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Theorem csbsngVD 34094
Description: Virtual deduction proof of csbsng 4074. 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 4074 is csbsngVD 34094 without virtual deductions and was automatically derived from csbsngVD 34094.
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 33745 . . . . . . . . 9  |-  (. A  e.  V  ->.  A  e.  V ).
2 sbceqg 3823 . . . . . . . . 9  |-  ( A  e.  V  ->  ( [. A  /  x ]. y  =  B  <->  [_ A  /  x ]_ y  =  [_ A  /  x ]_ B ) )
31, 2e1a 33807 . . . . . . . 8  |-  (. A  e.  V  ->.  ( [. A  /  x ]. y  =  B  <->  [_ A  /  x ]_ y  =  [_ A  /  x ]_ B ) ).
4 csbconstg 3433 . . . . . . . . . 10  |-  ( A  e.  V  ->  [_ A  /  x ]_ y  =  y )
51, 4e1a 33807 . . . . . . . . 9  |-  (. A  e.  V  ->.  [_ A  /  x ]_ y  =  y ).
6 eqeq1 2458 . . . . . . . . 9  |-  ( [_ A  /  x ]_ y  =  y  ->  ( [_ A  /  x ]_ y  =  [_ A  /  x ]_ B  <->  y  =  [_ A  /  x ]_ B
) )
75, 6e1a 33807 . . . . . . . 8  |-  (. A  e.  V  ->.  ( [_ A  /  x ]_ y  = 
[_ A  /  x ]_ B  <->  y  =  [_ A  /  x ]_ B
) ).
8 bibi1 325 . . . . . . . . 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 223 . . . . . . . 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 33868 . . . . . . 7  |-  (. A  e.  V  ->.  ( [. A  /  x ]. y  =  B  <->  y  =  [_ A  /  x ]_ B
) ).
1110gen11 33796 . . . . . 6  |-  (. A  e.  V  ->.  A. y ( [. A  /  x ]. y  =  B  <->  y  =  [_ A  /  x ]_ B
) ).
12 abbi 2585 . . . . . . 7  |-  ( A. y ( [. A  /  x ]. y  =  B  <->  y  =  [_ A  /  x ]_ B
)  <->  { y  |  [. A  /  x ]. y  =  B }  =  {
y  |  y  = 
[_ A  /  x ]_ B } )
1312biimpi 194 . . . . . 6  |-  ( A. y ( [. A  /  x ]. y  =  B  <->  y  =  [_ A  /  x ]_ B
)  ->  { y  |  [. A  /  x ]. y  =  B }  =  { y  |  y  =  [_ A  /  x ]_ B }
)
1411, 13e1a 33807 . . . . 5  |-  (. A  e.  V  ->.  { y  | 
[. A  /  x ]. y  =  B }  =  { y  |  y  =  [_ A  /  x ]_ B } ).
15 csbabgOLD 34015 . . . . . . 7  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  y  =  B }  =  { y  |  [. A  /  x ]. y  =  B } )
1615eqcomd 2462 . . . . . 6  |-  ( A  e.  V  ->  { y  |  [. A  /  x ]. y  =  B }  =  [_ A  /  x ]_ { y  |  y  =  B } )
171, 16e1a 33807 . . . . 5  |-  (. A  e.  V  ->.  { y  | 
[. A  /  x ]. y  =  B }  =  [_ A  /  x ]_ { y  |  y  =  B } ).
18 eqeq1 2458 . . . . . 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 224 . . . . 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 33868 . . . 4  |-  (. A  e.  V  ->.  [_ A  /  x ]_ { y  |  y  =  B }  =  { y  |  y  =  [_ A  /  x ]_ B } ).
21 df-sn 4017 . . . . . 6  |-  { B }  =  { y  |  y  =  B }
2221ax-gen 1623 . . . . 5  |-  A. x { B }  =  {
y  |  y  =  B }
23 csbeq2gOLD 33716 . . . . 5  |-  ( A  e.  V  ->  ( A. x { B }  =  { y  |  y  =  B }  ->  [_ A  /  x ]_ { B }  =  [_ A  /  x ]_ {
y  |  y  =  B } ) )
241, 22, 23e10 33874 . . . 4  |-  (. A  e.  V  ->.  [_ A  /  x ]_ { B }  =  [_ A  /  x ]_ { y  |  y  =  B } ).
25 eqeq2 2469 . . . . 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 207 . . . 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 33868 . . 3  |-  (. A  e.  V  ->.  [_ A  /  x ]_ { B }  =  { y  |  y  =  [_ A  /  x ]_ B } ).
28 df-sn 4017 . . 3  |-  { [_ A  /  x ]_ B }  =  { y  |  y  =  [_ A  /  x ]_ B }
29 eqeq2 2469 . . . 4  |-  ( {
[_ A  /  x ]_ B }  =  {
y  |  y  = 
[_ A  /  x ]_ B }  ->  ( [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B }  <->  [_ A  /  x ]_ { B }  =  { y  |  y  =  [_ A  /  x ]_ B } ) )
3029biimprcd 225 . . 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 33874 . 2  |-  (. A  e.  V  ->.  [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B } ).
3231in1 33742 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1396    = wceq 1398    e. wcel 1823   {cab 2439   [.wsbc 3324   [_csb 3420   {csn 4016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-sbc 3325  df-csb 3421  df-sn 4017  df-vd1 33741
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator