Metamath Proof Explorer

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 x x A x B


Step Hyp Ref Expression
1 axextb y = z u u y u z
2 axextb t = t v v t v t
3 1 2 df-cleq A = B x x A x B