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Theorem dfid3 4491
Description: A stronger version of df-id 4490 that doesn't require  x and  y to be distinct. Ordinarily, we wouldn't use this as a definition, since non-distinct dummy variables would make soundness verification more difficult (as the proof here shows). The proof can be instructive in showing how distinct variable requirements may be eliminated, a task that is not necessarily obvious. (Contributed by NM, 5-Feb-2008.) (Revised by Mario Carneiro, 18-Nov-2016.)
Assertion
Ref Expression
dfid3  |-  _I  =  { <. x ,  y
>.  |  x  =  y }

Proof of Theorem dfid3
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-id 4490 . 2  |-  _I  =  { <. x ,  z
>.  |  x  =  z }
2 ancom 438 . . . . . . . . . . 11  |-  ( ( w  =  <. x ,  z >.  /\  x  =  z )  <->  ( x  =  z  /\  w  =  <. x ,  z
>. ) )
3 equcom 1692 . . . . . . . . . . . 12  |-  ( x  =  z  <->  z  =  x )
43anbi1i 677 . . . . . . . . . . 11  |-  ( ( x  =  z  /\  w  =  <. x ,  z >. )  <->  ( z  =  x  /\  w  =  <. x ,  z
>. ) )
52, 4bitri 241 . . . . . . . . . 10  |-  ( ( w  =  <. x ,  z >.  /\  x  =  z )  <->  ( z  =  x  /\  w  =  <. x ,  z
>. ) )
65exbii 1592 . . . . . . . . 9  |-  ( E. z ( w  = 
<. x ,  z >.  /\  x  =  z
)  <->  E. z ( z  =  x  /\  w  =  <. x ,  z
>. ) )
7 vex 2951 . . . . . . . . . 10  |-  x  e. 
_V
8 opeq2 3977 . . . . . . . . . . 11  |-  ( z  =  x  ->  <. x ,  z >.  =  <. x ,  x >. )
98eqeq2d 2446 . . . . . . . . . 10  |-  ( z  =  x  ->  (
w  =  <. x ,  z >.  <->  w  =  <. x ,  x >. ) )
107, 9ceqsexv 2983 . . . . . . . . 9  |-  ( E. z ( z  =  x  /\  w  = 
<. x ,  z >.
)  <->  w  =  <. x ,  x >. )
11 equid 1688 . . . . . . . . . 10  |-  x  =  x
1211biantru 492 . . . . . . . . 9  |-  ( w  =  <. x ,  x >.  <-> 
( w  =  <. x ,  x >.  /\  x  =  x ) )
136, 10, 123bitri 263 . . . . . . . 8  |-  ( E. z ( w  = 
<. x ,  z >.  /\  x  =  z
)  <->  ( w  = 
<. x ,  x >.  /\  x  =  x ) )
1413exbii 1592 . . . . . . 7  |-  ( E. x E. z ( w  =  <. x ,  z >.  /\  x  =  z )  <->  E. x
( w  =  <. x ,  x >.  /\  x  =  x ) )
15 nfe1 1747 . . . . . . . 8  |-  F/ x E. x ( w  = 
<. x ,  x >.  /\  x  =  x )
161519.9 1797 . . . . . . 7  |-  ( E. x E. x ( w  =  <. x ,  x >.  /\  x  =  x )  <->  E. x
( w  =  <. x ,  x >.  /\  x  =  x ) )
1714, 16bitr4i 244 . . . . . 6  |-  ( E. x E. z ( w  =  <. x ,  z >.  /\  x  =  z )  <->  E. x E. x ( w  = 
<. x ,  x >.  /\  x  =  x ) )
18 opeq2 3977 . . . . . . . . . . 11  |-  ( x  =  y  ->  <. x ,  x >.  =  <. x ,  y >. )
1918eqeq2d 2446 . . . . . . . . . 10  |-  ( x  =  y  ->  (
w  =  <. x ,  x >.  <->  w  =  <. x ,  y >. )
)
20 equequ2 1698 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x  =  x  <->  x  =  y ) )
2119, 20anbi12d 692 . . . . . . . . 9  |-  ( x  =  y  ->  (
( w  =  <. x ,  x >.  /\  x  =  x )  <->  ( w  =  <. x ,  y
>.  /\  x  =  y ) ) )
2221sps 1770 . . . . . . . 8  |-  ( A. x  x  =  y  ->  ( ( w  = 
<. x ,  x >.  /\  x  =  x )  <-> 
( w  =  <. x ,  y >.  /\  x  =  y ) ) )
2322drex1 2055 . . . . . . 7  |-  ( A. x  x  =  y  ->  ( E. x ( w  =  <. x ,  x >.  /\  x  =  x )  <->  E. y
( w  =  <. x ,  y >.  /\  x  =  y ) ) )
2423drex2 2056 . . . . . 6  |-  ( A. x  x  =  y  ->  ( E. x E. x ( w  = 
<. x ,  x >.  /\  x  =  x )  <->  E. x E. y ( w  =  <. x ,  y >.  /\  x  =  y ) ) )
2517, 24syl5bb 249 . . . . 5  |-  ( A. x  x  =  y  ->  ( E. x E. z ( w  = 
<. x ,  z >.  /\  x  =  z
)  <->  E. x E. y
( w  =  <. x ,  y >.  /\  x  =  y ) ) )
26 nfnae 2044 . . . . . 6  |-  F/ x  -.  A. x  x  =  y
27 nfnae 2044 . . . . . . 7  |-  F/ y  -.  A. x  x  =  y
28 nfcvd 2572 . . . . . . . . 9  |-  ( -. 
A. x  x  =  y  ->  F/_ y w )
29 nfcvf2 2594 . . . . . . . . . 10  |-  ( -. 
A. x  x  =  y  ->  F/_ y x )
30 nfcvd 2572 . . . . . . . . . 10  |-  ( -. 
A. x  x  =  y  ->  F/_ y z )
3129, 30nfopd 3993 . . . . . . . . 9  |-  ( -. 
A. x  x  =  y  ->  F/_ y <.
x ,  z >.
)
3228, 31nfeqd 2585 . . . . . . . 8  |-  ( -. 
A. x  x  =  y  ->  F/ y  w  =  <. x ,  z >. )
3329, 30nfeqd 2585 . . . . . . . 8  |-  ( -. 
A. x  x  =  y  ->  F/ y  x  =  z )
3432, 33nfand 1843 . . . . . . 7  |-  ( -. 
A. x  x  =  y  ->  F/ y
( w  =  <. x ,  z >.  /\  x  =  z ) )
35 opeq2 3977 . . . . . . . . . 10  |-  ( z  =  y  ->  <. x ,  z >.  =  <. x ,  y >. )
3635eqeq2d 2446 . . . . . . . . 9  |-  ( z  =  y  ->  (
w  =  <. x ,  z >.  <->  w  =  <. x ,  y >.
) )
37 equequ2 1698 . . . . . . . . 9  |-  ( z  =  y  ->  (
x  =  z  <->  x  =  y ) )
3836, 37anbi12d 692 . . . . . . . 8  |-  ( z  =  y  ->  (
( w  =  <. x ,  z >.  /\  x  =  z )  <->  ( w  =  <. x ,  y
>.  /\  x  =  y ) ) )
3938a1i 11 . . . . . . 7  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  ( ( w  =  <. x ,  z >.  /\  x  =  z )  <->  ( w  =  <. x ,  y
>.  /\  x  =  y ) ) ) )
4027, 34, 39cbvexd 1988 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  ( E. z ( w  = 
<. x ,  z >.  /\  x  =  z
)  <->  E. y ( w  =  <. x ,  y
>.  /\  x  =  y ) ) )
4126, 40exbid 1789 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  ( E. x E. z ( w  =  <. x ,  z
>.  /\  x  =  z )  <->  E. x E. y
( w  =  <. x ,  y >.  /\  x  =  y ) ) )
4225, 41pm2.61i 158 . . . 4  |-  ( E. x E. z ( w  =  <. x ,  z >.  /\  x  =  z )  <->  E. x E. y ( w  = 
<. x ,  y >.  /\  x  =  y
) )
4342abbii 2547 . . 3  |-  { w  |  E. x E. z
( w  =  <. x ,  z >.  /\  x  =  z ) }  =  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  x  =  y ) }
44 df-opab 4259 . . 3  |-  { <. x ,  z >.  |  x  =  z }  =  { w  |  E. x E. z ( w  =  <. x ,  z
>.  /\  x  =  z ) }
45 df-opab 4259 . . 3  |-  { <. x ,  y >.  |  x  =  y }  =  { w  |  E. x E. y ( w  =  <. x ,  y
>.  /\  x  =  y ) }
4643, 44, 453eqtr4i 2465 . 2  |-  { <. x ,  z >.  |  x  =  z }  =  { <. x ,  y
>.  |  x  =  y }
471, 46eqtri 2455 1  |-  _I  =  { <. x ,  y
>.  |  x  =  y }
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549   E.wex 1550    = wceq 1652   {cab 2421   <.cop 3809   {copab 4257    _I cid 4485
This theorem is referenced by:  dfid2  4492  reli  4993  opabresid  5185  ider  6930  cnmptid  17681
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-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-opab 4259  df-id 4490
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