Theorem csbov123g 5701

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 2937	. . 3 |- ( y = A -> [_ y / x ]_ ( B F C ) = [_ A / x ]_ ( B F C ) )
2		csbeq1 2937	. . . 4 |- ( y = A -> [_ y / x ]_ F = [_ A / x ]_ F )
3		csbeq1 2937	. . . 4 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
4		csbeq1 2937	. . . 4 |- ( y = A -> [_ y / x ]_ C = [_ A / x ]_ C )
5	2, 3, 4	oveq123d 5687	. . 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 2103	. 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 2623	. . 3 |- y e. _V
8		nfcsb1v 2964	. . . 4 |- F/_ x [_ y / x ]_ B
9		nfcsb1v 2964	. . . 4 |- F/_ x [_ y / x ]_ F
10		nfcsb1v 2964	. . . 4 |- F/_ x [_ y / x ]_ C
11	8, 9, 10	nfov 5693	. . 3 |- F/_ x ( [_ y / x ]_ B [_ y / x ]_ F [_ y / x ]_ C )
12		csbeq1a 2942	. . . 4 |- ( x = y -> F = [_ y / x ]_ F )
13		csbeq1a 2942	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
14		csbeq1a 2942	. . . 4 |- ( x = y -> C = [_ y / x ]_ C )
15	12, 13, 14	oveq123d 5687	. . 3 |- ( x = y -> ( B F C ) = ( [_ y / x ]_ B [_ y / x ]_ F [_ y / x ]_ C ) )
16	7, 11, 15	csbief 2973	. 2 |- [_ y / x ]_ ( B F C ) = ( [_ y / x ]_ B [_ y / x ]_ F [_ y / x ]_ C )
17	6, 16	vtoclg 2680	1 |- ( A e. D -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   = wceq 1290   e. wcel 1439   [_csb 2934  (class class class)co 5666

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-v 2622  df-sbc 2842  df-csb 2935  df-un 3004  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-iota 4993  df-fv 5036  df-ov 5669

This theorem is referenced by:  csbov12g  5702