Theorem wl-dfcleq.basic

Description: This is a copy of dfcleq in main. It is not sufficient to avoid reproving ax-8 as shown in in-ax8 . (Contributed by NM, 15-Sep-1993) (Revised by BJ, 24-Jun-2019)

	Ref	Expression
Assertion	wl-dfcleq.basic	$$⊢A=B\leftrightarrow \forall x\left(x\in A\leftrightarrow x\in B\right)$$

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	wl-df.cleq	$$⊢A=B\leftrightarrow \forall x\left(x\in A\leftrightarrow x\in B\right)$$