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Theorem csbresgVD 37355
Description: Virtual deduction proof of csbresgOLD 37279. 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 37279 is csbresgVD 37355 without virtual deductions and was automatically derived from csbresgVD 37355.
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 37012 . . . . . . . . 9  |-  (. A  e.  V  ->.  A  e.  V ).
2 csbconstg 3362 . . . . . . . . 9  |-  ( A  e.  V  ->  [_ A  /  x ]_ _V  =  _V )
31, 2e1a 37074 . . . . . . . 8  |-  (. A  e.  V  ->.  [_ A  /  x ]_ _V  =  _V ).
4 xpeq2 4854 . . . . . . . 8  |-  ( [_ A  /  x ]_ _V  =  _V  ->  ( [_ A  /  x ]_ C  X.  [_ A  /  x ]_ _V )  =  (
[_ A  /  x ]_ C  X.  _V )
)
53, 4e1a 37074 . . . . . . 7  |-  (. A  e.  V  ->.  ( [_ A  /  x ]_ C  X.  [_ A  /  x ]_ _V )  =  ( [_ A  /  x ]_ C  X.  _V ) ).
6 csbxpgOLD 37277 . . . . . . . 8  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( C  X.  _V )  =  ( [_ A  /  x ]_ C  X.  [_ A  /  x ]_ _V ) )
71, 6e1a 37074 . . . . . . 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 212 . . . . . . 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 37135 . . . . . 6  |-  (. A  e.  V  ->.  [_ A  /  x ]_ ( C  X.  _V )  =  ( [_ A  /  x ]_ C  X.  _V ) ).
11 ineq2 3619 . . . . . 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 37074 . . . . 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 37278 . . . . . 6  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B  i^i  ( C  X.  _V ) )  =  (
[_ A  /  x ]_ B  i^i  [_ A  /  x ]_ ( C  X.  _V ) ) )
141, 13e1a 37074 . . . . 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 212 . . . . 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 37135 . . . 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 4851 . . . . . 6  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
1918ax-gen 1677 . . . . 5  |-  A. x
( B  |`  C )  =  ( B  i^i  ( C  X.  _V )
)
20 csbeq2gOLD 36986 . . . . 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 37141 . . . 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 212 . . . 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 37135 . . 3  |-  (. A  e.  V  ->.  [_ A  /  x ]_ ( B  |`  C )  =  ( [_ A  /  x ]_ B  i^i  ( [_ A  /  x ]_ C  X.  _V )
) ).
25 df-res 4851 . . 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 233 . . 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 37141 . 2  |-  (. A  e.  V  ->.  [_ A  /  x ]_ ( B  |`  C )  =  ( [_ A  /  x ]_ B  |`  [_ A  /  x ]_ C ) ).
2928in1 37009 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 1450    = wceq 1452    e. wcel 1904   _Vcvv 3031   [_csb 3349    i^i cin 3389    X. cxp 4837    |` cres 4841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-in 3397  df-opab 4455  df-xp 4845  df-res 4851  df-vd1 37008
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator