Theorem csbingOLD 37254

Description: Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.) Obsolete as of 18-Aug-2018. Use csbin 3810 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 3377	. . 3 |- ( y = A -> [_ y / x ]_ ( C i^i D ) = [_ A / x ]_ ( C i^i D ) )
2		csbeq1 3377	. . . 4 |- ( y = A -> [_ y / x ]_ C = [_ A / x ]_ C )
3		csbeq1 3377	. . . 4 |- ( y = A -> [_ y / x ]_ D = [_ A / x ]_ D )
4	2, 3	ineq12d 3646	. . 3 |- ( y = A -> ( [_ y / x ]_ C i^i [_ y / x ]_ D ) = ( [_ A / x ]_ C i^i [_ A / x ]_ D ) )
5	1, 4	eqeq12d 2476	. 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 3059	. . 3 |- y e. _V
7		nfcsb1v 3390	. . . 4 |- F/_ x [_ y / x ]_ C
8		nfcsb1v 3390	. . . 4 |- F/_ x [_ y / x ]_ D
9	7, 8	nfin 3650	. . 3 |- F/_ x ( [_ y / x ]_ C i^i [_ y / x ]_ D )
10		csbeq1a 3383	. . . 4 |- ( x = y -> C = [_ y / x ]_ C )
11		csbeq1a 3383	. . . 4 |- ( x = y -> D = [_ y / x ]_ D )
12	10, 11	ineq12d 3646	. . 3 |- ( x = y -> ( C i^i D ) = ( [_ y / x ]_ C i^i [_ y / x ]_ D ) )
13	6, 9, 12	csbief 3399	. 2 |- [_ y / x ]_ ( C i^i D ) = ( [_ y / x ]_ C i^i [_ y / x ]_ D )
14	5, 13	vtoclg 3118	1 |- ( A e. B -> [_ A / x ]_ ( C i^i D ) = ( [_ A / x ]_ C i^i [_ A / x ]_ D ) )

Syntax hints:   -> wi 4   = wceq 1454   e. wcel 1897   [_csb 3374   i^i cin 3414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-in 3422

This theorem is referenced by:  csbresgOLD  37255  onfrALTlem5VD  37321  onfrALTlem4VD  37322  csbresgVD  37331