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Theorem eusvobj2 6574
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 3867 . . 3  |-  ( E! x E. y  e.  A  x  =  B  <->  E. z { x  |  E. y  e.  A  x  =  B }  =  { z } )
2 eleq2 2496 . . . . . 6  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  (
x  e.  { x  |  E. y  e.  A  x  =  B }  <->  x  e.  { z } ) )
3 abid 2423 . . . . . 6  |-  ( x  e.  { x  |  E. y  e.  A  x  =  B }  <->  E. y  e.  A  x  =  B )
4 elsn 3821 . . . . . 6  |-  ( x  e.  { z }  <-> 
x  =  z )
52, 3, 43bitr3g 279 . . . . 5  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  ( E. y  e.  A  x  =  B  <->  x  =  z ) )
6 nfre1 2754 . . . . . . . . 9  |-  F/ y E. y  e.  A  x  =  B
76nfab 2575 . . . . . . . 8  |-  F/_ y { x  |  E. y  e.  A  x  =  B }
87nfeq1 2580 . . . . . . 7  |-  F/ y { x  |  E. y  e.  A  x  =  B }  =  {
z }
9 eusvobj1.1 . . . . . . . . 9  |-  B  e. 
_V
109elabrex 5977 . . . . . . . 8  |-  ( y  e.  A  ->  B  e.  { x  |  E. y  e.  A  x  =  B } )
11 eleq2 2496 . . . . . . . . 9  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  ( B  e.  { x  |  E. y  e.  A  x  =  B }  <->  B  e.  { z } ) )
129elsnc 3829 . . . . . . . . . 10  |-  ( B  e.  { z }  <-> 
B  =  z )
13 eqcom 2437 . . . . . . . . . 10  |-  ( B  =  z  <->  z  =  B )
1412, 13bitri 241 . . . . . . . . 9  |-  ( B  e.  { z }  <-> 
z  =  B )
1511, 14syl6bb 253 . . . . . . . 8  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  ( B  e.  { x  |  E. y  e.  A  x  =  B }  <->  z  =  B ) )
1610, 15syl5ib 211 . . . . . . 7  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  (
y  e.  A  -> 
z  =  B ) )
178, 16ralrimi 2779 . . . . . 6  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  A. y  e.  A  z  =  B )
18 eqeq1 2441 . . . . . . 7  |-  ( x  =  z  ->  (
x  =  B  <->  z  =  B ) )
1918ralbidv 2717 . . . . . 6  |-  ( x  =  z  ->  ( A. y  e.  A  x  =  B  <->  A. y  e.  A  z  =  B ) )
2017, 19syl5ibrcom 214 . . . . 5  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  (
x  =  z  ->  A. y  e.  A  x  =  B )
)
215, 20sylbid 207 . . . 4  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  ( E. y  e.  A  x  =  B  ->  A. y  e.  A  x  =  B ) )
2221exlimiv 1644 . . 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 188 . 2  |-  ( E! x E. y  e.  A  x  =  B  ->  ( E. y  e.  A  x  =  B  ->  A. y  e.  A  x  =  B )
)
24 euex 2303 . . 3  |-  ( E! x E. y  e.  A  x  =  B  ->  E. x E. y  e.  A  x  =  B )
25 rexn0 3722 . . . 4  |-  ( E. y  e.  A  x  =  B  ->  A  =/=  (/) )
2625exlimiv 1644 . . 3  |-  ( E. x E. y  e.  A  x  =  B  ->  A  =/=  (/) )
27 r19.2z 3709 . . . 4  |-  ( ( A  =/=  (/)  /\  A. y  e.  A  x  =  B )  ->  E. y  e.  A  x  =  B )
2827ex 424 . . 3  |-  ( A  =/=  (/)  ->  ( A. y  e.  A  x  =  B  ->  E. y  e.  A  x  =  B ) )
2924, 26, 283syl 19 . 2  |-  ( E! x E. y  e.  A  x  =  B  ->  ( A. y  e.  A  x  =  B  ->  E. y  e.  A  x  =  B )
)
3023, 29impbid 184 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 177   E.wex 1550    = wceq 1652    e. wcel 1725   E!weu 2280   {cab 2421    =/= wne 2598   A.wral 2697   E.wrex 2698   _Vcvv 2948   (/)c0 3620   {csn 3806
This theorem is referenced by:  eusvobj1  6575
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-nul 3621  df-sn 3812
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