Theorem wl-dfclel 37842

Description: Characterization of the elements of a class. (Contributed by BJ, 27-Jun-2019.) Base on wl-dfclel.just 37840. (Revised by Wolf Lammen, 13-Apr-2026.)

Assertion

Ref	Expression
wl-dfclel	⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))

Distinct variable group:   𝑥,𝐴,𝐵

Proof of Theorem wl-dfclel

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2741	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴))
2		eleq1w 2820	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
3	1, 2	anbi12d 633	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑦 = 𝐴 ∧ 𝑦 ∈ 𝐵)))
4	3	cbvexvw 2039	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝐵))
5	4	wl-dfclel.just 37840	1 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2729  df-clel 2812

This theorem is referenced by: (None)