Theorem elini 3797

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 3796	. 2 |- ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) )
4	1, 2, 3	mpbir2an 955	1 |- A e. ( B i^i C )

Syntax hints:   e. wcel 1990   i^i cin 3573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581

This theorem is referenced by:  recvs  22946  qcvs  22947  cnncvs  22959  0pwfi  39227  sge0rnn0  40585  sge0reuz  40664