Theorem csbresgOLD 39055

Description: Distribute proper substitution through the restriction of a class. csbresgOLD 39055 is derived from the virtual deduction proof csbresgVD 39131. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 23-Aug-2018. Use csbres 5399 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 39054	. . 3 |- ( A e. V -> [_ A / x ]_ ( B i^i ( C X. _V ) ) = ( [_ A / x ]_ B i^i [_ A / x ]_ ( C X. _V ) ) )
2		csbxpgOLD 39053	. . . . 5 |- ( A e. V -> [_ A / x ]_ ( C X. _V ) = ( [_ A / x ]_ C X. [_ A / x ]_ _V ) )
3		csbconstg 3546	. . . . . 6 |- ( A e. V -> [_ A / x ]_ _V = _V )
4	3	xpeq2d 5139	. . . . 5 |- ( A e. V -> ( [_ A / x ]_ C X. [_ A / x ]_ _V ) = ( [_ A / x ]_ C X. _V ) )
5	2, 4	eqtrd 2656	. . . 4 |- ( A e. V -> [_ A / x ]_ ( C X. _V ) = ( [_ A / x ]_ C X. _V ) )
6	5	ineq2d 3814	. . 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 2656	. 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 5126	. . 3 |- ( B |` C ) = ( B i^i ( C X. _V ) )
9	8	csbeq2i 3993	. 2 |- [_ A / x ]_ ( B |` C ) = [_ A / x ]_ ( B i^i ( C X. _V ) )
10		df-res 5126	. 2 |- ( [_ A / x ]_ B |` [_ A / x ]_ C ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) )
11	7, 9, 10	3eqtr4g 2681	1 |- ( A e. V -> [_ A / x ]_ ( B |` C ) = ( [_ A / x ]_ B |` [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   i^i cin 3573   X. cxp 5112   |` cres 5116

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  df-opab 4713  df-xp 5120  df-res 5126

This theorem is referenced by:  csbima12gALTOLD  39057  csbima12gALTVD  39133