Theorem eqtr2

Description: A transitive law for class equality. (Contributed by NM, 20-May-2005) (Proof shortened by Andrew Salmon, 25-May-2011)

	Ref	Expression
Assertion	eqtr2	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐴 = 𝐶 ) → 𝐵 = 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		eqcom	⊢ ( 𝐴 = 𝐵 ↔ 𝐵 = 𝐴 )
2		eqtr	⊢ ( ( 𝐵 = 𝐴 ∧ 𝐴 = 𝐶 ) → 𝐵 = 𝐶 )
3	1 2	sylanb	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐴 = 𝐶 ) → 𝐵 = 𝐶 )