Theorem csbres 5399

Description: Distribute proper substitution through the restriction of a class. (Contributed by Alan Sare, 10-Nov-2012.) (Revised by NM, 23-Aug-2018.)

Assertion

Ref	Expression
csbres	|- [_ A / x ]_ ( B |` C ) = ( [_ A / x ]_ B |` [_ A / x ]_ C )

Proof of Theorem csbres

Step	Hyp	Ref	Expression
1		df-res 5126	. . 3 |- ( B |` C ) = ( B i^i ( C X. _V ) )
2	1	csbeq2i 3993	. 2 |- [_ A / x ]_ ( B |` C ) = [_ A / x ]_ ( B i^i ( C X. _V ) )
3		csbxp 5200	. . . . . 6 |- [_ A / x ]_ ( C X. _V ) = ( [_ A / x ]_ C X. [_ A / x ]_ _V )
4		csbconstg 3546	. . . . . . 7 |- ( A e. _V -> [_ A / x ]_ _V = _V )
5	4	xpeq2d 5139	. . . . . 6 |- ( A e. _V -> ( [_ A / x ]_ C X. [_ A / x ]_ _V ) = ( [_ A / x ]_ C X. _V ) )
6	3, 5	syl5eq 2668	. . . . 5 |- ( A e. _V -> [_ A / x ]_ ( C X. _V ) = ( [_ A / x ]_ C X. _V ) )
7		0xp 5199	. . . . . . 7 |- ( (/) X. _V ) = (/)
8	7	a1i 11	. . . . . 6 |- ( -. A e. _V -> ( (/) X. _V ) = (/) )
9		csbprc 3980	. . . . . . 7 |- ( -. A e. _V -> [_ A / x ]_ C = (/) )
10	9	xpeq1d 5138	. . . . . 6 |- ( -. A e. _V -> ( [_ A / x ]_ C X. _V ) = ( (/) X. _V ) )
11		csbprc 3980	. . . . . 6 |- ( -. A e. _V -> [_ A / x ]_ ( C X. _V ) = (/) )
12	8, 10, 11	3eqtr4rd 2667	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ ( C X. _V ) = ( [_ A / x ]_ C X. _V ) )
13	6, 12	pm2.61i 176	. . . 4 |- [_ A / x ]_ ( C X. _V ) = ( [_ A / x ]_ C X. _V )
14	13	ineq2i 3811	. . 3 |- ( [_ A / x ]_ B i^i [_ A / x ]_ ( C X. _V ) ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) )
15		csbin 4010	. . 3 |- [_ A / x ]_ ( B i^i ( C X. _V ) ) = ( [_ A / x ]_ B i^i [_ A / x ]_ ( C X. _V ) )
16		df-res 5126	. . 3 |- ( [_ A / x ]_ B |` [_ A / x ]_ C ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) )
17	14, 15, 16	3eqtr4i 2654	. 2 |- [_ A / x ]_ ( B i^i ( C X. _V ) ) = ( [_ A / x ]_ B |` [_ A / x ]_ C )
18	2, 17	eqtri 2644	1 |- [_ A / x ]_ ( B |` C ) = ( [_ A / x ]_ B |` [_ A / x ]_ C )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   i^i cin 3573   (/)c0 3915   X. cxp 5112   |` cres 5116

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120  df-res 5126

This theorem is referenced by:  csbwrecsg  33173