Theorem csbresg 4977

Description: Distribute proper substitution through the restriction of a class. csbresg 4977 is derived from the virtual deduction proof csbresgVD in set.mm. (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 3463	. . 3 |- (A e. V -> [_A / x]_(B i^i (C X. _V)) = ([_A / x]_B i^i [_A / x]_(C X. _V)))
2		csbxpg 4814	. . . . 5 |- (A e. V -> [_A / x]_(C X. _V) = ([_A / x]_C X. [_A / x]__V))
3		csbconstg 3151	. . . . . 6 |- (A e. V -> [_A / x]__V = _V)
4	3	xpeq2d 4809	. . . . 5 |- (A e. V -> ([_A / x]_C X. [_A / x]__V) = ([_A / x]_C X. _V))
5	2, 4	eqtrd 2385	. . . 4 |- (A e. V -> [_A / x]_(C X. _V) = ([_A / x]_C X. _V))
6	5	ineq2d 3458	. . 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 2385	. 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 4789	. . 3 |- (B |` C) = (B i^i (C X. _V))
9	8	csbeq2i 3163	. 2 |- [_A / x]_(B |` C) = [_A / x]_(B i^i (C X. _V))
10		df-res 4789	. 2 |- ([_A / x]_B |` [_A / x]_C) = ([_A / x]_B i^i ([_A / x]_C X. _V))
11	7, 9, 10	3eqtr4g 2410	1 |- (A e. V -> [_A / x]_(B |` C) = ([_A / x]_B |` [_A / x]_C))

Syntax hints:   -> wi 4   = wceq 1642   e. wcel 1710  _Vcvv 2860  [_csb 3137   i^i cin 3209   X. cxp 4771   |` cres 4775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-opab 4624  df-xp 4785  df-res 4789

This theorem is referenced by: (None)