Theorem erALTVeq1

Description: Equality theorem for equivalence relation on domain quotient. (Contributed by Peter Mazsa, 25-Sep-2021)

	Ref	Expression
Assertion	erALTVeq1	$$⊢R=S\to \left(RErALTVA\leftrightarrow SErALTVA\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqvreleq	$$⊢R=S\to \left(EqvRelR\leftrightarrow EqvRelS\right)$$
2		dmqseqeq1	$$⊢R=S\to \left(\mathrm{dom}R/R=A\leftrightarrow \mathrm{dom}S/S=A\right)$$
3	1 2	anbi12d	$$⊢R=S\to \left(\left(EqvRelR\wedge \mathrm{dom}R/R=A\right)\leftrightarrow \left(EqvRelS\wedge \mathrm{dom}S/S=A\right)\right)$$
4		dferALTV2	$$⊢RErALTVA\leftrightarrow \left(EqvRelR\wedge \mathrm{dom}R/R=A\right)$$
5		dferALTV2	$$⊢SErALTVA\leftrightarrow \left(EqvRelS\wedge \mathrm{dom}S/S=A\right)$$
6	3 4 5	3bitr4g	$$⊢R=S\to \left(RErALTVA\leftrightarrow SErALTVA\right)$$