Theorem bj-csbprc 34877

Description: More direct proof of csbprc 3820 (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 3421	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
2		sbcex 3334	. . . . 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 1724	. . 3 |- ( -. A e. _V -> A. y -. [. A / x ]. y e. B )
5		bj-abfal 34875	. . 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 2507	1 |- ( -. A e. _V -> [_ A / x ]_ B = (/) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1396   = wceq 1398   e. wcel 1823   {cab 2439   _Vcvv 3106   [.wsbc 3324   [_csb 3420   (/)c0 3783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-nul 3784

This theorem is referenced by: (None)