Theorem elini 3320

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 3319	. 2 |- ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) )
4	1, 2, 3	mpbir2an 942	1 |- A e. ( B i^i C )

Syntax hints:   e. wcel 2148   i^i cin 3129

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-in 3136

This theorem is referenced by:  exmidonfinlem  7192  taupi  14823