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Theorem clel4 2978
 Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
Hypothesis
Ref Expression
clel4.1 B V
Assertion
Ref Expression
clel4 (A Bx(x = BA x))
Distinct variable groups:   x,A   x,B

Proof of Theorem clel4
StepHypRef Expression
1 clel4.1 . . 3 B V
2 eleq2 2414 . . 3 (x = B → (A xA B))
31, 2ceqsalv 2885 . 2 (x(x = BA x) ↔ A B)
43bicomi 193 1 (A Bx(x = BA x))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Vcvv 2859 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-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861 This theorem is referenced by:  intpr  3959
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