Theorem csbresgVD 31465

Description: Virtual deduction proof of csbresgOLD 5110. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbresgOLD 5110 is csbresgVD 31465 without virtual deductions and was automatically derived from csbresgVD 31465.

1::	|- (. A e. V ->. A e. V ).
2:1:	|- (. A e. V ->. [_ A / x ]_ _V = _V ).
3:2:	|- (. A e. V ->. ( [_ A / x ]_ C X. [_ A / x ]_ _V ) = ( [_ A / x ]_ C X. _V ) ).
4:1:	|- (. A e. V ->. [_ A / x ]_ ( C X. _V ) = ( [_ A / x ]_ C X. [_ A / x ]_ _V ) ).
5:3,4:	|- (. A e. V ->. [_ A / x ]_ ( C X. _V ) = ( [_ A / x ]_ C X. _V ) ).
6:5:	|- (. 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:	|- (. A e. V ->. [_ A / x ]_ ( B i^i ( C X. _V ) ) = ( [_ A / x ]_ B i^i [_ A / x ]_ ( C X. _V ) ) ).
8:6,7:	|- (. A e. V ->. [_ A / x ]_ ( B i^i ( C X. _V ) ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) ) ).
9::	|- ( B |` C ) = ( B i^i ( C X. _V ) )
10:9:	|- A. x ( B |` C ) = ( B i^i ( C X. _V ) )
11:1,10:	|- (. A e. V ->. [_ A / x ]_ ( B |` C ) = [_ A / x ]_ ( B i^i ( C X. _V ) ) ).
12:8,11:	|- (. A e. V ->. [_ A / x ]_ ( B |` C ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) ) ).
13::	|- ( [_ A / x ]_ B |` [_ A / x ]_ C ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) )
14:12,13:	|- (. A e. V ->. [_ A / x ]_ ( B |` C ) = ( [_ A / x ]_ B |` [_ A / x ]_ C ) ).
qed:14:	|- ( A e. V -> [_ A / x ]_ ( B |` C ) = ( [_ A / x ]_ B |` [_ A / x ]_ C ) )

(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem csbresgVD

Step	Hyp	Ref	Expression
1		idn1 31120	. . . . . . . . 9 |- (. A e. V ->. A e. V ).
2		csbconstg 3298	. . . . . . . . 9 |- ( A e. V -> [_ A / x ]_ _V = _V )
3	1, 2	e1_ 31183	. . . . . . . 8 |- (. A e. V ->. [_ A / x ]_ _V = _V ).
4		xpeq2 4851	. . . . . . . 8 |- ( [_ A / x ]_ _V = _V -> ( [_ A / x ]_ C X. [_ A / x ]_ _V ) = ( [_ A / x ]_ C X. _V ) )
5	3, 4	e1_ 31183	. . . . . . 7 |- (. A e. V ->. ( [_ A / x ]_ C X. [_ A / x ]_ _V ) = ( [_ A / x ]_ C X. _V ) ).
6		csbxpgOLD 4915	. . . . . . . 8 |- ( A e. V -> [_ A / x ]_ ( C X. _V ) = ( [_ A / x ]_ C X. [_ A / x ]_ _V ) )
7	1, 6	e1_ 31183	. . . . . . 7 |- (. A e. V ->. [_ A / x ]_ ( C X. _V ) = ( [_ A / x ]_ C X. [_ A / x ]_ _V ) ).
8		eqeq2 2450	. . . . . . . 8 |- ( ( [_ A / x ]_ C X. [_ A / x ]_ _V ) = ( [_ A / x ]_ C X. _V ) -> ( [_ A / x ]_ ( C X. _V ) = ( [_ A / x ]_ C X. [_ A / x ]_ _V ) <-> [_ A / x ]_ ( C X. _V ) = ( [_ A / x ]_ C X. _V ) ) )
9	8	biimpd 207	. . . . . . 7 |- ( ( [_ A / x ]_ C X. [_ A / x ]_ _V ) = ( [_ A / x ]_ C X. _V ) -> ( [_ A / x ]_ ( C X. _V ) = ( [_ A / x ]_ C X. [_ A / x ]_ _V ) -> [_ A / x ]_ ( C X. _V ) = ( [_ A / x ]_ C X. _V ) ) )
10	5, 7, 9	e11 31244	. . . . . 6 |- (. A e. V ->. [_ A / x ]_ ( C X. _V ) = ( [_ A / x ]_ C X. _V ) ).
11		ineq2 3543	. . . . . 6 |- ( [_ A / x ]_ ( C X. _V ) = ( [_ A / x ]_ C X. _V ) -> ( [_ A / x ]_ B i^i [_ A / x ]_ ( C X. _V ) ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) ) )
12	10, 11	e1_ 31183	. . . . 5 |- (. A e. V ->. ( [_ A / x ]_ B i^i [_ A / x ]_ ( C X. _V ) ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) ) ).
13		csbingOLD 3710	. . . . . 6 |- ( A e. V -> [_ A / x ]_ ( B i^i ( C X. _V ) ) = ( [_ A / x ]_ B i^i [_ A / x ]_ ( C X. _V ) ) )
14	1, 13	e1_ 31183	. . . . 5 |- (. A e. V ->. [_ A / x ]_ ( B i^i ( C X. _V ) ) = ( [_ A / x ]_ B i^i [_ A / x ]_ ( C X. _V ) ) ).
15		eqeq2 2450	. . . . . 6 |- ( ( [_ A / x ]_ B i^i [_ A / x ]_ ( C X. _V ) ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) ) -> ( [_ A / x ]_ ( B i^i ( C X. _V ) ) = ( [_ A / x ]_ B i^i [_ A / x ]_ ( C X. _V ) ) <-> [_ A / x ]_ ( B i^i ( C X. _V ) ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) ) ) )
16	15	biimpd 207	. . . . 5 |- ( ( [_ A / x ]_ B i^i [_ A / x ]_ ( C X. _V ) ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) ) -> ( [_ A / x ]_ ( B i^i ( C X. _V ) ) = ( [_ A / x ]_ B i^i [_ A / x ]_ ( C X. _V ) ) -> [_ A / x ]_ ( B i^i ( C X. _V ) ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) ) ) )
17	12, 14, 16	e11 31244	. . . 4 |- (. A e. V ->. [_ A / x ]_ ( B i^i ( C X. _V ) ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) ) ).
18		df-res 4848	. . . . . 6 |- ( B |` C ) = ( B i^i ( C X. _V ) )
19	18	ax-gen 1596	. . . . 5 |- A. x ( B |` C ) = ( B i^i ( C X. _V ) )
20		csbeq2gOLD 31091	. . . . 5 |- ( A e. V -> ( A. x ( B |` C ) = ( B i^i ( C X. _V ) ) -> [_ A / x ]_ ( B |` C ) = [_ A / x ]_ ( B i^i ( C X. _V ) ) ) )
21	1, 19, 20	e10 31250	. . . 4 |- (. A e. V ->. [_ A / x ]_ ( B |` C ) = [_ A / x ]_ ( B i^i ( C X. _V ) ) ).
22		eqeq2 2450	. . . . 5 |- ( [_ A / x ]_ ( B i^i ( C X. _V ) ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) ) -> ( [_ A / x ]_ ( B |` C ) = [_ A / x ]_ ( B i^i ( C X. _V ) ) <-> [_ A / x ]_ ( B |` C ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) ) ) )
23	22	biimpd 207	. . . 4 |- ( [_ A / x ]_ ( B i^i ( C X. _V ) ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) ) -> ( [_ A / x ]_ ( B |` C ) = [_ A / x ]_ ( B i^i ( C X. _V ) ) -> [_ A / x ]_ ( B |` C ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) ) ) )
24	17, 21, 23	e11 31244	. . 3 |- (. A e. V ->. [_ A / x ]_ ( B |` C ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) ) ).
25		df-res 4848	. . 3 |- ( [_ A / x ]_ B |` [_ A / x ]_ C ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) )
26		eqeq2 2450	. . . 4 |- ( ( [_ A / x ]_ B |` [_ A / x ]_ C ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) ) -> ( [_ A / x ]_ ( B |` C ) = ( [_ A / x ]_ B |` [_ A / x ]_ C ) <-> [_ A / x ]_ ( B |` C ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) ) ) )
27	26	biimprcd 225	. . 3 |- ( [_ A / x ]_ ( B |` C ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) ) -> ( ( [_ A / x ]_ B |` [_ A / x ]_ C ) = ( [_ A / x ]_ B i^i ( [_ A / x ]_ C X. _V ) ) -> [_ A / x ]_ ( B |` C ) = ( [_ A / x ]_ B |` [_ A / x ]_ C ) ) )
28	24, 25, 27	e10 31250	. 2 |- (. A e. V ->. [_ A / x ]_ ( B |` C ) = ( [_ A / x ]_ B |` [_ A / x ]_ C ) ).
29	28	in1 31117	1 |- ( A e. V -> [_ A / x ]_ ( B |` C ) = ( [_ A / x ]_ B |` [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   A.wal 1362   = wceq 1364   e. wcel 1761   _Vcvv 2970   [_csb 3285   i^i cin 3324   X. cxp 4834   |` cres 4838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-in 3332  df-opab 4348  df-xp 4842  df-res 4848  df-vd1 31116

This theorem is referenced by: (None)