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Metamath Proof Explorer

Theorem ideq2

Description: For sets, the identity binary relation is the same as equality. (Contributed by Peter Mazsa, 24-Jun-2020) (Revised by Peter Mazsa, 18-Dec-2021)

		Ref	Expression
	Assertion	ideq2	$⊢ A \in V \to (A I B \leftrightarrow A = B)$

Proof

Step	Hyp	Ref	Expression
1		brid	$⊢ A I B \leftrightarrow B I A$
2		ideqg	$⊢ A \in V \to (B I A \leftrightarrow B = A)$
3		eqcom	$⊢ B = A \leftrightarrow A = B$
4	2 3	bitrdi	$⊢ A \in V \to (B I A \leftrightarrow A = B)$
5	1 4	bitrid	$⊢ A \in V \to (A I B \leftrightarrow A = B)$