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Theorem bj-dfclel 32864
Description: Characterization of the elements of a class. Note: cleljust 1996 could be relabeled "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 1996 . . 3 (𝑢𝑣 ↔ ∃𝑤(𝑤 = 𝑢𝑤𝑣))
21gen2 1721 . 2 𝑢𝑣(𝑢𝑣 ↔ ∃𝑤(𝑤 = 𝑢𝑤𝑣))
32bj-df-clel 32863 1 (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1481  wex 1702  wcel 1988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703  df-clel 2616
This theorem is referenced by: (None)
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