Theorem csbprc 3980

Description: The proper substitution of a proper class for a set into a class results in the empty set. (Contributed by NM, 17-Aug-2018.) (Proof shortened by JJ, 27-Aug-2021.)

Assertion

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

Proof of Theorem csbprc

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex 3445	. . . 4 |- ( [. A / x ]. y e. B -> A e. _V )
2		falim 1498	. . . 4 |- ( F. -> A e. _V )
3	1, 2	pm5.21ni 367	. . 3 |- ( -. A e. _V -> ( [. A / x ]. y e. B <-> F. ) )
4	3	abbidv 2741	. 2 |- ( -. A e. _V -> { y | [. A / x ]. y e. B } = { y | F. } )
5		df-csb 3534	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
6		fal 1490	. . . 4 |- -. F.
7	6	abf 3978	. . 3 |- { y | F. } = (/)
8	7	eqcomi 2631	. 2 |- (/) = { y | F. }
9	4, 5, 8	3eqtr4g 2681	1 |- ( -. A e. _V -> [_ A / x ]_ B = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   F. wfal 1488   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-fal 1489  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:  csb0  3982  sbcel12  3983  sbcne12  3986  sbcel2  3989  csbidm  4002  csbun  4009  csbin  4010  csbif  4138  csbuni  4466  sbcbr123  4706  sbcbr  4707  csbexg  4792  csbopab  5008  csbxp  5200  csbres  5399  csbima12  5483  csbrn  5596  csbiota  5881  csbfv12  6231  csbfv  6233  csbriota  6623  csbov123  6687  csbov  6688  csbdif  33171