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	$$⊢\phi \to R\mathrm{Er}X$$
	erthi.2	$$⊢\phi \to ARB$$
Assertion	erthi	$$⊢\phi \to \left[A\right]R=\left[B\right]R$$

Proof

Step	Hyp	Ref	Expression
1		erthi.1	$$⊢\phi \to R\mathrm{Er}X$$
2		erthi.2	$$⊢\phi \to ARB$$
3	1 2	ercl	$$⊢\phi \to A\in X$$
4	1 3	erth	$$⊢\phi \to \left(ARB\leftrightarrow \left[A\right]R=\left[B\right]R\right)$$
5	2 4	mpbid	$$⊢\phi \to \left[A\right]R=\left[B\right]R$$