Theorem 3eqtr4d

Description: A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995) (Proof shortened by Andrew Salmon, 25-May-2011)

	Ref	Expression
Hypotheses	3eqtr4d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	3eqtr4d.2	⊢ ( 𝜑 → 𝐶 = 𝐴 )
	3eqtr4d.3	⊢ ( 𝜑 → 𝐷 = 𝐵 )
Assertion	3eqtr4d	⊢ ( 𝜑 → 𝐶 = 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		3eqtr4d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		3eqtr4d.2	⊢ ( 𝜑 → 𝐶 = 𝐴 )
3		3eqtr4d.3	⊢ ( 𝜑 → 𝐷 = 𝐵 )
4	3 1	eqtr4d	⊢ ( 𝜑 → 𝐷 = 𝐴 )
5	2 4	eqtr4d	⊢ ( 𝜑 → 𝐶 = 𝐷 )