Theorem csbresgOLD 37255

Description: Distribute proper substitution through the restriction of a class. csbresgOLD 37255 is derived from the virtual deduction proof csbresgVD 37331. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 23-Aug-2018. Use csbres 5126 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
csbresgOLD	|- ( A e. V -> [_ A / x ]_ ( B |` C ) = ( [_ A / x ]_ B |` [_ A / x ]_ C ) )

Proof of Theorem csbresgOLD

Step	Hyp	Ref	Expression
1		csbingOLD 37254	. . 3 |- ( A e. V -> [_ A / x ]_ ( B i^i ( C X. _V ) ) = ( [_ A / x ]_ B i^i [_ A / x ]_ ( C X. _V ) ) )
2		csbxpgOLD 37253	. . . . 5 |- ( A e. V -> [_ A / x ]_ ( C X. _V ) = ( [_ A / x ]_ C X. [_ A / x ]_ _V ) )
3		csbconstg 3387	. . . . . 6 |- ( A e. V -> [_ A / x ]_ _V = _V )
4	3	xpeq2d 4876	. . . . 5 |- ( A e. V -> ( [_ A / x ]_ C X. [_ A / x ]_ _V ) = ( [_ A / x ]_ C X. _V ) )
5	2, 4	eqtrd 2495	. . . 4 |- ( A e. V -> [_ A / x ]_ ( C X. _V ) = ( [_ A / x ]_ C X. _V ) )
6	5	ineq2d 3645	. . 3 |- ( A e. V -> ( [_ A / x ]_ B i^i [_ A / x ]_ ( C X. _V ) ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) ) )
7	1, 6	eqtrd 2495	. 2 |- ( A e. V -> [_ A / x ]_ ( B i^i ( C X. _V ) ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) ) )
8		df-res 4864	. . 3 |- ( B |` C ) = ( B i^i ( C X. _V ) )
9	8	csbeq2i 3793	. 2 |- [_ A / x ]_ ( B |` C ) = [_ A / x ]_ ( B i^i ( C X. _V ) )
10		df-res 4864	. 2 |- ( [_ A / x ]_ B |` [_ A / x ]_ C ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) )
11	7, 9, 10	3eqtr4g 2520	1 |- ( A e. V -> [_ A / x ]_ ( B |` C ) = ( [_ A / x ]_ B |` [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   = wceq 1454   e. wcel 1897   _Vcvv 3056   [_csb 3374   i^i cin 3414   X. cxp 4850   |` cres 4854

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  df-opab 4475  df-xp 4858  df-res 4864

This theorem is referenced by:  csbima12gALTOLD  37257  csbima12gALTVD  37333