Theorem 3eqtr3d

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

	Ref	Expression
Hypotheses	3eqtr3d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	3eqtr3d.2	⊢ ( 𝜑 → 𝐴 = 𝐶 )
	3eqtr3d.3	⊢ ( 𝜑 → 𝐵 = 𝐷 )
Assertion	3eqtr3d	⊢ ( 𝜑 → 𝐶 = 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		3eqtr3d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		3eqtr3d.2	⊢ ( 𝜑 → 𝐴 = 𝐶 )
3		3eqtr3d.3	⊢ ( 𝜑 → 𝐵 = 𝐷 )
4	1 2	eqtr3d	⊢ ( 𝜑 → 𝐵 = 𝐶 )
5	4 3	eqtr3d	⊢ ( 𝜑 → 𝐶 = 𝐷 )