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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 = ((BC) ∩ D)
Assertion
Ref Expression
elin3 (A X ↔ (A B A C A D))

Proof of Theorem elin3
StepHypRef Expression
1 elin 3219 . . 3 (A (BC) ↔ (A B A C))
21anbi1i 676 . 2 ((A (BC) A D) ↔ ((A B A C) A D))
3 elin3.x . . 3 X = ((BC) ∩ D)
43elin2 3446 . 2 (A X ↔ (A (BC) A D))
5 df-3an 936 . 2 ((A B A C A D) ↔ ((A B A C) A D))
62, 4, 53bitr4i 268 1 (A X ↔ (A B A C A D))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710   ∩ cin 3208 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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)
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