Theorem ertrd

Description: A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014)

	Ref	Expression
Hypotheses	ersymb.1	|- ( ph -> R Er X )
	ertrd.5	|- ( ph -> A R B )
	ertrd.6	|- ( ph -> B R C )
Assertion	ertrd	|- ( ph -> A R C )

Proof

Step	Hyp	Ref	Expression
1		ersymb.1	|- ( ph -> R Er X )
2		ertrd.5	|- ( ph -> A R B )
3		ertrd.6	|- ( ph -> B R C )
4	1	ertr	|- ( ph -> ( ( A R B /\ B R C ) -> A R C ) )
5	2 3 4	mp2and	|- ( ph -> A R C )