Theorem bj-dfclel 32864

Description: Characterization of the elements of a class. Note: cleljust 1996 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 1996	. . 3 ⊢ (𝑢 ∈ 𝑣 ↔ ∃𝑤(𝑤 = 𝑢 ∧ 𝑤 ∈ 𝑣))
2	1	gen2 1721	. 2 ⊢ ∀𝑢∀𝑣(𝑢 ∈ 𝑣 ↔ ∃𝑤(𝑤 = 𝑢 ∧ 𝑤 ∈ 𝑣))
3	2	bj-df-clel 32863	1 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1481  ∃wex 1702   ∈ wcel 1988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703  df-clel 2616

This theorem is referenced by: (None)