Theorem 3eqtr3g

Description: A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994)

	Ref	Expression
Hypotheses	3eqtr3g.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	3eqtr3g.2	⊢ 𝐴 = 𝐶
	3eqtr3g.3	⊢ 𝐵 = 𝐷
Assertion	3eqtr3g	⊢ ( 𝜑 → 𝐶 = 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		3eqtr3g.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		3eqtr3g.2	⊢ 𝐴 = 𝐶
3		3eqtr3g.3	⊢ 𝐵 = 𝐷
4	2 1	syl5eqr	⊢ ( 𝜑 → 𝐶 = 𝐵 )
5	4 3	syl6eq	⊢ ( 𝜑 → 𝐶 = 𝐷 )