Theorem csbresg 4949

Description: Distribute proper substitution through the restriction of a class. (Contributed by Alan Sare, 10-Nov-2012.)

Assertion

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

Proof of Theorem csbresg

Step	Hyp	Ref	Expression
1		csbing 3370	. . 3 |- ( A e. V -> [_ A / x ]_ ( B i^i ( C X. _V ) ) = ( [_ A / x ]_ B i^i [_ A / x ]_ ( C X. _V ) ) )
2		csbxpg 4744	. . . . 5 |- ( A e. V -> [_ A / x ]_ ( C X. _V ) = ( [_ A / x ]_ C X. [_ A / x ]_ _V ) )
3		csbconstg 3098	. . . . . 6 |- ( A e. V -> [_ A / x ]_ _V = _V )
4	3	xpeq2d 4687	. . . . 5 |- ( A e. V -> ( [_ A / x ]_ C X. [_ A / x ]_ _V ) = ( [_ A / x ]_ C X. _V ) )
5	2, 4	eqtrd 2229	. . . 4 |- ( A e. V -> [_ A / x ]_ ( C X. _V ) = ( [_ A / x ]_ C X. _V ) )
6	5	ineq2d 3364	. . 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 2229	. 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 4675	. . 3 |- ( B |` C ) = ( B i^i ( C X. _V ) )
9	8	csbeq2i 3111	. 2 |- [_ A / x ]_ ( B |` C ) = [_ A / x ]_ ( B i^i ( C X. _V ) )
10		df-res 4675	. 2 |- ( [_ A / x ]_ B |` [_ A / x ]_ C ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) )
11	7, 9, 10	3eqtr4g 2254	1 |- ( A e. V -> [_ A / x ]_ ( B |` C ) = ( [_ A / x ]_ B |` [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   _Vcvv 2763   [_csb 3084   i^i cin 3156   X. cxp 4661   |` cres 4665

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  df-opab 4095  df-xp 4669  df-res 4675

This theorem is referenced by: (None)