Theorem csbing 3173

Description: Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.)

Assertion

Ref	Expression
csbing	|- ( A e. B -> [_ A / x ]_ ( C i^i D ) = ( [_ A / x ]_ C i^i [_ A / x ]_ D ) )

Proof of Theorem csbing

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 2911	. . 3 |- ( y = A -> [_ y / x ]_ ( C i^i D ) = [_ A / x ]_ ( C i^i D ) )
2		csbeq1 2911	. . . 4 |- ( y = A -> [_ y / x ]_ C = [_ A / x ]_ C )
3		csbeq1 2911	. . . 4 |- ( y = A -> [_ y / x ]_ D = [_ A / x ]_ D )
4	2, 3	ineq12d 3168	. . 3 |- ( y = A -> ( [_ y / x ]_ C i^i [_ y / x ]_ D ) = ( [_ A / x ]_ C i^i [_ A / x ]_ D ) )
5	1, 4	eqeq12d 2095	. 2 |- ( y = A -> ( [_ y / x ]_ ( C i^i D ) = ( [_ y / x ]_ C i^i [_ y / x ]_ D ) <-> [_ A / x ]_ ( C i^i D ) = ( [_ A / x ]_ C i^i [_ A / x ]_ D ) ) )
6		vex 2604	. . 3 |- y e. _V
7		nfcsb1v 2938	. . . 4 |- F/_ x [_ y / x ]_ C
8		nfcsb1v 2938	. . . 4 |- F/_ x [_ y / x ]_ D
9	7, 8	nfin 3172	. . 3 |- F/_ x ( [_ y / x ]_ C i^i [_ y / x ]_ D )
10		csbeq1a 2916	. . . 4 |- ( x = y -> C = [_ y / x ]_ C )
11		csbeq1a 2916	. . . 4 |- ( x = y -> D = [_ y / x ]_ D )
12	10, 11	ineq12d 3168	. . 3 |- ( x = y -> ( C i^i D ) = ( [_ y / x ]_ C i^i [_ y / x ]_ D ) )
13	6, 9, 12	csbief 2947	. 2 |- [_ y / x ]_ ( C i^i D ) = ( [_ y / x ]_ C i^i [_ y / x ]_ D )
14	5, 13	vtoclg 2658	1 |- ( A e. B -> [_ A / x ]_ ( C i^i D ) = ( [_ A / x ]_ C i^i [_ A / x ]_ D ) )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   [_csb 2908   i^i cin 2972

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-in 2979

This theorem is referenced by:  csbresg  4633