Theorem eqtrd

Description: An equality transitivity deduction. (Contributed by NM, 21-Jun-1993)

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

Proof

Step	Hyp	Ref	Expression
1		eqtrd.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		eqtrd.2	⊢ ( 𝜑 → 𝐵 = 𝐶 )
3	2	eqeq2d	⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ 𝐴 = 𝐶 ) )
4	1 3	mpbid	⊢ ( 𝜑 → 𝐴 = 𝐶 )