Theorem csbing 3370

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 3087	. . 3 |- ( y = A -> [_ y / x ]_ ( C i^i D ) = [_ A / x ]_ ( C i^i D ) )
2		csbeq1 3087	. . . 4 |- ( y = A -> [_ y / x ]_ C = [_ A / x ]_ C )
3		csbeq1 3087	. . . 4 |- ( y = A -> [_ y / x ]_ D = [_ A / x ]_ D )
4	2, 3	ineq12d 3365	. . 3 |- ( y = A -> ( [_ y / x ]_ C i^i [_ y / x ]_ D ) = ( [_ A / x ]_ C i^i [_ A / x ]_ D ) )
5	1, 4	eqeq12d 2211	. 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 2766	. . 3 |- y e. _V
7		nfcsb1v 3117	. . . 4 |- F/_ x [_ y / x ]_ C
8		nfcsb1v 3117	. . . 4 |- F/_ x [_ y / x ]_ D
9	7, 8	nfin 3369	. . 3 |- F/_ x ( [_ y / x ]_ C i^i [_ y / x ]_ D )
10		csbeq1a 3093	. . . 4 |- ( x = y -> C = [_ y / x ]_ C )
11		csbeq1a 3093	. . . 4 |- ( x = y -> D = [_ y / x ]_ D )
12	10, 11	ineq12d 3365	. . 3 |- ( x = y -> ( C i^i D ) = ( [_ y / x ]_ C i^i [_ y / x ]_ D ) )
13	6, 9, 12	csbief 3129	. 2 |- [_ y / x ]_ ( C i^i D ) = ( [_ y / x ]_ C i^i [_ y / x ]_ D )
14	5, 13	vtoclg 2824	1 |- ( A e. B -> [_ A / x ]_ ( C i^i D ) = ( [_ A / x ]_ C i^i [_ A / x ]_ D ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   [_csb 3084   i^i cin 3156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-in 3163

This theorem is referenced by:  csbresg  4949