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Theorem eusvobj2 6283
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 4043 . . 3  |-  ( E! x E. y  e.  A  x  =  B  <->  E. z { x  |  E. y  e.  A  x  =  B }  =  { z } )
2 eleq2 2518 . . . . . 6  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  (
x  e.  { x  |  E. y  e.  A  x  =  B }  <->  x  e.  { z } ) )
3 abid 2439 . . . . . 6  |-  ( x  e.  { x  |  E. y  e.  A  x  =  B }  <->  E. y  e.  A  x  =  B )
4 elsn 3982 . . . . . 6  |-  ( x  e.  { z }  <-> 
x  =  z )
52, 3, 43bitr3g 291 . . . . 5  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  ( E. y  e.  A  x  =  B  <->  x  =  z ) )
6 nfre1 2848 . . . . . . . . 9  |-  F/ y E. y  e.  A  x  =  B
76nfab 2596 . . . . . . . 8  |-  F/_ y { x  |  E. y  e.  A  x  =  B }
87nfeq1 2605 . . . . . . 7  |-  F/ y { x  |  E. y  e.  A  x  =  B }  =  {
z }
9 eusvobj1.1 . . . . . . . . 9  |-  B  e. 
_V
109elabrex 6148 . . . . . . . 8  |-  ( y  e.  A  ->  B  e.  { x  |  E. y  e.  A  x  =  B } )
11 eleq2 2518 . . . . . . . . 9  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  ( B  e.  { x  |  E. y  e.  A  x  =  B }  <->  B  e.  { z } ) )
129elsnc 3992 . . . . . . . . . 10  |-  ( B  e.  { z }  <-> 
B  =  z )
13 eqcom 2458 . . . . . . . . . 10  |-  ( B  =  z  <->  z  =  B )
1412, 13bitri 253 . . . . . . . . 9  |-  ( B  e.  { z }  <-> 
z  =  B )
1511, 14syl6bb 265 . . . . . . . 8  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  ( B  e.  { x  |  E. y  e.  A  x  =  B }  <->  z  =  B ) )
1610, 15syl5ib 223 . . . . . . 7  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  (
y  e.  A  -> 
z  =  B ) )
178, 16ralrimi 2788 . . . . . 6  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  A. y  e.  A  z  =  B )
18 eqeq1 2455 . . . . . . 7  |-  ( x  =  z  ->  (
x  =  B  <->  z  =  B ) )
1918ralbidv 2827 . . . . . 6  |-  ( x  =  z  ->  ( A. y  e.  A  x  =  B  <->  A. y  e.  A  z  =  B ) )
2017, 19syl5ibrcom 226 . . . . 5  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  (
x  =  z  ->  A. y  e.  A  x  =  B )
)
215, 20sylbid 219 . . . 4  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  ( E. y  e.  A  x  =  B  ->  A. y  e.  A  x  =  B ) )
2221exlimiv 1776 . . 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 199 . 2  |-  ( E! x E. y  e.  A  x  =  B  ->  ( E. y  e.  A  x  =  B  ->  A. y  e.  A  x  =  B )
)
24 euex 2323 . . 3  |-  ( E! x E. y  e.  A  x  =  B  ->  E. x E. y  e.  A  x  =  B )
25 rexn0 3872 . . . 4  |-  ( E. y  e.  A  x  =  B  ->  A  =/=  (/) )
2625exlimiv 1776 . . 3  |-  ( E. x E. y  e.  A  x  =  B  ->  A  =/=  (/) )
27 r19.2z 3858 . . . 4  |-  ( ( A  =/=  (/)  /\  A. y  e.  A  x  =  B )  ->  E. y  e.  A  x  =  B )
2827ex 436 . . 3  |-  ( A  =/=  (/)  ->  ( A. y  e.  A  x  =  B  ->  E. y  e.  A  x  =  B ) )
2924, 26, 283syl 18 . 2  |-  ( E! x E. y  e.  A  x  =  B  ->  ( A. y  e.  A  x  =  B  ->  E. y  e.  A  x  =  B )
)
3023, 29impbid 194 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 setvar class
Syntax hints:    -> wi 4    <-> wb 188    = wceq 1444   E.wex 1663    e. wcel 1887   E!weu 2299   {cab 2437    =/= wne 2622   A.wral 2737   E.wrex 2738   _Vcvv 3045   (/)c0 3731   {csn 3968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-nul 3732  df-sn 3969
This theorem is referenced by:  eusvobj1  6284
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