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Theorem clel4 2657
 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 2083 . . 3 (x = B → (A xA B))
31, 2ceqsalv 2561 . 2 (x(x = BA x) ↔ A B)
43bicomi 123 1 (A Bx(x = BA x))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1226   = wceq 1228   ∈ wcel 1374  Vcvv 2535 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 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-v 2537 This theorem is referenced by:  intpr  3621
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