Theorem bj-csbprc 32714

Description: More direct proof of csbprc 3774 (fewer essential steps). (Contributed by BJ, 24-Jul-2019.)

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 3390	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
2		sbcex 3297	. . . . 5 |- ( [. A / x ]. y e. B -> A e. _V )
3	2	con3i 135	. . . 4 |- ( -. A e. _V -> -. [. A / x ]. y e. B )
4	3	alrimiv 1686	. . 3 |- ( -. A e. _V -> A. y -. [. A / x ]. y e. B )
5		bj-abfal 32712	. . 3 |- ( A. y -. [. A / x ]. y e. B -> { y | [. A / x ]. y e. B } = (/) )
6	4, 5	syl 16	. 2 |- ( -. A e. _V -> { y | [. A / x ]. y e. B } = (/) )
7	1, 6	syl5eq 2504	1 |- ( -. A e. _V -> [_ A / x ]_ B = (/) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1368   = wceq 1370   e. wcel 1758   {cab 2436   _Vcvv 3071   [.wsbc 3287   [_csb 3389   (/)c0 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-nul 3739

This theorem is referenced by: (None)