Theorem csbresg 5041

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 3428	. . 3 |- ( A e. V -> [_ A / x ]_ ( B i^i ( C X. _V ) ) = ( [_ A / x ]_ B i^i [_ A / x ]_ ( C X. _V ) ) )
2		csbxpg 4831	. . . . 5 |- ( A e. V -> [_ A / x ]_ ( C X. _V ) = ( [_ A / x ]_ C X. [_ A / x ]_ _V ) )
3		csbconstg 3152	. . . . . 6 |- ( A e. V -> [_ A / x ]_ _V = _V )
4	3	xpeq2d 4773	. . . . 5 |- ( A e. V -> ( [_ A / x ]_ C X. [_ A / x ]_ _V ) = ( [_ A / x ]_ C X. _V ) )
5	2, 4	eqtrd 2265	. . . 4 |- ( A e. V -> [_ A / x ]_ ( C X. _V ) = ( [_ A / x ]_ C X. _V ) )
6	5	ineq2d 3422	. . 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 2265	. 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 4761	. . 3 |- ( B |` C ) = ( B i^i ( C X. _V ) )
9	8	csbeq2i 3165	. 2 |- [_ A / x ]_ ( B |` C ) = [_ A / x ]_ ( B i^i ( C X. _V ) )
10		df-res 4761	. 2 |- ( [_ A / x ]_ B |` [_ A / x ]_ C ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) )
11	7, 9, 10	3eqtr4g 2290	1 |- ( A e. V -> [_ A / x ]_ ( B |` C ) = ( [_ A / x ]_ B |` [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   _Vcvv 2813   [_csb 3138   i^i cin 3210   X. cxp 4747   |` cres 4751

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-in 3217  df-opab 4172  df-xp 4755  df-res 4761

This theorem is referenced by: (None)