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Theorem clel4 3653
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 (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐵𝐴𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem clel4
StepHypRef Expression
1 clel4.1 . . 3 𝐵 ∈ V
2 eleq2 2898 . . 3 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
31, 2ceqsalv 3530 . 2 (∀𝑥(𝑥 = 𝐵𝐴𝑥) ↔ 𝐴𝐵)
43bicomi 225 1 (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐵𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1526   = wceq 1528  wcel 2105  Vcvv 3492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1081  df-ex 1772  df-nf 1776  df-cleq 2811  df-clel 2890
This theorem is referenced by:  intpr  4900
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