Theorem eqvreltrd

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

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

Proof

Step	Hyp	Ref	Expression
1		eqvreltrd.1	⊢ ( 𝜑 → EqvRel 𝑅 )
2		eqvreltrd.2	⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
3		eqvreltrd.3	⊢ ( 𝜑 → 𝐵 𝑅 𝐶 )
4	1	eqvreltr	⊢ ( 𝜑 → ( ( 𝐴 𝑅 𝐵 ∧ 𝐵 𝑅 𝐶 ) → 𝐴 𝑅 𝐶 ) )
5	2 3 4	mp2and	⊢ ( 𝜑 → 𝐴 𝑅 𝐶 )