Theorem elin2 3156

Description: Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)

Hypothesis

Ref	Expression
elin2.x	|- X = ( B i^i C )

Assertion

Ref	Expression
elin2	|- ( A e. X <-> ( A e. B /\ A e. C ) )

Proof of Theorem elin2

Step	Hyp	Ref	Expression
1		elin2.x	. . 3 |- X = ( B i^i C )
2	1	eleq2i 2145	. 2 |- ( A e. X <-> A e. ( B i^i C ) )
3		elin 3155	. 2 |- ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) )
4	2, 3	bitri 182	1 |- ( A e. X <-> ( A e. B /\ A e. C ) )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   i^i cin 2972

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-v 2603  df-in 2979

This theorem is referenced by:  elin3  3157  fnres  5035  funfvima  5411