Theorem eqvreltr4d

Description: A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014) (Revised by Peter Mazsa, 2-Jun-2019)

	Ref	Expression
Hypotheses	eqvreltr4d.1	⊢ ( 𝜑 → EqvRel 𝑅 )
	eqvreltr4d.2	⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
	eqvreltr4d.3	⊢ ( 𝜑 → 𝐶 𝑅 𝐵 )
Assertion	eqvreltr4d	⊢ ( 𝜑 → 𝐴 𝑅 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		eqvreltr4d.1	⊢ ( 𝜑 → EqvRel 𝑅 )
2		eqvreltr4d.2	⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
3		eqvreltr4d.3	⊢ ( 𝜑 → 𝐶 𝑅 𝐵 )
4	1 3	eqvrelsym	⊢ ( 𝜑 → 𝐵 𝑅 𝐶 )
5	1 2 4	eqvreltrd	⊢ ( 𝜑 → 𝐴 𝑅 𝐶 )