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Theorem clel3g 2976
 Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.)
Assertion
Ref Expression
clel3g (B V → (A Bx(x = B A x)))
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   V(x)

Proof of Theorem clel3g
StepHypRef Expression
1 eleq2 2414 . . 3 (x = B → (A xA B))
21ceqsexgv 2971 . 2 (B V → (x(x = B A x) ↔ A B))
32bicomd 192 1 (B V → (A Bx(x = B A x)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710 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-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 This theorem is referenced by:  clel3  2977  dfiun2g  3999
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