Theorem csbingOLD 39054

Description: Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.) Obsolete as of 18-Aug-2018. Use csbin 4010 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem csbingOLD

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3536	. . 3 |- ( y = A -> [_ y / x ]_ ( C i^i D ) = [_ A / x ]_ ( C i^i D ) )
2		csbeq1 3536	. . . 4 |- ( y = A -> [_ y / x ]_ C = [_ A / x ]_ C )
3		csbeq1 3536	. . . 4 |- ( y = A -> [_ y / x ]_ D = [_ A / x ]_ D )
4	2, 3	ineq12d 3815	. . 3 |- ( y = A -> ( [_ y / x ]_ C i^i [_ y / x ]_ D ) = ( [_ A / x ]_ C i^i [_ A / x ]_ D ) )
5	1, 4	eqeq12d 2637	. 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 3203	. . 3 |- y e. _V
7		nfcsb1v 3549	. . . 4 |- F/_ x [_ y / x ]_ C
8		nfcsb1v 3549	. . . 4 |- F/_ x [_ y / x ]_ D
9	7, 8	nfin 3820	. . 3 |- F/_ x ( [_ y / x ]_ C i^i [_ y / x ]_ D )
10		csbeq1a 3542	. . . 4 |- ( x = y -> C = [_ y / x ]_ C )
11		csbeq1a 3542	. . . 4 |- ( x = y -> D = [_ y / x ]_ D )
12	10, 11	ineq12d 3815	. . 3 |- ( x = y -> ( C i^i D ) = ( [_ y / x ]_ C i^i [_ y / x ]_ D ) )
13	6, 9, 12	csbief 3558	. 2 |- [_ y / x ]_ ( C i^i D ) = ( [_ y / x ]_ C i^i [_ y / x ]_ D )
14	5, 13	vtoclg 3266	1 |- ( A e. B -> [_ A / x ]_ ( C i^i D ) = ( [_ A / x ]_ C i^i [_ A / x ]_ D ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   [_csb 3533   i^i cin 3573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-in 3581

This theorem is referenced by:  csbresgOLD  39055  onfrALTlem5VD  39121  onfrALTlem4VD  39122  csbresgVD  39131