Theorem elind 3307

Description: Deduce membership in an intersection of two classes. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

Ref	Expression
elind.1	|- ( ph -> X e. A )
elind.2	|- ( ph -> X e. B )

Assertion

Ref	Expression
elind	|- ( ph -> X e. ( A i^i B ) )

Proof of Theorem elind

Step	Hyp	Ref	Expression
1		elind.1	. 2 |- ( ph -> X e. A )
2		elind.2	. 2 |- ( ph -> X e. B )
3		elin 3305	. 2 |- ( X e. ( A i^i B ) <-> ( X e. A /\ X e. B ) )
4	1, 2, 3	sylanbrc 414	1 |- ( ph -> X e. ( A i^i B ) )

Syntax hints:   -> wi 4   e. wcel 2136   i^i cin 3115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122

This theorem is referenced by:  elfir  6938  infpwfidom  7154  nninfdclemcl  12381  nninfdclemp1  12383  strslfv2d  12436  baspartn  12698  bastg  12711  isopn3  12775  restbasg  12818  lmss  12896  metrest  13156  tgioo  13196  dvmulxxbr  13316  pilem3  13354  2sqlem7  13607