Theorem elin2 3338

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 2256	. 2 |- ( A e. X <-> A e. ( B i^i C ) )
3		elin 3333	. 2 |- ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) )
4	2, 3	bitri 184	1 |- ( A e. X <-> ( A e. B /\ A e. C ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2160   i^i cin 3143

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 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150

This theorem is referenced by:  elin3  3341  fnres  5351  funfvima  5769  isabl  13244  lmres  14225