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Theorem bj-dfclel 32082
Description: Characterization of the elements of a class. Note: cleljust 1985 could be relabeled as clelhyp. (Contributed by BJ, 27-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-dfclel (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem bj-dfclel
Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cleljust 1985 . 2 (𝑢𝑣 ↔ ∃𝑤(𝑤 = 𝑢𝑤𝑣))
21bj-df-clel 32081 1 (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383   = wceq 1475  wex 1695  wcel 1977
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-8 1979
This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-clel 2606
This theorem is referenced by: (None)
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