Theorem dmqseqeq1d

Description: Equality theorem for domain quotient set, deduction version. (Contributed by Peter Mazsa, 26-Sep-2021)

	Ref	Expression
Hypothesis	dmqseqeq1d.1	$$⊢\phi \to R=S$$
Assertion	dmqseqeq1d	$$⊢\phi \to \left(\mathrm{dom}R/R=A\leftrightarrow \mathrm{dom}S/S=A\right)$$

Proof

Step	Hyp	Ref	Expression
1		dmqseqeq1d.1	$$⊢\phi \to R=S$$
2		dmqseqeq1	$$⊢R=S\to \left(\mathrm{dom}R/R=A\leftrightarrow \mathrm{dom}S/S=A\right)$$
3	1 2	syl	$$⊢\phi \to \left(\mathrm{dom}R/R=A\leftrightarrow \mathrm{dom}S/S=A\right)$$