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Theorem csbsngVD 28669
Description: Virtual deduction proof of csbsng 3692. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbsng 3692 is csbsngVD 28669 without virtual deductions and was automatically derived from csbsngVD 28669.
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 28342 . . . . . . . . 9  |-  (. A  e.  V  ->.  A  e.  V ).
2 sbceqg 3097 . . . . . . . . 9  |-  ( A  e.  V  ->  ( [. A  /  x ]. y  =  B  <->  [_ A  /  x ]_ y  =  [_ A  /  x ]_ B ) )
31, 2e1_ 28399 . . . . . . . 8  |-  (. A  e.  V  ->.  ( [. A  /  x ]. y  =  B  <->  [_ A  /  x ]_ y  =  [_ A  /  x ]_ B ) ).
4 csbconstg 3095 . . . . . . . . . 10  |-  ( A  e.  V  ->  [_ A  /  x ]_ y  =  y )
51, 4e1_ 28399 . . . . . . . . 9  |-  (. A  e.  V  ->.  [_ A  /  x ]_ y  =  y ).
6 eqeq1 2289 . . . . . . . . 9  |-  ( [_ A  /  x ]_ y  =  y  ->  ( [_ A  /  x ]_ y  =  [_ A  /  x ]_ B  <->  y  =  [_ A  /  x ]_ B
) )
75, 6e1_ 28399 . . . . . . . 8  |-  (. A  e.  V  ->.  ( [_ A  /  x ]_ y  = 
[_ A  /  x ]_ B  <->  y  =  [_ A  /  x ]_ B
) ).
8 bibi1 317 . . . . . . . . 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 214 . . . . . . . 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 28460 . . . . . . 7  |-  (. A  e.  V  ->.  ( [. A  /  x ]. y  =  B  <->  y  =  [_ A  /  x ]_ B
) ).
1110gen11 28388 . . . . . 6  |-  (. A  e.  V  ->.  A. y ( [. A  /  x ]. y  =  B  <->  y  =  [_ A  /  x ]_ B
) ).
12 abbi 2393 . . . . . . 7  |-  ( A. y ( [. A  /  x ]. y  =  B  <->  y  =  [_ A  /  x ]_ B
)  <->  { y  |  [. A  /  x ]. y  =  B }  =  {
y  |  y  = 
[_ A  /  x ]_ B } )
1312biimpi 186 . . . . . 6  |-  ( A. y ( [. A  /  x ]. y  =  B  <->  y  =  [_ A  /  x ]_ B
)  ->  { y  |  [. A  /  x ]. y  =  B }  =  { y  |  y  =  [_ A  /  x ]_ B }
)
1411, 13e1_ 28399 . . . . 5  |-  (. A  e.  V  ->.  { y  | 
[. A  /  x ]. y  =  B }  =  { y  |  y  =  [_ A  /  x ]_ B } ).
15 csbabg 3142 . . . . . . 7  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  y  =  B }  =  { y  |  [. A  /  x ]. y  =  B } )
1615eqcomd 2288 . . . . . 6  |-  ( A  e.  V  ->  { y  |  [. A  /  x ]. y  =  B }  =  [_ A  /  x ]_ { y  |  y  =  B } )
171, 16e1_ 28399 . . . . 5  |-  (. A  e.  V  ->.  { y  | 
[. A  /  x ]. y  =  B }  =  [_ A  /  x ]_ { y  |  y  =  B } ).
18 eqeq1 2289 . . . . . 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 215 . . . . 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 28460 . . . 4  |-  (. A  e.  V  ->.  [_ A  /  x ]_ { y  |  y  =  B }  =  { y  |  y  =  [_ A  /  x ]_ B } ).
21 df-sn 3646 . . . . . 6  |-  { B }  =  { y  |  y  =  B }
2221ax-gen 1533 . . . . 5  |-  A. x { B }  =  {
y  |  y  =  B }
23 csbeq2g 28315 . . . . 5  |-  ( A  e.  V  ->  ( A. x { B }  =  { y  |  y  =  B }  ->  [_ A  /  x ]_ { B }  =  [_ A  /  x ]_ {
y  |  y  =  B } ) )
241, 22, 23e10 28467 . . . 4  |-  (. A  e.  V  ->.  [_ A  /  x ]_ { B }  =  [_ A  /  x ]_ { y  |  y  =  B } ).
25 eqeq2 2292 . . . . 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 198 . . . 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 28460 . . 3  |-  (. A  e.  V  ->.  [_ A  /  x ]_ { B }  =  { y  |  y  =  [_ A  /  x ]_ B } ).
28 df-sn 3646 . . 3  |-  { [_ A  /  x ]_ B }  =  { y  |  y  =  [_ A  /  x ]_ B }
29 eqeq2 2292 . . . 4  |-  ( {
[_ A  /  x ]_ B }  =  {
y  |  y  = 
[_ A  /  x ]_ B }  ->  ( [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B }  <->  [_ A  /  x ]_ { B }  =  { y  |  y  =  [_ A  /  x ]_ B } ) )
3029biimprcd 216 . . 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 28467 . 2  |-  (. A  e.  V  ->.  [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B } ).
3231in1 28339 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527    = wceq 1623    e. wcel 1684   {cab 2269   [.wsbc 2991   [_csb 3081   {csn 3640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992  df-csb 3082  df-sn 3646  df-vd1 28338
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