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	$$⊢\phi \to EqvRelR$$
	eqvreltrd.2	$$⊢\phi \to ARB$$
	eqvreltrd.3	$$⊢\phi \to BRC$$
Assertion	eqvreltrd	$$⊢\phi \to ARC$$

Proof

Step	Hyp	Ref	Expression
1		eqvreltrd.1	$$⊢\phi \to EqvRelR$$
2		eqvreltrd.2	$$⊢\phi \to ARB$$
3		eqvreltrd.3	$$⊢\phi \to BRC$$
4	1	eqvreltr	$$⊢\phi \to \left(\left(ARB\wedge BRC\right)\to ARC\right)$$
5	2 3 4	mp2and	$$⊢\phi \to ARC$$