Theorem elin2 3395

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 2298	. 2 |- ( A e. X <-> A e. ( B i^i C ) )
3		elin 3390	. 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 1397   e. wcel 2202   i^i cin 3199

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206

This theorem is referenced by:  elin3  3398  fnres  5449  funfvima  5885  isabl  13874  isidom  14289  lmres  14971