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Theorem eliminable2a 32034
Description: A theorem used to prove the base case of the Eliminability Theorem (see section comment). (Contributed by BJ, 19-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Ref Expression
eliminable2a (𝑥 = {𝑦𝜑} ↔ ∀𝑧(𝑧𝑥𝑧 ∈ {𝑦𝜑}))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧   𝜑,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem eliminable2a
StepHypRef Expression
1 dfcleq 2604 1 (𝑥 = {𝑦𝜑} ↔ ∀𝑧(𝑧𝑥𝑧 ∈ {𝑦𝜑}))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wal 1473   = wceq 1475  wcel 1977  {cab 2596
This theorem was proved from axioms:  ax-ext 2590
This theorem depends on definitions:  df-cleq 2603
This theorem is referenced by: (None)
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