Theorem disjeq1d

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

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

Proof

Step	Hyp	Ref	Expression
1		disjeq1d.1	$$⊢\phi \to A=B$$
2		disjeq1	$$⊢A=B\to \left(\underset{x\in A}{Disj}C\leftrightarrow \underset{x\in B}{Disj}C\right)$$
3	1 2	syl	$$⊢\phi \to \left(\underset{x\in A}{Disj}C\leftrightarrow \underset{x\in B}{Disj}C\right)$$