Theorem bj-csbprc

Description: More direct proof of csbprc (fewer essential steps). (Contributed by BJ, 24-Jul-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-csbprc	|- ( -. A e. _V -> [_ A / x ]_ B = (/) )

Proof

Step	Hyp	Ref	Expression
1		df-csb	|- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
2		sbcex	|- ( [. A / x ]. y e. B -> A e. _V )
3	2	con3i	|- ( -. A e. _V -> -. [. A / x ]. y e. B )
4	3	alrimiv	|- ( -. A e. _V -> A. y -. [. A / x ]. y e. B )
5		bj-ab0	|- ( A. y -. [. A / x ]. y e. B -> { y | [. A / x ]. y e. B } = (/) )
6	4 5	syl	|- ( -. A e. _V -> { y | [. A / x ]. y e. B } = (/) )
7	1 6	syl5eq	|- ( -. A e. _V -> [_ A / x ]_ B = (/) )