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Theorem csbsngVD 33426
Description: Virtual deduction proof of csbsng 4073. 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 4073 is csbsngVD 33426 without virtual deductions and was automatically derived from csbsngVD 33426.
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 33084 . . . . . . . . 9  |-  (. A  e.  V  ->.  A  e.  V ).
2 sbceqg 3811 . . . . . . . . 9  |-  ( A  e.  V  ->  ( [. A  /  x ]. y  =  B  <->  [_ A  /  x ]_ y  =  [_ A  /  x ]_ B ) )
31, 2e1a 33146 . . . . . . . 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 33146 . . . . . . . . 9  |-  (. A  e.  V  ->.  [_ A  /  x ]_ y  =  y ).
6 eqeq1 2447 . . . . . . . . 9  |-  ( [_ A  /  x ]_ y  =  y  ->  ( [_ A  /  x ]_ y  =  [_ A  /  x ]_ B  <->  y  =  [_ A  /  x ]_ B
) )
75, 6e1a 33146 . . . . . . . 8  |-  (. A  e.  V  ->.  ( [_ A  /  x ]_ y  = 
[_ A  /  x ]_ B  <->  y  =  [_ A  /  x ]_ B
) ).
8 bibi1 327 . . . . . . . . 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 33207 . . . . . . 7  |-  (. A  e.  V  ->.  ( [. A  /  x ]. y  =  B  <->  y  =  [_ A  /  x ]_ B
) ).
1110gen11 33135 . . . . . 6  |-  (. A  e.  V  ->.  A. y ( [. A  /  x ]. y  =  B  <->  y  =  [_ A  /  x ]_ B
) ).
12 abbi 2574 . . . . . . 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 33146 . . . . 5  |-  (. A  e.  V  ->.  { y  | 
[. A  /  x ]. y  =  B }  =  { y  |  y  =  [_ A  /  x ]_ B } ).
15 csbabgOLD 3842 . . . . . . 7  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  y  =  B }  =  { y  |  [. A  /  x ]. y  =  B } )
1615eqcomd 2451 . . . . . 6  |-  ( A  e.  V  ->  { y  |  [. A  /  x ]. y  =  B }  =  [_ A  /  x ]_ { y  |  y  =  B } )
171, 16e1a 33146 . . . . 5  |-  (. A  e.  V  ->.  { y  | 
[. A  /  x ]. y  =  B }  =  [_ A  /  x ]_ { y  |  y  =  B } ).
18 eqeq1 2447 . . . . . 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 33207 . . . 4  |-  (. A  e.  V  ->.  [_ A  /  x ]_ { y  |  y  =  B }  =  { y  |  y  =  [_ A  /  x ]_ B } ).
21 df-sn 4015 . . . . . 6  |-  { B }  =  { y  |  y  =  B }
2221ax-gen 1605 . . . . 5  |-  A. x { B }  =  {
y  |  y  =  B }
23 csbeq2gOLD 33055 . . . . 5  |-  ( A  e.  V  ->  ( A. x { B }  =  { y  |  y  =  B }  ->  [_ A  /  x ]_ { B }  =  [_ A  /  x ]_ {
y  |  y  =  B } ) )
241, 22, 23e10 33213 . . . 4  |-  (. A  e.  V  ->.  [_ A  /  x ]_ { B }  =  [_ A  /  x ]_ { y  |  y  =  B } ).
25 eqeq2 2458 . . . . 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 33207 . . 3  |-  (. A  e.  V  ->.  [_ A  /  x ]_ { B }  =  { y  |  y  =  [_ A  /  x ]_ B } ).
28 df-sn 4015 . . 3  |-  { [_ A  /  x ]_ B }  =  { y  |  y  =  [_ A  /  x ]_ B }
29 eqeq2 2458 . . . 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 33213 . 2  |-  (. A  e.  V  ->.  [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B } ).
3231in1 33081 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1381    = wceq 1383    e. wcel 1804   {cab 2428   [.wsbc 3313   [_csb 3420   {csn 4014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-v 3097  df-sbc 3314  df-csb 3421  df-sn 4015  df-vd1 33080
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator