Theorem bj-dfclel 32889

Description: Characterization of the elements of a class. Note: cleljust 1998 could be relabeled "clelhyp". (Contributed by BJ, 27-Jun-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-dfclel	|- ( A e. B <-> E. x ( x = A /\ x e. 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 1998	. . 3 |- ( u e. v <-> E. w ( w = u /\ w e. v ) )
2	1	gen2 1723	. 2 |- A. u A. v ( u e. v <-> E. w ( w = u /\ w e. v ) )
3	2	bj-df-clel 32888	1 |- ( A e. B <-> E. x ( x = A /\ x e. B ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-clel 2618

This theorem is referenced by: (None)