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Theorem bj-dfclel 33218
 Description: Characterization of the elements of a class. Note: cleljust 2153 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 2153 . . 3 (𝑢𝑣 ↔ ∃𝑤(𝑤 = 𝑢𝑤𝑣))
21gen2 1871 . 2 𝑢𝑣(𝑢𝑣 ↔ ∃𝑤(𝑤 = 𝑢𝑤𝑣))
32bj-df-clel 33217 1 (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 382   = wceq 1631  ∃wex 1852   ∈ wcel 2145 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147 This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-clel 2767 This theorem is referenced by: (None)
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