Theorem bj-dfclel 33218

Description: Characterization of the elements of a class. Note: cleljust 2153 could be relabeled "clelhyp". (Contributed by BJ, 27-Jun-2019.) (Proof modification is discouraged.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem bj-dfclel

Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cleljust 2153	. . 3 ⊢ (𝑢 ∈ 𝑣 ↔ ∃𝑤(𝑤 = 𝑢 ∧ 𝑤 ∈ 𝑣))
2	1	gen2 1871	. 2 ⊢ ∀𝑢∀𝑣(𝑢 ∈ 𝑣 ↔ ∃𝑤(𝑤 = 𝑢 ∧ 𝑤 ∈ 𝑣))
3	2	bj-df-clel 33217	1 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))

Syntax hints:   ↔ wb 196   ∧ wa 382   = wceq 1631  ∃wex 1852   ∈ wcel 2145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147

This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-clel 2767

This theorem is referenced by: (None)