Theorem csbov 6688

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

Assertion

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

Distinct variable groups:   x, B   x, C

Allowed substitution hints:   A( x)   F( x)

Proof of Theorem csbov

Step	Hyp	Ref	Expression
1		csbov123 6687	. 2 |- [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C )
2		csbconstg 3546	. . . 4 |- ( A e. _V -> [_ A / x ]_ B = B )
3		csbconstg 3546	. . . 4 |- ( A e. _V -> [_ A / x ]_ C = C )
4	2, 3	oveq12d 6668	. . 3 |- ( A e. _V -> ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) = ( B [_ A / x ]_ F C ) )
5		0fv 6227	. . . . 5 |- ( (/) ` <. B , C >. ) = (/)
6		df-ov 6653	. . . . 5 |- ( B (/) C ) = ( (/) ` <. B , C >. )
7		df-ov 6653	. . . . . 6 |- ( [_ A / x ]_ B (/) [_ A / x ]_ C ) = ( (/) ` <. [_ A / x ]_ B , [_ A / x ]_ C >. )
8		0fv 6227	. . . . . 6 |- ( (/) ` <. [_ A / x ]_ B , [_ A / x ]_ C >. ) = (/)
9	7, 8	eqtri 2644	. . . . 5 |- ( [_ A / x ]_ B (/) [_ A / x ]_ C ) = (/)
10	5, 6, 9	3eqtr4ri 2655	. . . 4 |- ( [_ A / x ]_ B (/) [_ A / x ]_ C ) = ( B (/) C )
11		csbprc 3980	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ F = (/) )
12	11	oveqd 6667	. . . 4 |- ( -. A e. _V -> ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) = ( [_ A / x ]_ B (/) [_ A / x ]_ C ) )
13	11	oveqd 6667	. . . 4 |- ( -. A e. _V -> ( B [_ A / x ]_ F C ) = ( B (/) C ) )
14	10, 12, 13	3eqtr4a 2682	. . 3 |- ( -. A e. _V -> ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) = ( B [_ A / x ]_ F C ) )
15	4, 14	pm2.61i 176	. 2 |- ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) = ( B [_ A / x ]_ F C )
16	1, 15	eqtri 2644	1 |- [_ A / x ]_ ( B F C ) = ( B [_ A / x ]_ F 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:  mptcoe1matfsupp  20607  mp2pm2mplem4  20614