Theorem elini 3182

Description: Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
elini.1	|- A e. B
elini.2	|- A e. C

Assertion

Ref	Expression
elini	|- A e. ( B i^i C )

Proof of Theorem elini

Step	Hyp	Ref	Expression
1		elini.1	. 2 |- A e. B
2		elini.2	. 2 |- A e. C
3		elin 3181	. 2 |- ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) )
4	1, 2, 3	mpbir2an 888	1 |- A e. ( B i^i C )

Syntax hints:   e. wcel 1438   i^i cin 2996

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3003

This theorem is referenced by: (None)