Theorem disjeq2dv

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

	Ref	Expression
Hypothesis	disjeq2dv.1	|- ( ( ph /\ x e. A ) -> B = C )
Assertion	disjeq2dv	|- ( ph -> ( Disj_ x e. A B <-> Disj_ x e. A C ) )

Proof

Step	Hyp	Ref	Expression
1		disjeq2dv.1	|- ( ( ph /\ x e. A ) -> B = C )
2	1	ralrimiva	|- ( ph -> A. x e. A B = C )
3		disjeq2	|- ( A. x e. A B = C -> ( Disj_ x e. A B <-> Disj_ x e. A C ) )
4	2 3	syl	|- ( ph -> ( Disj_ x e. A B <-> Disj_ x e. A C ) )