Theorem elin3 3173

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 3165	. . 3 |- ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) )
2	1	anbi1i 446	. 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 3170	. 2 |- ( A e. X <-> ( A e. ( B i^i C ) /\ A e. D ) )
5		df-3an 922	. 2 |- ( ( A e. B /\ A e. C /\ A e. D ) <-> ( ( A e. B /\ A e. C ) /\ A e. D ) )
6	2, 4, 5	3bitr4i 210	1 |- ( A e. X <-> ( A e. B /\ A e. C /\ A e. D ) )

Syntax hints:   /\ wa 102   <-> wb 103   /\ w3a 920   = wceq 1285   e. wcel 1434   i^i cin 2981

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-in 2988

This theorem is referenced by: (None)