Theorem elintg 3878

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 2256	. 2 |- ( y = A -> ( y e. |^| B <-> A e. |^| B ) )
2		eleq1 2256	. . 3 |- ( y = A -> ( y e. x <-> A e. x ) )
3	2	ralbidv 2494	. 2 |- ( y = A -> ( A. x e. B y e. x <-> A. x e. B A e. x ) )
4		vex 2763	. . 3 |- y e. _V
5	4	elint2 3877	. 2 |- ( y e. |^| B <-> A. x e. B y e. x )
6	1, 3, 5	vtoclbg 2821	1 |- ( A e. V -> ( A e. |^| B <-> A. x e. B A e. x ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2164   A.wral 2472   |^|cint 3870

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-int 3871

This theorem is referenced by:  elinti  3879  elrint  3910  peano2  4627  pitonn  7908  peano1nnnn  7912  peano2nnnn  7913  1nn  8993  peano2nn  8994  subgintm  13268  subrngintm  13708  subrgintm  13739  lssintclm  13880