Theorem elin3 3364

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 3356	. . 3 |- ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) )
2	1	anbi1i 458	. 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 3361	. 2 |- ( A e. X <-> ( A e. ( B i^i C ) /\ A e. D ) )
5		df-3an 983	. 2 |- ( ( A e. B /\ A e. C /\ A e. D ) <-> ( ( A e. B /\ A e. C ) /\ A e. D ) )
6	2, 4, 5	3bitr4i 212	1 |- ( A e. X <-> ( A e. B /\ A e. C /\ A e. D ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2176   i^i cin 3165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172

This theorem is referenced by: (None)