Theorem csbprc 3571

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 3148	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
2		sbcex 3060	. . . . . . 7 |- ( [. A / x ]. y e. B -> A e. _V )
3	2	con3i 641	. . . . . 6 |- ( -. A e. _V -> -. [. A / x ]. y e. B )
4	3	pm2.21d 628	. . . . 5 |- ( -. A e. _V -> ( [. A / x ]. y e. B -> F. ) )
5		falim 1416	. . . . 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 2358	. . 3 |- ( -. A e. _V -> { y | [. A / x ]. y e. B } = { y | F. } )
8		fal 1409	. . . 4 |- -. F.
9	8	abf 3569	. . 3 |- { y | F. } = (/)
10	7, 9	eqtrdi 2287	. 2 |- ( -. A e. _V -> { y | [. A / x ]. y e. B } = (/) )
11	1, 10	eqtrid 2283	1 |- ( -. A e. _V -> [_ A / x ]_ B = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1402   F. wfal 1407   e. wcel 2209   {cab 2224   _Vcvv 2821   [.wsbc 3051   [_csb 3147   (/)c0 3520

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521

This theorem is referenced by: (None)