Theorem csbov123g 5802

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 3001	. . 3 |- ( y = A -> [_ y / x ]_ ( B F C ) = [_ A / x ]_ ( B F C ) )
2		csbeq1 3001	. . . 4 |- ( y = A -> [_ y / x ]_ F = [_ A / x ]_ F )
3		csbeq1 3001	. . . 4 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
4		csbeq1 3001	. . . 4 |- ( y = A -> [_ y / x ]_ C = [_ A / x ]_ C )
5	2, 3, 4	oveq123d 5788	. . 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 2152	. 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 2684	. . 3 |- y e. _V
8		nfcsb1v 3030	. . . 4 |- F/_ x [_ y / x ]_ B
9		nfcsb1v 3030	. . . 4 |- F/_ x [_ y / x ]_ F
10		nfcsb1v 3030	. . . 4 |- F/_ x [_ y / x ]_ C
11	8, 9, 10	nfov 5794	. . 3 |- F/_ x ( [_ y / x ]_ B [_ y / x ]_ F [_ y / x ]_ C )
12		csbeq1a 3007	. . . 4 |- ( x = y -> F = [_ y / x ]_ F )
13		csbeq1a 3007	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
14		csbeq1a 3007	. . . 4 |- ( x = y -> C = [_ y / x ]_ C )
15	12, 13, 14	oveq123d 5788	. . 3 |- ( x = y -> ( B F C ) = ( [_ y / x ]_ B [_ y / x ]_ F [_ y / x ]_ C ) )
16	7, 11, 15	csbief 3039	. 2 |- [_ y / x ]_ ( B F C ) = ( [_ y / x ]_ B [_ y / x ]_ F [_ y / x ]_ C )
17	6, 16	vtoclg 2741	1 |- ( A e. D -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   [_csb 2998  (class class class)co 5767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-iota 5083  df-fv 5126  df-ov 5770

This theorem is referenced by:  csbov12g  5803