Theorem erthi

Description: Basic property of equivalence relations. Part of Lemma 3N of Enderton p. 57. (Contributed by NM, 30-Jul-1995) (Revised by Mario Carneiro, 9-Jul-2014)

	Ref	Expression
Hypotheses	erthi.1	|- ( ph -> R Er X )
	erthi.2	|- ( ph -> A R B )
Assertion	erthi	|- ( ph -> [ A ] R = [ B ] R )

Proof

Step	Hyp	Ref	Expression
1		erthi.1	|- ( ph -> R Er X )
2		erthi.2	|- ( ph -> A R B )
3	1 2	ercl	|- ( ph -> A e. X )
4	1 3	erth	|- ( ph -> ( A R B <-> [ A ] R = [ B ] R ) )
5	2 4	mpbid	|- ( ph -> [ A ] R = [ B ] R )