Theorem cleljustab 2710

Description: Extension of cleljust 2113 from formulas of the form "setvar ∈ setvar" to formulas of the form "setvar ∈ class abstraction". This is an instance of dfclel 2809 where the containing class is a class abstraction. The same remarks as for eleq1ab 2709 apply. (Contributed by BJ, 8-Nov-2021.) (Proof shortened by Steven Nguyen, 19-May-2023.)

Assertion

Ref	Expression
cleljustab	⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑}))

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧   𝜑,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem cleljustab

Step	Hyp	Ref	Expression
1		eleq1ab 2709	. . 3 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ 𝑥 ∈ {𝑦 ∣ 𝜑}))
2	1	equsexvw 2006	. 2 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑}) ↔ 𝑥 ∈ {𝑦 ∣ 𝜑})
3	2	bicomi 223	1 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑}))

Syntax hints:   ↔ wb 205   ∧ wa 394  ∃wex 1779   ∈ wcel 2104  {cab 2707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1780  df-sb 2066  df-clab 2708

This theorem is referenced by: (None)