Theorem csbov123g 5956

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 3083	. . 3 |- ( y = A -> [_ y / x ]_ ( B F C ) = [_ A / x ]_ ( B F C ) )
2		csbeq1 3083	. . . 4 |- ( y = A -> [_ y / x ]_ F = [_ A / x ]_ F )
3		csbeq1 3083	. . . 4 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
4		csbeq1 3083	. . . 4 |- ( y = A -> [_ y / x ]_ C = [_ A / x ]_ C )
5	2, 3, 4	oveq123d 5939	. . 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 2208	. 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 2763	. . 3 |- y e. _V
8		nfcsb1v 3113	. . . 4 |- F/_ x [_ y / x ]_ B
9		nfcsb1v 3113	. . . 4 |- F/_ x [_ y / x ]_ F
10		nfcsb1v 3113	. . . 4 |- F/_ x [_ y / x ]_ C
11	8, 9, 10	nfov 5948	. . 3 |- F/_ x ( [_ y / x ]_ B [_ y / x ]_ F [_ y / x ]_ C )
12		csbeq1a 3089	. . . 4 |- ( x = y -> F = [_ y / x ]_ F )
13		csbeq1a 3089	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
14		csbeq1a 3089	. . . 4 |- ( x = y -> C = [_ y / x ]_ C )
15	12, 13, 14	oveq123d 5939	. . 3 |- ( x = y -> ( B F C ) = ( [_ y / x ]_ B [_ y / x ]_ F [_ y / x ]_ C ) )
16	7, 11, 15	csbief 3125	. 2 |- [_ y / x ]_ ( B F C ) = ( [_ y / x ]_ B [_ y / x ]_ F [_ y / x ]_ C )
17	6, 16	vtoclg 2820	1 |- ( A e. D -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   [_csb 3080  (class class class)co 5918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fv 5262  df-ov 5921

This theorem is referenced by:  csbov12g  5957