Theorem bj-dfcleq 32894

Description: Proof of class extensionality from the axiom of set extensionality (ax-ext 2602) and the definition of class equality (bj-df-cleq 32893). (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-dfcleq	|- ( A = B <-> A. x ( x e. A <-> x e. B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem bj-dfcleq

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

Step	Hyp	Ref	Expression
1		bj-cleqhyp 32892	. . 3 |- ( u = v <-> A. w ( w e. u <-> w e. v ) )
2	1	gen2 1723	. 2 |- A. u A. v ( u = v <-> A. w ( w e. u <-> w e. v ) )
3	2	bj-df-cleq 32893	1 |- ( A = B <-> A. x ( x e. A <-> x e. B ) )

Syntax hints:   <-> wb 196   A.wal 1481   = wceq 1483   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-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615

This theorem is referenced by: (None)