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Theorem clel3g 2655
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 2083 . . 3 (x = B → (A xA B))
21ceqsexgv 2650 . 2 (B 𝑉 → (x(x = B A x) ↔ A B))
32bicomd 129 1 (B 𝑉 → (A Bx(x = B A x)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228  wex 1362   wcel 1374
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-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537
This theorem is referenced by:  clel3  2656  dfiun2g  3663
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