Theorem elind 3358

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 3356	. 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 2176   i^i cin 3165

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172

This theorem is referenced by:  elfir  7075  infpwfidom  7306  nninfdclemcl  12819  nninfdclemp1  12821  strslfv2d  12875  insubm  13317  2idl0  14274  2idl1  14275  baspartn  14522  bastg  14533  isopn3  14597  restbasg  14640  lmss  14718  metrest  14978  tgioo  15026  dvmulxxbr  15174  elply2  15207  pilem3  15255  2sqlem7  15598