3, 5	vtoclbg 2750	1 |- ( A e. V -> ( A e. |^| B <-> A. x e. B A e. x ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   e. wcel 1481   A.wral 2417   |^|cint 3779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-int 3780

This theorem is referenced by:  elinti  3788  elrint  3819  peano2  4517  pitonn  7680  peano1nnnn  7684  peano2nnnn  7685  1nn  8755  peano2nn  8756