Theorem csbprc 3537

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.)

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		df-csb 3125	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
2		sbcex 3037	. . . . . . 7 |- ( [. A / x ]. y e. B -> A e. _V )
3	2	con3i 635	. . . . . 6 |- ( -. A e. _V -> -. [. A / x ]. y e. B )
4	3	pm2.21d 622	. . . . 5 |- ( -. A e. _V -> ( [. A / x ]. y e. B -> F. ) )
5		falim 1409	. . . . 5 |- ( F. -> [. A / x ]. y e. B )
6	4, 5	impbid1 142	. . . 4 |- ( -. A e. _V -> ( [. A / x ]. y e. B <-> F. ) )
7	6	abbidv 2347	. . 3 |- ( -. A e. _V -> { y | [. A / x ]. y e. B } = { y | F. } )
8		fal 1402	. . . 4 |- -. F.
9	8	abf 3535	. . 3 |- { y | F. } = (/)
10	7, 9	eqtrdi 2278	. 2 |- ( -. A e. _V -> { y | [. A / x ]. y e. B } = (/) )
11	1, 10	eqtrid 2274	1 |- ( -. A e. _V -> [_ A / x ]_ B = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1395   F. wfal 1400   e. wcel 2200   {cab 2215   _Vcvv 2799   [.wsbc 3028   [_csb 3124   (/)c0 3491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-in 3203  df-ss 3210  df-nul 3492

This theorem is referenced by: (None)