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Theorem eusvobj2 6585
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 3877 . . 3  |-  ( E! x E. y  e.  A  x  =  B  <->  E. z { x  |  E. y  e.  A  x  =  B }  =  { z } )
2 eleq2 2499 . . . . . 6  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  (
x  e.  { x  |  E. y  e.  A  x  =  B }  <->  x  e.  { z } ) )
3 abid 2426 . . . . . 6  |-  ( x  e.  { x  |  E. y  e.  A  x  =  B }  <->  E. y  e.  A  x  =  B )
4 elsn 3831 . . . . . 6  |-  ( x  e.  { z }  <-> 
x  =  z )
52, 3, 43bitr3g 280 . . . . 5  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  ( E. y  e.  A  x  =  B  <->  x  =  z ) )
6 nfre1 2764 . . . . . . . . 9  |-  F/ y E. y  e.  A  x  =  B
76nfab 2578 . . . . . . . 8  |-  F/_ y { x  |  E. y  e.  A  x  =  B }
87nfeq1 2583 . . . . . . 7  |-  F/ y { x  |  E. y  e.  A  x  =  B }  =  {
z }
9 eusvobj1.1 . . . . . . . . 9  |-  B  e. 
_V
109elabrex 5988 . . . . . . . 8  |-  ( y  e.  A  ->  B  e.  { x  |  E. y  e.  A  x  =  B } )
11 eleq2 2499 . . . . . . . . 9  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  ( B  e.  { x  |  E. y  e.  A  x  =  B }  <->  B  e.  { z } ) )
129elsnc 3839 . . . . . . . . . 10  |-  ( B  e.  { z }  <-> 
B  =  z )
13 eqcom 2440 . . . . . . . . . 10  |-  ( B  =  z  <->  z  =  B )
1412, 13bitri 242 . . . . . . . . 9  |-  ( B  e.  { z }  <-> 
z  =  B )
1511, 14syl6bb 254 . . . . . . . 8  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  ( B  e.  { x  |  E. y  e.  A  x  =  B }  <->  z  =  B ) )
1610, 15syl5ib 212 . . . . . . 7  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  (
y  e.  A  -> 
z  =  B ) )
178, 16ralrimi 2789 . . . . . 6  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  A. y  e.  A  z  =  B )
18 eqeq1 2444 . . . . . . 7  |-  ( x  =  z  ->  (
x  =  B  <->  z  =  B ) )
1918ralbidv 2727 . . . . . 6  |-  ( x  =  z  ->  ( A. y  e.  A  x  =  B  <->  A. y  e.  A  z  =  B ) )
2017, 19syl5ibrcom 215 . . . . 5  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  (
x  =  z  ->  A. y  e.  A  x  =  B )
)
215, 20sylbid 208 . . . 4  |-  ( { x  |  E. y  e.  A  x  =  B }  =  {
z }  ->  ( E. y  e.  A  x  =  B  ->  A. y  e.  A  x  =  B ) )
2221exlimiv 1645 . . 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 189 . 2  |-  ( E! x E. y  e.  A  x  =  B  ->  ( E. y  e.  A  x  =  B  ->  A. y  e.  A  x  =  B )
)
24 euex 2306 . . 3  |-  ( E! x E. y  e.  A  x  =  B  ->  E. x E. y  e.  A  x  =  B )
25 rexn0 3732 . . . 4  |-  ( E. y  e.  A  x  =  B  ->  A  =/=  (/) )
2625exlimiv 1645 . . 3  |-  ( E. x E. y  e.  A  x  =  B  ->  A  =/=  (/) )
27 r19.2z 3719 . . . 4  |-  ( ( A  =/=  (/)  /\  A. y  e.  A  x  =  B )  ->  E. y  e.  A  x  =  B )
2827ex 425 . . 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 185 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 178   E.wex 1551    = wceq 1653    e. wcel 1726   E!weu 2283   {cab 2424    =/= wne 2601   A.wral 2707   E.wrex 2708   _Vcvv 2958   (/)c0 3630   {csn 3816
This theorem is referenced by:  eusvobj1  6586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-nul 3631  df-sn 3822
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