Theorem elini 3260

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 3259	. 2 |- ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) )
4	1, 2, 3	mpbir2an 926	1 |- A e. ( B i^i C )

Syntax hints:   e. wcel 1480   i^i cin 3070

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077

This theorem is referenced by:  exmidonfinlem  7049  taupi  13239