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Theorem cleljustab 2805
 Description: Extension of cleljust 2124 from formulas of the form "setvar ∈ setvar" to formulas of the form "setvar ∈ class abstraction". This is an instance of dfclel 2897 where the containing class is a class abstraction. The same remarks as for eleq1ab 2804 apply. (Contributed by BJ, 8-Nov-2021.) (Proof shortened by Steven Nguyen, 19-May-2023.)
Assertion
Ref Expression
cleljustab (𝑥 ∈ {𝑦𝜑} ↔ ∃𝑧(𝑧 = 𝑥𝑧 ∈ {𝑦𝜑}))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧   𝜑,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem cleljustab
StepHypRef Expression
1 eleq1ab 2804 . . 3 (𝑧 = 𝑥 → (𝑧 ∈ {𝑦𝜑} ↔ 𝑥 ∈ {𝑦𝜑}))
21equsexvw 2012 . 2 (∃𝑧(𝑧 = 𝑥𝑧 ∈ {𝑦𝜑}) ↔ 𝑥 ∈ {𝑦𝜑})
32bicomi 227 1 (𝑥 ∈ {𝑦𝜑} ↔ ∃𝑧(𝑧 = 𝑥𝑧 ∈ {𝑦𝜑}))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399  ∃wex 1781   ∈ wcel 2115  {cab 2802 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803 This theorem is referenced by: (None)
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