Theorem extep

Description: Property of epsilon relation, see also extid , extssr and the comment of df-ssr . (Contributed by Peter Mazsa, 10-Jul-2019)

	Ref	Expression
Assertion	extep	$$⊢\left(A\in V\wedge B\in W\right)\to \left(\left[A\right]{\mathrm{E}}^{-1}=\left[B\right]{\mathrm{E}}^{-1}\leftrightarrow A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		eccnvep	$$⊢A\in V\to \left[A\right]{\mathrm{E}}^{-1}=A$$
2		eccnvep	$$⊢B\in W\to \left[B\right]{\mathrm{E}}^{-1}=B$$
3	1 2	eqeqan12d	$$⊢\left(A\in V\wedge B\in W\right)\to \left(\left[A\right]{\mathrm{E}}^{-1}=\left[B\right]{\mathrm{E}}^{-1}\leftrightarrow A=B\right)$$