Theorem disjeq1

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

	Ref	Expression
Assertion	disjeq1	$$⊢A=B\to \left(\underset{x\in A}{Disj}C\leftrightarrow \underset{x\in B}{Disj}C\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqimss2	$$⊢A=B\to B\subseteq A$$
2		disjss1	$$⊢B\subseteq A\to \left(\underset{x\in A}{Disj}C\to \underset{x\in B}{Disj}C\right)$$
3	1 2	syl	$$⊢A=B\to \left(\underset{x\in A}{Disj}C\to \underset{x\in B}{Disj}C\right)$$
4		eqimss	$$⊢A=B\to A\subseteq B$$
5		disjss1	$$⊢A\subseteq B\to \left(\underset{x\in B}{Disj}C\to \underset{x\in A}{Disj}C\right)$$
6	4 5	syl	$$⊢A=B\to \left(\underset{x\in B}{Disj}C\to \underset{x\in A}{Disj}C\right)$$
7	3 6	impbid	$$⊢A=B\to \left(\underset{x\in A}{Disj}C\leftrightarrow \underset{x\in B}{Disj}C\right)$$