Theorem elintg 3883

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 2259	. 2 |- ( y = A -> ( y e. |^| B <-> A e. |^| B ) )
2		eleq1 2259	. . 3 |- ( y = A -> ( y e. x <-> A e. x ) )
3	2	ralbidv 2497	. 2 |- ( y = A -> ( A. x e. B y e. x <-> A. x e. B A e. x ) )
4		vex 2766	. . 3 |- y e. _V
5	4	elint2 3882	. 2 |- ( y e. |^| B <-> A. x e. B y e. x )
6	1, 3, 5	vtoclbg 2825	1 |- ( A e. V -> ( A e. |^| B <-> A. x e. B A e. x ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2167   A.wral 2475   |^|cint 3875

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-int 3876

This theorem is referenced by:  elinti  3884  elrint  3915  peano2  4632  pitonn  7932  peano1nnnn  7936  peano2nnnn  7937  1nn  9018  peano2nn  9019  subgintm  13404  subrngintm  13844  subrgintm  13875  lssintclm  14016