Theorem elin3 3318

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 3310	. . 3 |- ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) )
2	1	anbi1i 455	. 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 3315	. 2 |- ( A e. X <-> ( A e. ( B i^i C ) /\ A e. D ) )
5		df-3an 975	. 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 973   = wceq 1348   e. wcel 2141   i^i cin 3120

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127

This theorem is referenced by: (None)