Theorem csbaovg 41260

Description: Move class substitution in and out of an operation. (Contributed by Alexander van der Vekens, 26-May-2017.)

Assertion

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

Proof of Theorem csbaovg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3536	. . 3 |- ( y = A -> [_ y / x ]_ (( B F C)) = [_ A / x ]_ (( B F C)) )
2		csbeq1 3536	. . . 4 |- ( y = A -> [_ y / x ]_ F = [_ A / x ]_ F )
3		csbeq1 3536	. . . 4 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
4		csbeq1 3536	. . . 4 |- ( y = A -> [_ y / x ]_ C = [_ A / x ]_ C )
5	2, 3, 4	aoveq123d 41258	. . 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 2637	. 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 3203	. . 3 |- y e. _V
8		nfcsb1v 3549	. . . 4 |- F/_ x [_ y / x ]_ B
9		nfcsb1v 3549	. . . 4 |- F/_ x [_ y / x ]_ F
10		nfcsb1v 3549	. . . 4 |- F/_ x [_ y / x ]_ C
11	8, 9, 10	nfaov 41259	. . 3 |- F/_ x (( [_ y / x ]_ B [_ y / x ]_ F [_ y / x ]_ C))
12		csbeq1a 3542	. . . 4 |- ( x = y -> F = [_ y / x ]_ F )
13		csbeq1a 3542	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
14		csbeq1a 3542	. . . 4 |- ( x = y -> C = [_ y / x ]_ C )
15	12, 13, 14	aoveq123d 41258	. . 3 |- ( x = y -> (( B F C)) = (( [_ y / x ]_ B [_ y / x ]_ F [_ y / x ]_ C)) )
16	7, 11, 15	csbief 3558	. 2 |- [_ y / x ]_ (( B F C)) = (( [_ y / x ]_ B [_ y / x ]_ F [_ y / x ]_ C))
17	6, 16	vtoclg 3266	1 |- ( A e. D -> [_ A / x ]_ (( B F C)) = (( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C)) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   [_csb 3533   ((caov 41195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-dfat 41196  df-afv 41197  df-aov 41198

This theorem is referenced by: (None)