Theorem elin2 3417

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 2305	. 2 |- ( A e. X <-> A e. ( B i^i C ) )
3		elin 3412	. 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 1402   e. wcel 2209   i^i cin 3219

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226

This theorem is referenced by:  elin3  3420  fnres  5495  funfvima  5940  isabl  14068  isidom  14558  isfld  14587  lmres  15272