Theorem csbcog 37941

Description: Distribute proper substitution through a composition of relations. (Contributed by RP, 28-Jun-2020.)

Assertion

Ref	Expression
csbcog	|- ( A e. V -> [_ A / x ]_ ( B o. C ) = ( [_ A / x ]_ B o. [_ A / x ]_ C ) )

Proof of Theorem csbcog

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3536	. . 3 |- ( y = A -> [_ y / x ]_ ( B o. C ) = [_ A / x ]_ ( B o. C ) )
2		csbeq1 3536	. . . 4 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
3		csbeq1 3536	. . . 4 |- ( y = A -> [_ y / x ]_ C = [_ A / x ]_ C )
4	2, 3	coeq12d 5286	. . 3 |- ( y = A -> ( [_ y / x ]_ B o. [_ y / x ]_ C ) = ( [_ A / x ]_ B o. [_ A / x ]_ C ) )
5	1, 4	eqeq12d 2637	. 2 |- ( y = A -> ( [_ y / x ]_ ( B o. C ) = ( [_ y / x ]_ B o. [_ y / x ]_ C ) <-> [_ A / x ]_ ( B o. C ) = ( [_ A / x ]_ B o. [_ A / x ]_ C ) ) )
6		vex 3203	. . 3 |- y e. _V
7		nfcsb1v 3549	. . . 4 |- F/_ x [_ y / x ]_ B
8		nfcsb1v 3549	. . . 4 |- F/_ x [_ y / x ]_ C
9	7, 8	nfco 5287	. . 3 |- F/_ x ( [_ y / x ]_ B o. [_ y / x ]_ C )
10		csbeq1a 3542	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
11		csbeq1a 3542	. . . 4 |- ( x = y -> C = [_ y / x ]_ C )
12	10, 11	coeq12d 5286	. . 3 |- ( x = y -> ( B o. C ) = ( [_ y / x ]_ B o. [_ y / x ]_ C ) )
13	6, 9, 12	csbief 3558	. 2 |- [_ y / x ]_ ( B o. C ) = ( [_ y / x ]_ B o. [_ y / x ]_ C )
14	5, 13	vtoclg 3266	1 |- ( A e. V -> [_ A / x ]_ ( B o. C ) = ( [_ A / x ]_ B o. [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   [_csb 3533   o. ccom 5118

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-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-br 4654  df-opab 4713  df-co 5123

This theorem is referenced by:  brtrclfv2  38019