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Theorem eusvobj2 5638
Description: Specify the same property in two ways when class  B ( y ) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
eusvobj1.1  |-  B  e. 
_V
Assertion
Ref Expression
eusvobj2  |-  ( E! x E. y  e.  A  x  =  B  ->  ( E. y  e.  A  x  =  B 
<-> 
A. y  e.  A  x  =  B )
)
Distinct variable groups:    x, y, A   
x, B
Allowed substitution hint:    B( y)

Proof of Theorem eusvobj2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 euabsn2 3511 . . 3  |-  ( E! x E. y  e.  A  x  =  B  <->  E. z { x  |  E. y  e.  A  x  =  B }  =  { z } )
2 eleq2 2151 . . . . . 6  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  (
x  e.  { x  |  E. y  e.  A  x  =  B }  <->  x  e.  { z } ) )
3 abid 2076 . . . . . 6  |-  ( x  e.  { x  |  E. y  e.  A  x  =  B }  <->  E. y  e.  A  x  =  B )
4 velsn 3463 . . . . . 6  |-  ( x  e.  { z }  <-> 
x  =  z )
52, 3, 43bitr3g 220 . . . . 5  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  ( E. y  e.  A  x  =  B  <->  x  =  z ) )
6 nfre1 2419 . . . . . . . . 9  |-  F/ y E. y  e.  A  x  =  B
76nfab 2233 . . . . . . . 8  |-  F/_ y { x  |  E. y  e.  A  x  =  B }
87nfeq1 2238 . . . . . . 7  |-  F/ y { x  |  E. y  e.  A  x  =  B }  =  {
z }
9 eusvobj1.1 . . . . . . . . 9  |-  B  e. 
_V
109elabrex 5537 . . . . . . . 8  |-  ( y  e.  A  ->  B  e.  { x  |  E. y  e.  A  x  =  B } )
11 eleq2 2151 . . . . . . . . 9  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  ( B  e.  { x  |  E. y  e.  A  x  =  B }  <->  B  e.  { z } ) )
129elsn 3462 . . . . . . . . . 10  |-  ( B  e.  { z }  <-> 
B  =  z )
13 eqcom 2090 . . . . . . . . . 10  |-  ( B  =  z  <->  z  =  B )
1412, 13bitri 182 . . . . . . . . 9  |-  ( B  e.  { z }  <-> 
z  =  B )
1511, 14syl6bb 194 . . . . . . . 8  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  ( B  e.  { x  |  E. y  e.  A  x  =  B }  <->  z  =  B ) )
1610, 15syl5ib 152 . . . . . . 7  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  (
y  e.  A  -> 
z  =  B ) )
178, 16ralrimi 2444 . . . . . 6  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  A. y  e.  A  z  =  B )
18 eqeq1 2094 . . . . . . 7  |-  ( x  =  z  ->  (
x  =  B  <->  z  =  B ) )
1918ralbidv 2380 . . . . . 6  |-  ( x  =  z  ->  ( A. y  e.  A  x  =  B  <->  A. y  e.  A  z  =  B ) )
2017, 19syl5ibrcom 155 . . . . 5  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  (
x  =  z  ->  A. y  e.  A  x  =  B )
)
215, 20sylbid 148 . . . 4  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  ( E. y  e.  A  x  =  B  ->  A. y  e.  A  x  =  B ) )
2221exlimiv 1534 . . 3  |-  ( E. z { x  |  E. y  e.  A  x  =  B }  =  { z }  ->  ( E. y  e.  A  x  =  B  ->  A. y  e.  A  x  =  B ) )
231, 22sylbi 119 . 2  |-  ( E! x E. y  e.  A  x  =  B  ->  ( E. y  e.  A  x  =  B  ->  A. y  e.  A  x  =  B )
)
24 euex 1978 . . 3  |-  ( E! x E. y  e.  A  x  =  B  ->  E. x E. y  e.  A  x  =  B )
25 rexm 3381 . . . 4  |-  ( E. y  e.  A  x  =  B  ->  E. y 
y  e.  A )
2625exlimiv 1534 . . 3  |-  ( E. x E. y  e.  A  x  =  B  ->  E. y  y  e.  A )
27 r19.2m 3369 . . . 4  |-  ( ( E. y  y  e.  A  /\  A. y  e.  A  x  =  B )  ->  E. y  e.  A  x  =  B )
2827ex 113 . . 3  |-  ( E. y  y  e.  A  ->  ( A. y  e.  A  x  =  B  ->  E. y  e.  A  x  =  B )
)
2924, 26, 283syl 17 . 2  |-  ( E! x E. y  e.  A  x  =  B  ->  ( A. y  e.  A  x  =  B  ->  E. y  e.  A  x  =  B )
)
3023, 29impbid 127 1  |-  ( E! x E. y  e.  A  x  =  B  ->  ( E. y  e.  A  x  =  B 
<-> 
A. y  e.  A  x  =  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   E!weu 1948   {cab 2074   A.wral 2359   E.wrex 2360   _Vcvv 2619   {csn 3446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-csb 2934  df-sn 3452
This theorem is referenced by:  eusvobj1  5639
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