Theorem csbin 3863

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 3433	. . . 4 |- ( y = A -> [_ y / x ]_ ( B i^i C ) = [_ A / x ]_ ( B i^i C ) )
2		csbeq1 3433	. . . . 5 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
3		csbeq1 3433	. . . . 5 |- ( y = A -> [_ y / x ]_ C = [_ A / x ]_ C )
4	2, 3	ineq12d 3697	. . . 4 |- ( y = A -> ( [_ y / x ]_ B i^i [_ y / x ]_ C ) = ( [_ A / x ]_ B i^i [_ A / x ]_ C ) )
5	1, 4	eqeq12d 2479	. . 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 3112	. . . 4 |- y e. _V
7		nfcsb1v 3446	. . . . 5 |- F/_ x [_ y / x ]_ B
8		nfcsb1v 3446	. . . . 5 |- F/_ x [_ y / x ]_ C
9	7, 8	nfin 3701	. . . 4 |- F/_ x ( [_ y / x ]_ B i^i [_ y / x ]_ C )
10		csbeq1a 3439	. . . . 5 |- ( x = y -> B = [_ y / x ]_ B )
11		csbeq1a 3439	. . . . 5 |- ( x = y -> C = [_ y / x ]_ C )
12	10, 11	ineq12d 3697	. . . 4 |- ( x = y -> ( B i^i C ) = ( [_ y / x ]_ B i^i [_ y / x ]_ C ) )
13	6, 9, 12	csbief 3455	. . 3 |- [_ y / x ]_ ( B i^i C ) = ( [_ y / x ]_ B i^i [_ y / x ]_ C )
14	5, 13	vtoclg 3167	. 2 |- ( A e. _V -> [_ A / x ]_ ( B i^i C ) = ( [_ A / x ]_ B i^i [_ A / x ]_ C ) )
15		csbprc 3830	. . 3 |- ( -. A e. _V -> [_ A / x ]_ ( B i^i C ) = (/) )
16		csbprc 3830	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ B = (/) )
17		csbprc 3830	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ C = (/) )
18	16, 17	ineq12d 3697	. . . 4 |- ( -. A e. _V -> ( [_ A / x ]_ B i^i [_ A / x ]_ C ) = ( (/) i^i (/) ) )
19		in0 3820	. . . 4 |- ( (/) i^i (/) ) = (/)
20	18, 19	syl6req 2515	. . 3 |- ( -. A e. _V -> (/) = ( [_ A / x ]_ B i^i [_ A / x ]_ C ) )
21	15, 20	eqtrd 2498	. 2 |- ( -. A e. _V -> [_ A / x ]_ ( B i^i C ) = ( [_ A / x ]_ B i^i [_ A / x ]_ C ) )
22	14, 21	pm2.61i 164	1 |- [_ A / x ]_ ( B i^i C ) = ( [_ A / x ]_ B i^i [_ A / x ]_ C )

Syntax hints:   -. wn 3   = wceq 1395   e. wcel 1819   _Vcvv 3109   [_csb 3430   i^i cin 3470   (/)c0 3793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-in 3478  df-ss 3485  df-nul 3794

This theorem is referenced by:  csbres  5286  disjxpin  27587  onfrALTlem5  33457  onfrALTlem4  33458