Theorem bj-csbprc 32904

Description: More direct proof of csbprc 3980 (fewer essential steps). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem bj-csbprc

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3534	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
2		sbcex 3445	. . . . 5 |- ( [. A / x ]. y e. B -> A e. _V )
3	2	con3i 150	. . . 4 |- ( -. A e. _V -> -. [. A / x ]. y e. B )
4	3	alrimiv 1855	. . 3 |- ( -. A e. _V -> A. y -. [. A / x ]. y e. B )
5		bj-ab0 32902	. . 3 |- ( A. y -. [. A / x ]. y e. B -> { y | [. A / x ]. y e. B } = (/) )
6	4, 5	syl 17	. 2 |- ( -. A e. _V -> { y | [. A / x ]. y e. B } = (/) )
7	1, 6	syl5eq 2668	1 |- ( -. A e. _V -> [_ A / x ]_ B = (/) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   [.wsbc 3435   [_csb 3533   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-nul 3916

This theorem is referenced by: (None)