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	$$⊢A=B\leftrightarrow \forall x\left(x\in A\leftrightarrow x\in B\right)$$

Proof

Step	Hyp	Ref	Expression
1		axextb	$$⊢y=z\leftrightarrow \forall u\left(u\in y\leftrightarrow u\in z\right)$$
2		axextb	$$⊢t=t\leftrightarrow \forall v\left(v\in t\leftrightarrow v\in t\right)$$
3	1 2	df-cleq	$$⊢A=B\leftrightarrow \forall x\left(x\in A\leftrightarrow x\in B\right)$$