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Theorem csbresgVD 33176
Description: Virtual deduction proof of csbresgOLD 5283. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbresgOLD 5283 is csbresgVD 33176 without virtual deductions and was automatically derived from csbresgVD 33176.
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
StepHypRef Expression
1 idn1 32832 . . . . . . . . 9  |-  (. A  e.  V  ->.  A  e.  V ).
2 csbconstg 3453 . . . . . . . . 9  |-  ( A  e.  V  ->  [_ A  /  x ]_ _V  =  _V )
31, 2e1a 32894 . . . . . . . 8  |-  (. A  e.  V  ->.  [_ A  /  x ]_ _V  =  _V ).
4 xpeq2 5020 . . . . . . . 8  |-  ( [_ A  /  x ]_ _V  =  _V  ->  ( [_ A  /  x ]_ C  X.  [_ A  /  x ]_ _V )  =  (
[_ A  /  x ]_ C  X.  _V )
)
53, 4e1a 32894 . . . . . . 7  |-  (. A  e.  V  ->.  ( [_ A  /  x ]_ C  X.  [_ A  /  x ]_ _V )  =  ( [_ A  /  x ]_ C  X.  _V ) ).
6 csbxpgOLD 5088 . . . . . . . 8  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( C  X.  _V )  =  ( [_ A  /  x ]_ C  X.  [_ A  /  x ]_ _V ) )
71, 6e1a 32894 . . . . . . 7  |-  (. A  e.  V  ->.  [_ A  /  x ]_ ( C  X.  _V )  =  ( [_ A  /  x ]_ C  X.  [_ A  /  x ]_ _V ) ).
8 eqeq2 2482 . . . . . . . 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 ) ) )
98biimpd 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 ) ) )
105, 7, 9e11 32955 . . . . . 6  |-  (. A  e.  V  ->.  [_ A  /  x ]_ ( C  X.  _V )  =  ( [_ A  /  x ]_ C  X.  _V ) ).
11 ineq2 3699 . . . . . 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 ) ) )
1210, 11e1a 32894 . . . . 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 3866 . . . . . 6  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B  i^i  ( C  X.  _V ) )  =  (
[_ A  /  x ]_ B  i^i  [_ A  /  x ]_ ( C  X.  _V ) ) )
141, 13e1a 32894 . . . . 5  |-  (. A  e.  V  ->.  [_ A  /  x ]_ ( B  i^i  ( C  X.  _V ) )  =  ( [_ A  /  x ]_ B  i^i  [_ A  /  x ]_ ( C  X.  _V )
) ).
15 eqeq2 2482 . . . . . 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 )
) ) )
1615biimpd 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 )
) ) )
1712, 14, 16e11 32955 . . . 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 5017 . . . . . 6  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
1918ax-gen 1601 . . . . 5  |-  A. x
( B  |`  C )  =  ( B  i^i  ( C  X.  _V )
)
20 csbeq2gOLD 32803 . . . . 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 )
) ) )
211, 19, 20e10 32961 . . . 4  |-  (. A  e.  V  ->.  [_ A  /  x ]_ ( B  |`  C )  =  [_ A  /  x ]_ ( B  i^i  ( C  X.  _V )
) ).
22 eqeq2 2482 . . . . 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 )
) ) )
2322biimpd 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 )
) ) )
2417, 21, 23e11 32955 . . 3  |-  (. A  e.  V  ->.  [_ A  /  x ]_ ( B  |`  C )  =  ( [_ A  /  x ]_ B  i^i  ( [_ A  /  x ]_ C  X.  _V )
) ).
25 df-res 5017 . . 3  |-  ( [_ A  /  x ]_ B  |` 
[_ A  /  x ]_ C )  =  (
[_ A  /  x ]_ B  i^i  ( [_ A  /  x ]_ C  X.  _V )
)
26 eqeq2 2482 . . . 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 )
) ) )
2726biimprcd 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 ) ) )
2824, 25, 27e10 32961 . 2  |-  (. A  e.  V  ->.  [_ A  /  x ]_ ( B  |`  C )  =  ( [_ A  /  x ]_ B  |`  [_ A  /  x ]_ C ) ).
2928in1 32829 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B  |`  C )  =  (
[_ A  /  x ]_ B  |`  [_ A  /  x ]_ C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1377    = wceq 1379    e. wcel 1767   _Vcvv 3118   [_csb 3440    i^i cin 3480    X. cxp 5003    |` cres 5007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-in 3488  df-opab 4512  df-xp 5011  df-res 5017  df-vd1 32828
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator