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Metamath Proof Explorer

Theorem eqcom

Description: Commutative law for class equality. Theorem 6.5 of Quine p. 41. (Contributed by NM, 26-May-1993) (Proof shortened by Wolf Lammen, 19-Nov-2019)

		Ref	Expression
	Assertion	eqcom	$⊢ A = B \leftrightarrow B = A$

Proof

Step	Hyp	Ref	Expression
1		id	$⊢ A = B \to A = B$
2	1	eqcomd	$⊢ A = B \to B = A$
3		id	$⊢ B = A \to B = A$
4	3	eqcomd	$⊢ B = A \to A = B$
5	2 4	impbii	$⊢ A = B \leftrightarrow B = A$