Theorem disjeq1i

Description: Equality theorem for disjoint collection. Inference version. (Contributed by GG, 1-Sep-2025)

	Ref	Expression
Hypothesis	disjeq1i.1	|- A = B
Assertion	disjeq1i	|- ( Disj_ x e. A C <-> Disj_ x e. B C )

Proof

Step	Hyp	Ref	Expression
1		disjeq1i.1	|- A = B
2	1	rmoeqi	|- ( E* x e. A t e. C <-> E* x e. B t e. C )
3	2	albii	|- ( A. t E* x e. A t e. C <-> A. t E* x e. B t e. C )
4		df-disj	|- ( Disj_ x e. A C <-> A. t E* x e. A t e. C )
5		df-disj	|- ( Disj_ x e. B C <-> A. t E* x e. B t e. C )
6	3 4 5	3bitr4i	|- ( Disj_ x e. A C <-> Disj_ x e. B C )