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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 ↔ 𝐵 = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( 𝐴 = 𝐵 → 𝐴 = 𝐵 )
2	1	eqcomd	⊢ ( 𝐴 = 𝐵 → 𝐵 = 𝐴 )
3		id	⊢ ( 𝐵 = 𝐴 → 𝐵 = 𝐴 )
4	3	eqcomd	⊢ ( 𝐵 = 𝐴 → 𝐴 = 𝐵 )
5	2 4	impbii	⊢ ( 𝐴 = 𝐵 ↔ 𝐵 = 𝐴 )