Theorem csbin 3803

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 3352	. . . 4 |- ( y = A -> [_ y / x ]_ ( B i^i C ) = [_ A / x ]_ ( B i^i C ) )
2		csbeq1 3352	. . . . 5 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
3		csbeq1 3352	. . . . 5 |- ( y = A -> [_ y / x ]_ C = [_ A / x ]_ C )
4	2, 3	ineq12d 3626	. . . 4 |- ( y = A -> ( [_ y / x ]_ B i^i [_ y / x ]_ C ) = ( [_ A / x ]_ B i^i [_ A / x ]_ C ) )
5	1, 4	eqeq12d 2486	. . 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 3034	. . . 4 |- y e. _V
7		nfcsb1v 3365	. . . . 5 |- F/_ x [_ y / x ]_ B
8		nfcsb1v 3365	. . . . 5 |- F/_ x [_ y / x ]_ C
9	7, 8	nfin 3630	. . . 4 |- F/_ x ( [_ y / x ]_ B i^i [_ y / x ]_ C )
10		csbeq1a 3358	. . . . 5 |- ( x = y -> B = [_ y / x ]_ B )
11		csbeq1a 3358	. . . . 5 |- ( x = y -> C = [_ y / x ]_ C )
12	10, 11	ineq12d 3626	. . . 4 |- ( x = y -> ( B i^i C ) = ( [_ y / x ]_ B i^i [_ y / x ]_ C ) )
13	6, 9, 12	csbief 3374	. . 3 |- [_ y / x ]_ ( B i^i C ) = ( [_ y / x ]_ B i^i [_ y / x ]_ C )
14	5, 13	vtoclg 3093	. 2 |- ( A e. _V -> [_ A / x ]_ ( B i^i C ) = ( [_ A / x ]_ B i^i [_ A / x ]_ C ) )
15		csbprc 3774	. . 3 |- ( -. A e. _V -> [_ A / x ]_ ( B i^i C ) = (/) )
16		csbprc 3774	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ B = (/) )
17		csbprc 3774	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ C = (/) )
18	16, 17	ineq12d 3626	. . . 4 |- ( -. A e. _V -> ( [_ A / x ]_ B i^i [_ A / x ]_ C ) = ( (/) i^i (/) ) )
19		in0 3763	. . . 4 |- ( (/) i^i (/) ) = (/)
20	18, 19	syl6req 2522	. . 3 |- ( -. A e. _V -> (/) = ( [_ A / x ]_ B i^i [_ A / x ]_ C ) )
21	15, 20	eqtrd 2505	. 2 |- ( -. A e. _V -> [_ A / x ]_ ( B i^i C ) = ( [_ A / x ]_ B i^i [_ A / x ]_ C ) )
22	14, 21	pm2.61i 169	1 |- [_ A / x ]_ ( B i^i C ) = ( [_ A / x ]_ B i^i [_ A / x ]_ C )

Syntax hints:   -. wn 3   = wceq 1452   e. wcel 1904   _Vcvv 3031   [_csb 3349   i^i cin 3389   (/)c0 3722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-in 3397  df-ss 3404  df-nul 3723

This theorem is referenced by:  csbres  5114  disjxpin  28275  csbpredg  31797  onfrALTlem5  36978  onfrALTlem4  36979  disjinfi  37539