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Theorem elning 3217
 Description: Membership in anti-intersection. (Contributed by SF, 10-Jan-2015.)
Assertion
Ref Expression
elning (A V → (A (BC) ↔ (A B A C)))

Proof of Theorem elning
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eleq1 2413 . . 3 (x = A → (x BA B))
2 eleq1 2413 . . 3 (x = A → (x CA C))
31, 2nanbi12d 1303 . 2 (x = A → ((x B x C) ↔ (A B A C)))
4 df-nin 3211 . 2 (BC) = {x (x B x C)}
53, 4elab2g 2987 1 (A V → (A (BC) ↔ (A B A C)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ⊼ wnan 1287   = wceq 1642   ∈ wcel 1710   ⩃ cnin 3204 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-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 This theorem is referenced by:  elcomplg  3218  elin  3219  elun  3220  elnin  3224
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