Theorem wl-dfclel.just 37840

Description: Add a hypotheses to wl-dfclel.basic 37839, that allows alpha-renaming. (Contributed by Wolf Lammen, 7-Apr-2026.)

Hypothesis

Ref	Expression
wl-dfclel.just.1	⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝐵))

Assertion

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

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem wl-dfclel.just

Step	Hyp	Ref	Expression
1		wl-dfclel.basic 37839	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

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

This theorem is referenced by:  wl-dfclel  37842