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Theorem bj-dfcleq 32086
 Description: Proof of class extensionality from the axiom of set extensionality (ax-ext 2590) and the definition of class equality (bj-df-cleq 32085). (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-dfcleq (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem bj-dfcleq
Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-cleqhyp 32084 . 2 (𝑢 = 𝑣 ↔ ∀𝑤(𝑤𝑢𝑤𝑣))
21bj-df-cleq 32085 1 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ 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-9 1986  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603 This theorem is referenced by: (None)
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