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Theorem bj-df-clel 32081
 Description: Candidate definition for df-clel 2606 (the need for it is exposed in bj-ax8 32080). The similarity of the hypothesis and the conclusion, together with all possible dv conditions, makes it clear that this definition merely extends to class variables something that is true for setvar variables, hence is conservative. This definition should be directly referenced only by bj-dfclel 32082, which should be used instead. The proof is irrelevant since this is a proposal for an axiom. Note: the current definition df-clel 2606 already mentions cleljust 1985 as a justification; here, we merely propose to put it as a hypothesis to make things clearer. (Contributed by BJ, 27-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
bj-df-clel.1 (𝑢𝑣 ↔ ∃𝑤(𝑤 = 𝑢𝑤𝑣))
Assertion
Ref Expression
bj-df-clel (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
Distinct variable groups:   𝑣,𝑢,𝑤,𝑥,𝐴   𝑢,𝐵,𝑣,𝑤,𝑥

Proof of Theorem bj-df-clel
StepHypRef Expression
1 df-clel 2606 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 depends on definitions:  df-clel 2606 This theorem is referenced by:  bj-dfclel  32082
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