Theorem disjeq2dv

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

	Ref	Expression
Hypothesis	disjeq2dv.1	$$⊢\left(\phi \wedge x\in A\right)\to B=C$$
Assertion	disjeq2dv	$$⊢\phi \to \left(\underset{x\in A}{Disj}B\leftrightarrow \underset{x\in A}{Disj}C\right)$$

Proof

Step	Hyp	Ref	Expression
1		disjeq2dv.1	$$⊢\left(\phi \wedge x\in A\right)\to B=C$$
2	1	ralrimiva	$$⊢\phi \to \forall x\in AB=C$$
3		disjeq2	$$⊢\forall x\in AB=C\to \left(\underset{x\in A}{Disj}B\leftrightarrow \underset{x\in A}{Disj}C\right)$$
4	2 3	syl	$$⊢\phi \to \left(\underset{x\in A}{Disj}B\leftrightarrow \underset{x\in A}{Disj}C\right)$$