Theorem csbov123g 5543

Description: Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)

Assertion

Ref	Expression
csbov123g	|- ( A e. D -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) )

Proof of Theorem csbov123g

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 2855	. . 3 |- ( y = A -> [_ y / x ]_ ( B F C ) = [_ A / x ]_ ( B F C ) )
2		csbeq1 2855	. . . 4 |- ( y = A -> [_ y / x ]_ F = [_ A / x ]_ F )
3		csbeq1 2855	. . . 4 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
4		csbeq1 2855	. . . 4 |- ( y = A -> [_ y / x ]_ C = [_ A / x ]_ C )
5	2, 3, 4	oveq123d 5533	. . 3 |- ( y = A -> ( [_ y / x ]_ B [_ y / x ]_ F [_ y / x ]_ C ) = ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) )
6	1, 5	eqeq12d 2054	. 2 |- ( 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 2560	. . 3 |- y e. _V
8		nfcsb1v 2882	. . . 4 |- F/_ x [_ y / x ]_ B
9		nfcsb1v 2882	. . . 4 |- F/_ x [_ y / x ]_ F
10		nfcsb1v 2882	. . . 4 |- F/_ x [_ y / x ]_ C
11	8, 9, 10	nfov 5535	. . 3 |- F/_ x ( [_ y / x ]_ B [_ y / x ]_ F [_ y / x ]_ C )
12		csbeq1a 2860	. . . 4 |- ( x = y -> F = [_ y / x ]_ F )
13		csbeq1a 2860	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
14		csbeq1a 2860	. . . 4 |- ( x = y -> C = [_ y / x ]_ C )
15	12, 13, 14	oveq123d 5533	. . 3 |- ( x = y -> ( B F C ) = ( [_ y / x ]_ B [_ y / x ]_ F [_ y / x ]_ C ) )
16	7, 11, 15	csbief 2891	. 2 |- [_ y / x ]_ ( B F C ) = ( [_ y / x ]_ B [_ y / x ]_ F [_ y / x ]_ C )
17	6, 16	vtoclg 2613	1 |- ( A e. D -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393   [_csb 2852  (class class class)co 5512

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515

This theorem is referenced by:  csbov12g  5544