Theorem csbresg 5061

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 3438	. . 3 |- ( A e. V -> [_ A / x ]_ ( B i^i ( C X. _V ) ) = ( [_ A / x ]_ B i^i [_ A / x ]_ ( C X. _V ) ) )
2		csbxpg 4851	. . . . 5 |- ( A e. V -> [_ A / x ]_ ( C X. _V ) = ( [_ A / x ]_ C X. [_ A / x ]_ _V ) )
3		csbconstg 3161	. . . . . 6 |- ( A e. V -> [_ A / x ]_ _V = _V )
4	3	xpeq2d 4793	. . . . 5 |- ( A e. V -> ( [_ A / x ]_ C X. [_ A / x ]_ _V ) = ( [_ A / x ]_ C X. _V ) )
5	2, 4	eqtrd 2271	. . . 4 |- ( A e. V -> [_ A / x ]_ ( C X. _V ) = ( [_ A / x ]_ C X. _V ) )
6	5	ineq2d 3432	. . 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 2271	. 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 4781	. . 3 |- ( B |` C ) = ( B i^i ( C X. _V ) )
9	8	csbeq2i 3174	. 2 |- [_ A / x ]_ ( B |` C ) = [_ A / x ]_ ( B i^i ( C X. _V ) )
10		df-res 4781	. 2 |- ( [_ A / x ]_ B |` [_ A / x ]_ C ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) )
11	7, 9, 10	3eqtr4g 2296	1 |- ( A e. V -> [_ A / x ]_ ( B |` C ) = ( [_ A / x ]_ B |` [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   [_csb 3147   i^i cin 3219   X. cxp 4767   |` cres 4771

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-in 3226  df-opab 4188  df-xp 4775  df-res 4781

This theorem is referenced by: (None)