Theorem elind 3392

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 3390	. 2 |- ( X e. ( A i^i B ) <-> ( X e. A /\ X e. B ) )
4	1, 2, 3	sylanbrc 417	1 |- ( ph -> X e. ( A i^i B ) )

Syntax hints:   -> wi 4   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:  fnfvimad  5890  elfir  7172  infpwfidom  7409  nninfdclemcl  13087  nninfdclemp1  13089  strslfv2d  13143  bassetsnn  13157  insubm  13586  2idl0  14545  2idl1  14546  baspartn  14793  bastg  14804  isopn3  14868  restbasg  14911  lmss  14989  metrest  15249  tgioo  15297  dvmulxxbr  15445  elply2  15478  pilem3  15526  2sqlem7  15869