Theorem dfcleq

Description: The defining characterization of class equality. It is proved, over Tarski's FOL, from the axiom of (set) extensionality ( ax-ext ) and the definition of class equality ( df-cleq ). Its forward implication is called "class extensionality". Remark: the proof uses axextb to prove also the hypothesis of df-cleq that is a degenerate instance, but it could be proved also from minimal propositional calculus and { ax-gen , equid }. (Contributed by NM, 15-Sep-1993) (Revised by BJ, 24-Jun-2019)

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

Proof

Step	Hyp	Ref	Expression
1		axextb	|- ( y = z <-> A. u ( u e. y <-> u e. z ) )
2		axextb	|- ( t = t <-> A. v ( v e. t <-> v e. t ) )
3	1 2	df-cleq	|- ( A = B <-> A. x ( x e. A <-> x e. B ) )