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	|- ( ph -> EqvRel R )
	eqvreltrd.2	|- ( ph -> A R B )
	eqvreltrd.3	|- ( ph -> B R C )
Assertion	eqvreltrd	|- ( ph -> A R C )

Proof

Step	Hyp	Ref	Expression
1		eqvreltrd.1	|- ( ph -> EqvRel R )
2		eqvreltrd.2	|- ( ph -> A R B )
3		eqvreltrd.3	|- ( ph -> B R C )
4	1	eqvreltr	|- ( ph -> ( ( A R B /\ B R C ) -> A R C ) )
5	2 3 4	mp2and	|- ( ph -> A R C )