Theorem elind 3312

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 3310	. 2 |- ( X e. ( A i^i B ) <-> ( X e. A /\ X e. B ) )
4	1, 2, 3	sylanbrc 415	1 |- ( ph -> X e. ( A i^i B ) )

Syntax hints:   -> wi 4   e. wcel 2141   i^i cin 3120

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127

This theorem is referenced by:  elfir  6950  infpwfidom  7175  nninfdclemcl  12403  nninfdclemp1  12405  strslfv2d  12458  insubm  12703  baspartn  12842  bastg  12855  isopn3  12919  restbasg  12962  lmss  13040  metrest  13300  tgioo  13340  dvmulxxbr  13460  pilem3  13498  2sqlem7  13751