Theorem csbing 3278

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 3001	. . 3 |- ( y = A -> [_ y / x ]_ ( C i^i D ) = [_ A / x ]_ ( C i^i D ) )
2		csbeq1 3001	. . . 4 |- ( y = A -> [_ y / x ]_ C = [_ A / x ]_ C )
3		csbeq1 3001	. . . 4 |- ( y = A -> [_ y / x ]_ D = [_ A / x ]_ D )
4	2, 3	ineq12d 3273	. . 3 |- ( y = A -> ( [_ y / x ]_ C i^i [_ y / x ]_ D ) = ( [_ A / x ]_ C i^i [_ A / x ]_ D ) )
5	1, 4	eqeq12d 2152	. 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 2684	. . 3 |- y e. _V
7		nfcsb1v 3030	. . . 4 |- F/_ x [_ y / x ]_ C
8		nfcsb1v 3030	. . . 4 |- F/_ x [_ y / x ]_ D
9	7, 8	nfin 3277	. . 3 |- F/_ x ( [_ y / x ]_ C i^i [_ y / x ]_ D )
10		csbeq1a 3007	. . . 4 |- ( x = y -> C = [_ y / x ]_ C )
11		csbeq1a 3007	. . . 4 |- ( x = y -> D = [_ y / x ]_ D )
12	10, 11	ineq12d 3273	. . 3 |- ( x = y -> ( C i^i D ) = ( [_ y / x ]_ C i^i [_ y / x ]_ D ) )
13	6, 9, 12	csbief 3039	. 2 |- [_ y / x ]_ ( C i^i D ) = ( [_ y / x ]_ C i^i [_ y / x ]_ D )
14	5, 13	vtoclg 2741	1 |- ( A e. B -> [_ A / x ]_ ( C i^i D ) = ( [_ A / x ]_ C i^i [_ A / x ]_ D ) )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   [_csb 2998   i^i cin 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-in 3072

This theorem is referenced by:  csbresg  4817