Theorem bj-df-cleq 32085

Description: Candidate definition for df-cleq 2603 (the need for it is exposed in bj-ax9 32083). The similarity of the hypothesis and the conclusion 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-dfcleq 32086, which should be used instead. The proof is irrelevant since this is a proposal for an axiom.

Another definition, which would give finer control, is actually the pair of definitions where each has one implication of the present biconditional as hypothesis and conclusion. They assert that extensionality (respectively, the left-substitution axiom for the membership predicate) extends from setvars to classes. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bj-df-cleq.1	⊢ (𝑢 = 𝑣 ↔ ∀𝑤(𝑤 ∈ 𝑢 ↔ 𝑤 ∈ 𝑣))

Assertion

Ref	Expression
bj-df-cleq	⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Distinct variable groups:   𝑣,𝑢,𝑤,𝑥,𝐴   𝑢,𝐵,𝑣,𝑤,𝑥

Proof of Theorem bj-df-cleq

Step	Hyp	Ref	Expression
1		dfcleq 2604	1 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Syntax hints:   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977

This theorem was proved from axioms:  ax-ext 2590

This theorem depends on definitions:  df-cleq 2603

This theorem is referenced by:  bj-dfcleq  32086