Theorem eqtr2d

Description: An equality transitivity deduction. (Contributed by NM, 18-Oct-1999)

	Ref	Expression
Hypotheses	eqtr2d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	eqtr2d.2	⊢ ( 𝜑 → 𝐵 = 𝐶 )
Assertion	eqtr2d	⊢ ( 𝜑 → 𝐶 = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		eqtr2d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		eqtr2d.2	⊢ ( 𝜑 → 𝐵 = 𝐶 )
3	1 2	eqtrd	⊢ ( 𝜑 → 𝐴 = 𝐶 )
4	3	eqcomd	⊢ ( 𝜑 → 𝐶 = 𝐴 )