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Theorem clel2 2822
 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 (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem clel2
StepHypRef Expression
1 clel2.1 . . 3 𝐴 ∈ V
2 eleq1 2203 . . 3 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
31, 2ceqsalv 2719 . 2 (∀𝑥(𝑥 = 𝐴𝑥𝐵) ↔ 𝐴𝐵)
43bicomi 131 1 (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1330   = wceq 1332   ∈ wcel 1481  Vcvv 2689 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-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-v 2691 This theorem is referenced by:  snss  3657
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