Theorem 3eqtr4a

Description: A chained equality inference, useful for converting to definitions. (Contributed by NM, 2-Feb-2007) (Proof shortened by Andrew Salmon, 25-May-2011)

	Ref	Expression
Hypotheses	3eqtr4a.1	⊢ 𝐴 = 𝐵
	3eqtr4a.2	⊢ ( 𝜑 → 𝐶 = 𝐴 )
	3eqtr4a.3	⊢ ( 𝜑 → 𝐷 = 𝐵 )
Assertion	3eqtr4a	⊢ ( 𝜑 → 𝐶 = 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		3eqtr4a.1	⊢ 𝐴 = 𝐵
2		3eqtr4a.2	⊢ ( 𝜑 → 𝐶 = 𝐴 )
3		3eqtr4a.3	⊢ ( 𝜑 → 𝐷 = 𝐵 )
4	2 1	syl6eq	⊢ ( 𝜑 → 𝐶 = 𝐵 )
5	4 3	eqtr4d	⊢ ( 𝜑 → 𝐶 = 𝐷 )