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Theorem clel2 2741
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 2147 . . 3 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
31, 2ceqsalv 2643 . 2 (∀𝑥(𝑥 = 𝐴𝑥𝐵) ↔ 𝐴𝐵)
43bicomi 130 1 (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1285   = wceq 1287  wcel 1436  Vcvv 2615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-v 2617
This theorem is referenced by:  snss  3549
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