Theorem 3eqtr3rd

Description: A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006)

	Ref	Expression
Hypotheses	3eqtr3d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	3eqtr3d.2	⊢ ( 𝜑 → 𝐴 = 𝐶 )
	3eqtr3d.3	⊢ ( 𝜑 → 𝐵 = 𝐷 )
Assertion	3eqtr3rd	⊢ ( 𝜑 → 𝐷 = 𝐶 )

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	3 4	eqtr3d	⊢ ( 𝜑 → 𝐷 = 𝐶 )