Theorem elin3 3313

Description: Membership in a class defined as a ternary intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)

Hypothesis

Ref	Expression
elin3.x	|- X = ( ( B i^i C ) i^i D )

Assertion

Ref	Expression
elin3	|- ( A e. X <-> ( A e. B /\ A e. C /\ A e. D ) )

Proof of Theorem elin3

Step	Hyp	Ref	Expression
1		elin 3305	. . 3 |- ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) )
2	1	anbi1i 454	. 2 |- ( ( A e. ( B i^i C ) /\ A e. D ) <-> ( ( A e. B /\ A e. C ) /\ A e. D ) )
3		elin3.x	. . 3 |- X = ( ( B i^i C ) i^i D )
4	3	elin2 3310	. 2 |- ( A e. X <-> ( A e. ( B i^i C ) /\ A e. D ) )
5		df-3an 970	. 2 |- ( ( A e. B /\ A e. C /\ A e. D ) <-> ( ( A e. B /\ A e. C ) /\ A e. D ) )
6	2, 4, 5	3bitr4i 211	1 |- ( A e. X <-> ( A e. B /\ A e. C /\ A e. D ) )

Syntax hints:   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   i^i cin 3115

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122

This theorem is referenced by: (None)