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Theorem clel3g 2814
Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.)
Assertion
Ref Expression
clel3g (𝐵𝑉 → (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐵𝐴𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem clel3g
StepHypRef Expression
1 eleq2 2201 . . 3 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
21ceqsexgv 2809 . 2 (𝐵𝑉 → (∃𝑥(𝑥 = 𝐵𝐴𝑥) ↔ 𝐴𝐵))
32bicomd 140 1 (𝐵𝑉 → (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐵𝐴𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wex 1468  wcel 1480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683
This theorem is referenced by:  clel3  2815  dfiun2g  3840
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