Theorem csbov123g 5915

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 3062	. . 3 |- ( y = A -> [_ y / x ]_ ( B F C ) = [_ A / x ]_ ( B F C ) )
2		csbeq1 3062	. . . 4 |- ( y = A -> [_ y / x ]_ F = [_ A / x ]_ F )
3		csbeq1 3062	. . . 4 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
4		csbeq1 3062	. . . 4 |- ( y = A -> [_ y / x ]_ C = [_ A / x ]_ C )
5	2, 3, 4	oveq123d 5898	. . 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 2192	. 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 2742	. . 3 |- y e. _V
8		nfcsb1v 3092	. . . 4 |- F/_ x [_ y / x ]_ B
9		nfcsb1v 3092	. . . 4 |- F/_ x [_ y / x ]_ F
10		nfcsb1v 3092	. . . 4 |- F/_ x [_ y / x ]_ C
11	8, 9, 10	nfov 5907	. . 3 |- F/_ x ( [_ y / x ]_ B [_ y / x ]_ F [_ y / x ]_ C )
12		csbeq1a 3068	. . . 4 |- ( x = y -> F = [_ y / x ]_ F )
13		csbeq1a 3068	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
14		csbeq1a 3068	. . . 4 |- ( x = y -> C = [_ y / x ]_ C )
15	12, 13, 14	oveq123d 5898	. . 3 |- ( x = y -> ( B F C ) = ( [_ y / x ]_ B [_ y / x ]_ F [_ y / x ]_ C ) )
16	7, 11, 15	csbief 3103	. 2 |- [_ y / x ]_ ( B F C ) = ( [_ y / x ]_ B [_ y / x ]_ F [_ y / x ]_ C )
17	6, 16	vtoclg 2799	1 |- ( A e. D -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   [_csb 3059  (class class class)co 5877

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-ov 5880

This theorem is referenced by:  csbov12g  5916