Theorem elintg 3853

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 2240	. 2 |- ( y = A -> ( y e. |^| B <-> A e. |^| B ) )
2		eleq1 2240	. . 3 |- ( y = A -> ( y e. x <-> A e. x ) )
3	2	ralbidv 2477	. 2 |- ( y = A -> ( A. x e. B y e. x <-> A. x e. B A e. x ) )
4		vex 2741	. . 3 |- y e. _V
5	4	elint2 3852	. 2 |- ( y e. |^| B <-> A. x e. B y e. x )
6	1, 3, 5	vtoclbg 2799	1 |- ( A e. V -> ( A e. |^| B <-> A. x e. B A e. x ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   e. wcel 2148   A.wral 2455   |^|cint 3845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2740  df-int 3846

This theorem is referenced by:  elinti  3854  elrint  3885  peano2  4595  pitonn  7847  peano1nnnn  7851  peano2nnnn  7852  1nn  8930  peano2nn  8931  subgintm  13058  subrgintm  13364