Theorem elin3 3447

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 3219	. . 3 |- (A e. (B i^i C) <-> (A e. B /\ A e. C))
2	1	anbi1i 676	. 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 3446	. 2 |- (A e. X <-> (A e. (B i^i C) /\ A e. D))
5		df-3an 936	. 2 |- ((A e. B /\ A e. C /\ A e. D) <-> ((A e. B /\ A e. C) /\ A e. D))
6	2, 4, 5	3bitr4i 268	1 |- (A e. X <-> (A e. B /\ A e. C /\ A e. D))

Syntax hints:   <-> wb 176   /\ wa 358   /\ w3a 934   = wceq 1642   e. wcel 1710   i^i cin 3208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213

This theorem is referenced by: (None)