Theorem opideq

Description: Equality conditions for ordered pairs <. A , A >. and <. B , B >. . (Contributed by Peter Mazsa, 22-Jul-2019) (Revised by Thierry Arnoux, 16-Feb-2022)

	Ref	Expression
Assertion	opideq	$$⊢A\in V\to \left(⟨A,A⟩=⟨B,B⟩\leftrightarrow A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		opthg	$$⊢\left(A\in V\wedge A\in V\right)\to \left(⟨A,A⟩=⟨B,B⟩\leftrightarrow \left(A=B\wedge A=B\right)\right)$$
2	1	anidms	$$⊢A\in V\to \left(⟨A,A⟩=⟨B,B⟩\leftrightarrow \left(A=B\wedge A=B\right)\right)$$
3		anidm	$$⊢\left(A=B\wedge A=B\right)\leftrightarrow A=B$$
4	2 3	bitrdi	$$⊢A\in V\to \left(⟨A,A⟩=⟨B,B⟩\leftrightarrow A=B\right)$$