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Theorem clel4 3312
 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 2677 . . 3 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
31, 2ceqsalv 3206 . 2 (∀𝑥(𝑥 = 𝐵𝐴𝑥) ↔ 𝐴𝐵)
43bicomi 213 1 (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐵𝐴𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977  Vcvv 3173 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175 This theorem is referenced by:  intpr  4445
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