Theorem disjdmqseqeq1

Description: Lemma for the equality theorem for partition parteq1 . (Contributed by Peter Mazsa, 5-Oct-2021)

	Ref	Expression
Assertion	disjdmqseqeq1	$$⊢R=S\to \left(\left(DisjR\wedge \mathrm{dom}R/R=A\right)\leftrightarrow \left(DisjS\wedge \mathrm{dom}S/S=A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		disjeq	$$⊢R=S\to \left(DisjR\leftrightarrow DisjS\right)$$
2		dmqseqeq1	$$⊢R=S\to \left(\mathrm{dom}R/R=A\leftrightarrow \mathrm{dom}S/S=A\right)$$
3	1 2	anbi12d	$$⊢R=S\to \left(\left(DisjR\wedge \mathrm{dom}R/R=A\right)\leftrightarrow \left(DisjS\wedge \mathrm{dom}S/S=A\right)\right)$$