Theorem elind 3414

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 3412	. 2 |- ( X e. ( A i^i B ) <-> ( X e. A /\ X e. B ) )
4	1, 2, 3	sylanbrc 421	1 |- ( ph -> X e. ( A i^i B ) )

Syntax hints:   -> wi 4   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:  fnfvimad  5944  elfir  7297  infpwfidom  7540  hashfibclem  11260  ballotfilem2  13206  nninfdclemcl  13317  nninfdclemp1  13319  strslfv2d  13373  bassetsnn  13387  insubm  13769  2idl0  14821  2idl1  14822  baspartn  15074  bastg  15085  isopn3  15149  restbasg  15192  lmss  15270  metrest  15530  tgioo  15578  dvmulxxbr  15726  elply2  15759  pilem3  15807  2sqlem7  16154