Theorem csbov123 6687

Description: Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) (Revised by NM, 23-Aug-2018.)

Assertion

Ref	Expression
csbov123	|- [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C )

Proof of Theorem csbov123

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3536	. . . 4 |- ( y = A -> [_ y / x ]_ ( B F C ) = [_ A / x ]_ ( B F C ) )
2		csbeq1 3536	. . . . 5 |- ( y = A -> [_ y / x ]_ F = [_ A / x ]_ F )
3		csbeq1 3536	. . . . 5 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
4		csbeq1 3536	. . . . 5 |- ( y = A -> [_ y / x ]_ C = [_ A / x ]_ C )
5	2, 3, 4	oveq123d 6671	. . . 4 |- ( y = A -> ( [_ y / x ]_ B [_ y / x ]_ F [_ y / x ]_ C ) = ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) )
6	1, 5	eqeq12d 2637	. . 3 |- ( y = A -> ( [_ y / x ]_ ( B F C ) = ( [_ y / x ]_ B [_ y / x ]_ F [_ y / x ]_ C ) <-> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) ) )
7		vex 3203	. . . 4 |- y e. _V
8		nfcsb1v 3549	. . . . 5 |- F/_ x [_ y / x ]_ B
9		nfcsb1v 3549	. . . . 5 |- F/_ x [_ y / x ]_ F
10		nfcsb1v 3549	. . . . 5 |- F/_ x [_ y / x ]_ C
11	8, 9, 10	nfov 6676	. . . 4 |- F/_ x ( [_ y / x ]_ B [_ y / x ]_ F [_ y / x ]_ C )
12		csbeq1a 3542	. . . . 5 |- ( x = y -> F = [_ y / x ]_ F )
13		csbeq1a 3542	. . . . 5 |- ( x = y -> B = [_ y / x ]_ B )
14		csbeq1a 3542	. . . . 5 |- ( x = y -> C = [_ y / x ]_ C )
15	12, 13, 14	oveq123d 6671	. . . 4 |- ( x = y -> ( B F C ) = ( [_ y / x ]_ B [_ y / x ]_ F [_ y / x ]_ C ) )
16	7, 11, 15	csbief 3558	. . 3 |- [_ y / x ]_ ( B F C ) = ( [_ y / x ]_ B [_ y / x ]_ F [_ y / x ]_ C )
17	6, 16	vtoclg 3266	. 2 |- ( A e. _V -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) )
18		csbprc 3980	. . 3 |- ( -. A e. _V -> [_ A / x ]_ ( B F C ) = (/) )
19		df-ov 6653	. . . 4 |- ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) = ( [_ A / x ]_ F ` <. [_ A / x ]_ B , [_ A / x ]_ C >. )
20		csbprc 3980	. . . . . 6 |- ( -. A e. _V -> [_ A / x ]_ F = (/) )
21	20	fveq1d 6193	. . . . 5 |- ( -. A e. _V -> ( [_ A / x ]_ F ` <. [_ A / x ]_ B , [_ A / x ]_ C >. ) = ( (/) ` <. [_ A / x ]_ B , [_ A / x ]_ C >. ) )
22		0fv 6227	. . . . 5 |- ( (/) ` <. [_ A / x ]_ B , [_ A / x ]_ C >. ) = (/)
23	21, 22	syl6eq 2672	. . . 4 |- ( -. A e. _V -> ( [_ A / x ]_ F ` <. [_ A / x ]_ B , [_ A / x ]_ C >. ) = (/) )
24	19, 23	syl5req 2669	. . 3 |- ( -. A e. _V -> (/) = ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) )
25	18, 24	eqtrd 2656	. 2 |- ( -. A e. _V -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) )
26	17, 25	pm2.61i 176	1 |- [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   (/)c0 3915   <.cop 4183   ` cfv 5888  (class class class)co 6650

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789  ax-pow 4843

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  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-uni 4437  df-br 4654  df-dm 5124  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  csbov  6688  csbov12g  6689  relowlpssretop  33212  rdgeqoa  33218