Theorem disjeq12d

Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016)

	Ref	Expression
Hypotheses	disjeq1d.1	$$⊢\phi \to A=B$$
	disjeq12d.1	$$⊢\phi \to C=D$$
Assertion	disjeq12d	$$⊢\phi \to \left(\underset{x\in A}{Disj}C\leftrightarrow \underset{x\in B}{Disj}D\right)$$

Proof

Step	Hyp	Ref	Expression
1		disjeq1d.1	$$⊢\phi \to A=B$$
2		disjeq12d.1	$$⊢\phi \to C=D$$
3	1	disjeq1d	$$⊢\phi \to \left(\underset{x\in A}{Disj}C\leftrightarrow \underset{x\in B}{Disj}C\right)$$
4	2	adantr	$$⊢\left(\phi \wedge x\in B\right)\to C=D$$
5	4	disjeq2dv	$$⊢\phi \to \left(\underset{x\in B}{Disj}C\leftrightarrow \underset{x\in B}{Disj}D\right)$$
6	3 5	bitrd	$$⊢\phi \to \left(\underset{x\in A}{Disj}C\leftrightarrow \underset{x\in B}{Disj}D\right)$$