Theorem csbin 4010

Description: Distribute proper substitution into a class through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.) (Revised by NM, 18-Aug-2018.)

Assertion

Ref	Expression
csbin	|- [_ A / x ]_ ( B i^i C ) = ( [_ A / x ]_ B i^i [_ A / x ]_ C )

Proof of Theorem csbin

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3536	. . . 4 |- ( y = A -> [_ y / x ]_ ( B i^i C ) = [_ A / x ]_ ( B i^i C ) )
2		csbeq1 3536	. . . . 5 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
3		csbeq1 3536	. . . . 5 |- ( y = A -> [_ y / x ]_ C = [_ A / x ]_ C )
4	2, 3	ineq12d 3815	. . . 4 |- ( y = A -> ( [_ y / x ]_ B i^i [_ y / x ]_ C ) = ( [_ A / x ]_ B i^i [_ A / x ]_ C ) )
5	1, 4	eqeq12d 2637	. . 3 |- ( y = A -> ( [_ y / x ]_ ( B i^i C ) = ( [_ y / x ]_ B i^i [_ y / x ]_ C ) <-> [_ A / x ]_ ( B i^i C ) = ( [_ A / x ]_ B i^i [_ A / x ]_ C ) ) )
6		vex 3203	. . . 4 |- y e. _V
7		nfcsb1v 3549	. . . . 5 |- F/_ x [_ y / x ]_ B
8		nfcsb1v 3549	. . . . 5 |- F/_ x [_ y / x ]_ C
9	7, 8	nfin 3820	. . . 4 |- F/_ x ( [_ y / x ]_ B i^i [_ y / x ]_ C )
10		csbeq1a 3542	. . . . 5 |- ( x = y -> B = [_ y / x ]_ B )
11		csbeq1a 3542	. . . . 5 |- ( x = y -> C = [_ y / x ]_ C )
12	10, 11	ineq12d 3815	. . . 4 |- ( x = y -> ( B i^i C ) = ( [_ y / x ]_ B i^i [_ y / x ]_ C ) )
13	6, 9, 12	csbief 3558	. . 3 |- [_ y / x ]_ ( B i^i C ) = ( [_ y / x ]_ B i^i [_ y / x ]_ C )
14	5, 13	vtoclg 3266	. 2 |- ( A e. _V -> [_ A / x ]_ ( B i^i C ) = ( [_ A / x ]_ B i^i [_ A / x ]_ C ) )
15		csbprc 3980	. . 3 |- ( -. A e. _V -> [_ A / x ]_ ( B i^i C ) = (/) )
16		csbprc 3980	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ B = (/) )
17		csbprc 3980	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ C = (/) )
18	16, 17	ineq12d 3815	. . . 4 |- ( -. A e. _V -> ( [_ A / x ]_ B i^i [_ A / x ]_ C ) = ( (/) i^i (/) ) )
19		in0 3968	. . . 4 |- ( (/) i^i (/) ) = (/)
20	18, 19	syl6req 2673	. . 3 |- ( -. A e. _V -> (/) = ( [_ A / x ]_ B i^i [_ A / x ]_ C ) )
21	15, 20	eqtrd 2656	. 2 |- ( -. A e. _V -> [_ A / x ]_ ( B i^i C ) = ( [_ A / x ]_ B i^i [_ A / x ]_ C ) )
22	14, 21	pm2.61i 176	1 |- [_ A / x ]_ ( B i^i C ) = ( [_ A / x ]_ B i^i [_ A / x ]_ C )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   i^i cin 3573   (/)c0 3915

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-fal 1489  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-in 3581  df-nul 3916

This theorem is referenced by:  csbres  5399  disjxpin  29401  csbpredg  33172  onfrALTlem5  38757  onfrALTlem4  38758  disjinfi  39380