Theorem 3eqtrrd

Description: A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006) (Proof shortened by Andrew Salmon, 25-May-2011)

	Ref	Expression
Hypotheses	3eqtrd.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	3eqtrd.2	⊢ ( 𝜑 → 𝐵 = 𝐶 )
	3eqtrd.3	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	3eqtrrd	⊢ ( 𝜑 → 𝐷 = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		3eqtrd.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		3eqtrd.2	⊢ ( 𝜑 → 𝐵 = 𝐶 )
3		3eqtrd.3	⊢ ( 𝜑 → 𝐶 = 𝐷 )
4	1 2	eqtrd	⊢ ( 𝜑 → 𝐴 = 𝐶 )
5	4 3	eqtr2d	⊢ ( 𝜑 → 𝐷 = 𝐴 )