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Theorem clel2 2671
 Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
Hypothesis
Ref Expression
clel2.1 A V
Assertion
Ref Expression
clel2 (A Bx(x = Ax B))
Distinct variable groups:   x,A   x,B

Proof of Theorem clel2
StepHypRef Expression
1 clel2.1 . . 3 A V
2 eleq1 2097 . . 3 (x = A → (x BA B))
31, 2ceqsalv 2578 . 2 (x(x = Ax B) ↔ A B)
43bicomi 123 1 (A Bx(x = Ax B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240   = wceq 1242   ∈ wcel 1390  Vcvv 2551 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553 This theorem is referenced by:  snss  3485
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