Theorem 3eqtr2d

Description: A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006)

	Ref	Expression
Hypotheses	3eqtr2d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	3eqtr2d.2	⊢ ( 𝜑 → 𝐶 = 𝐵 )
	3eqtr2d.3	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	3eqtr2d	⊢ ( 𝜑 → 𝐴 = 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		3eqtr2d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		3eqtr2d.2	⊢ ( 𝜑 → 𝐶 = 𝐵 )
3		3eqtr2d.3	⊢ ( 𝜑 → 𝐶 = 𝐷 )
4	1 2	eqtr4d	⊢ ( 𝜑 → 𝐴 = 𝐶 )
5	4 3	eqtrd	⊢ ( 𝜑 → 𝐴 = 𝐷 )