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Theorem csbsngVD 37195
Description: Virtual deduction proof of csbsng 4056. 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. csbsng 4056 is csbsngVD 37195 without virtual deductions and was automatically derived from csbsngVD 37195.
1::  |-  (. A  e.  V  ->.  A  e.  V ).
2:1:  |-  (. A  e.  V  ->.  ( [. A  /  x ]. y  =  B  <->  [_ A  /  x ]_ y  =  [_ A  /  x ]_ B ) ).
3:1:  |-  (. A  e.  V  ->.  [_ A  /  x ]_ y  =  y ).
4:3:  |-  (. A  e.  V  ->.  ( [_ A  /  x ]_ y  =  [_ A  /  x ]_ B  <->  y  =  [_ A  /  x ]_ B ) ).
5:2,4:  |-  (. A  e.  V  ->.  ( [. A  /  x ]. y  =  B  <->  y  =  [_ A  /  x ]_ B ) ).
6:5:  |-  (. A  e.  V  ->.  A. y ( [. A  /  x ]. y  =  B  <->  y  =  [_ A  /  x ]_ B ) ).
7:6:  |-  (. A  e.  V  ->.  { y  |  [. A  /  x ]. y  =  B }  =  { y  |  y  =  [_ A  /  x ]_ B } ).
8:1:  |-  (. A  e.  V  ->.  { y  |  [. A  /  x ]. y  =  B }  =  [_ A  /  x ]_ { y  |  y  =  B } ).
9:7,8:  |-  (. A  e.  V  ->.  [_ A  /  x ]_ { y  |  y  =  B }  =  { y  |  y  =  [_ A  /  x ]_ B } ).
10::  |-  { B }  =  { y  |  y  =  B }
11:10:  |-  A. x { B }  =  { y  |  y  =  B }
12:1,11:  |-  (. A  e.  V  ->.  [_ A  /  x ]_ { B }  =  [_  A  /  x ]_ { y  |  y  =  B } ).
13:9,12:  |-  (. A  e.  V  ->.  [_ A  /  x ]_ { B }  =  {  y  |  y  =  [_ A  /  x ]_ B } ).
14::  |-  { [_ A  /  x ]_ B }  =  { y  |  y  =  [_ A  /  x ]_ B }
15:13,14:  |-  (. A  e.  V  ->.  [_ A  /  x ]_ { B }  =  {  [_ A  /  x ]_ B } ).
qed:15:  |-  ( A  e.  V  ->  [_ A  /  x ]_ { B }  =  { [_  A  /  x ]_ B } )
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
csbsngVD  |-  ( A  e.  V  ->  [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B }
)

Proof of Theorem csbsngVD
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 idn1 36847 . . . . . . . . 9  |-  (. A  e.  V  ->.  A  e.  V ).
2 sbceqg 3802 . . . . . . . . 9  |-  ( A  e.  V  ->  ( [. A  /  x ]. y  =  B  <->  [_ A  /  x ]_ y  =  [_ A  /  x ]_ B ) )
31, 2e1a 36909 . . . . . . . 8  |-  (. A  e.  V  ->.  ( [. A  /  x ]. y  =  B  <->  [_ A  /  x ]_ y  =  [_ A  /  x ]_ B ) ).
4 csbconstg 3409 . . . . . . . . . 10  |-  ( A  e.  V  ->  [_ A  /  x ]_ y  =  y )
51, 4e1a 36909 . . . . . . . . 9  |-  (. A  e.  V  ->.  [_ A  /  x ]_ y  =  y ).
6 eqeq1 2427 . . . . . . . . 9  |-  ( [_ A  /  x ]_ y  =  y  ->  ( [_ A  /  x ]_ y  =  [_ A  /  x ]_ B  <->  y  =  [_ A  /  x ]_ B
) )
75, 6e1a 36909 . . . . . . . 8  |-  (. A  e.  V  ->.  ( [_ A  /  x ]_ y  = 
[_ A  /  x ]_ B  <->  y  =  [_ A  /  x ]_ B
) ).
8 bibi1 329 . . . . . . . . 9  |-  ( (
[. A  /  x ]. y  =  B  <->  [_ A  /  x ]_ y  =  [_ A  /  x ]_ B )  -> 
( ( [. A  /  x ]. y  =  B  <->  y  =  [_ A  /  x ]_ B
)  <->  ( [_ A  /  x ]_ y  = 
[_ A  /  x ]_ B  <->  y  =  [_ A  /  x ]_ B
) ) )
98biimprd 227 . . . . . . . 8  |-  ( (
[. A  /  x ]. y  =  B  <->  [_ A  /  x ]_ y  =  [_ A  /  x ]_ B )  -> 
( ( [_ A  /  x ]_ y  = 
[_ A  /  x ]_ B  <->  y  =  [_ A  /  x ]_ B
)  ->  ( [. A  /  x ]. y  =  B  <->  y  =  [_ A  /  x ]_ B
) ) )
103, 7, 9e11 36970 . . . . . . 7  |-  (. A  e.  V  ->.  ( [. A  /  x ]. y  =  B  <->  y  =  [_ A  /  x ]_ B
) ).
1110gen11 36898 . . . . . 6  |-  (. A  e.  V  ->.  A. y ( [. A  /  x ]. y  =  B  <->  y  =  [_ A  /  x ]_ B
) ).
12 abbi 2554 . . . . . . 7  |-  ( A. y ( [. A  /  x ]. y  =  B  <->  y  =  [_ A  /  x ]_ B
)  <->  { y  |  [. A  /  x ]. y  =  B }  =  {
y  |  y  = 
[_ A  /  x ]_ B } )
1312biimpi 198 . . . . . 6  |-  ( A. y ( [. A  /  x ]. y  =  B  <->  y  =  [_ A  /  x ]_ B
)  ->  { y  |  [. A  /  x ]. y  =  B }  =  { y  |  y  =  [_ A  /  x ]_ B }
)
1411, 13e1a 36909 . . . . 5  |-  (. A  e.  V  ->.  { y  | 
[. A  /  x ]. y  =  B }  =  { y  |  y  =  [_ A  /  x ]_ B } ).
15 csbabgOLD 37116 . . . . . . 7  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  y  =  B }  =  { y  |  [. A  /  x ]. y  =  B } )
1615eqcomd 2431 . . . . . 6  |-  ( A  e.  V  ->  { y  |  [. A  /  x ]. y  =  B }  =  [_ A  /  x ]_ { y  |  y  =  B } )
171, 16e1a 36909 . . . . 5  |-  (. A  e.  V  ->.  { y  | 
[. A  /  x ]. y  =  B }  =  [_ A  /  x ]_ { y  |  y  =  B } ).
18 eqeq1 2427 . . . . . 6  |-  ( { y  |  [. A  /  x ]. y  =  B }  =  [_ A  /  x ]_ {
y  |  y  =  B }  ->  ( { y  |  [. A  /  x ]. y  =  B }  =  {
y  |  y  = 
[_ A  /  x ]_ B }  <->  [_ A  /  x ]_ { y  |  y  =  B }  =  { y  |  y  =  [_ A  /  x ]_ B } ) )
1918biimpcd 228 . . . . 5  |-  ( { y  |  [. A  /  x ]. y  =  B }  =  {
y  |  y  = 
[_ A  /  x ]_ B }  ->  ( { y  |  [. A  /  x ]. y  =  B }  =  [_ A  /  x ]_ {
y  |  y  =  B }  ->  [_ A  /  x ]_ { y  |  y  =  B }  =  { y  |  y  =  [_ A  /  x ]_ B } ) )
2014, 17, 19e11 36970 . . . 4  |-  (. A  e.  V  ->.  [_ A  /  x ]_ { y  |  y  =  B }  =  { y  |  y  =  [_ A  /  x ]_ B } ).
21 df-sn 3998 . . . . . 6  |-  { B }  =  { y  |  y  =  B }
2221ax-gen 1666 . . . . 5  |-  A. x { B }  =  {
y  |  y  =  B }
23 csbeq2gOLD 36818 . . . . 5  |-  ( A  e.  V  ->  ( A. x { B }  =  { y  |  y  =  B }  ->  [_ A  /  x ]_ { B }  =  [_ A  /  x ]_ {
y  |  y  =  B } ) )
241, 22, 23e10 36976 . . . 4  |-  (. A  e.  V  ->.  [_ A  /  x ]_ { B }  =  [_ A  /  x ]_ { y  |  y  =  B } ).
25 eqeq2 2438 . . . . 5  |-  ( [_ A  /  x ]_ {
y  |  y  =  B }  =  {
y  |  y  = 
[_ A  /  x ]_ B }  ->  ( [_ A  /  x ]_ { B }  =  [_ A  /  x ]_ { y  |  y  =  B }  <->  [_ A  /  x ]_ { B }  =  { y  |  y  =  [_ A  /  x ]_ B } ) )
2625biimpd 211 . . . 4  |-  ( [_ A  /  x ]_ {
y  |  y  =  B }  =  {
y  |  y  = 
[_ A  /  x ]_ B }  ->  ( [_ A  /  x ]_ { B }  =  [_ A  /  x ]_ { y  |  y  =  B }  ->  [_ A  /  x ]_ { B }  =  {
y  |  y  = 
[_ A  /  x ]_ B } ) )
2720, 24, 26e11 36970 . . 3  |-  (. A  e.  V  ->.  [_ A  /  x ]_ { B }  =  { y  |  y  =  [_ A  /  x ]_ B } ).
28 df-sn 3998 . . 3  |-  { [_ A  /  x ]_ B }  =  { y  |  y  =  [_ A  /  x ]_ B }
29 eqeq2 2438 . . . 4  |-  ( {
[_ A  /  x ]_ B }  =  {
y  |  y  = 
[_ A  /  x ]_ B }  ->  ( [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B }  <->  [_ A  /  x ]_ { B }  =  { y  |  y  =  [_ A  /  x ]_ B } ) )
3029biimprcd 229 . . 3  |-  ( [_ A  /  x ]_ { B }  =  {
y  |  y  = 
[_ A  /  x ]_ B }  ->  ( { [_ A  /  x ]_ B }  =  {
y  |  y  = 
[_ A  /  x ]_ B }  ->  [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B }
) )
3127, 28, 30e10 36976 . 2  |-  (. A  e.  V  ->.  [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B } ).
3231in1 36844 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188   A.wal 1436    = wceq 1438    e. wcel 1869   {cab 2408   [.wsbc 3300   [_csb 3396   {csn 3997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-v 3084  df-sbc 3301  df-csb 3397  df-sn 3998  df-vd1 36843
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator