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Theorem eqer 6391
Description: Equivalence relation involving equality of dependent classes  A ( x ) and  B ( y ). (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
eqer.1  |-  ( x  =  y  ->  A  =  B )
eqer.2  |-  R  =  { <. x ,  y
>.  |  A  =  B }
Assertion
Ref Expression
eqer  |-  R  Er  _V
Distinct variable groups:    x, y    y, A    x, B
Allowed substitution hints:    A( x)    B( y)    R( x, y)

Proof of Theorem eqer
Dummy variables  w  z  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqer.2 . . . . 5  |-  R  =  { <. x ,  y
>.  |  A  =  B }
21relopabi 4603 . . . 4  |-  Rel  R
32a1i 9 . . 3  |-  ( T. 
->  Rel  R )
4 id 19 . . . . . 6  |-  ( [_ z  /  x ]_ A  =  [_ w  /  x ]_ A  ->  [_ z  /  x ]_ A  = 
[_ w  /  x ]_ A )
54eqcomd 2105 . . . . 5  |-  ( [_ z  /  x ]_ A  =  [_ w  /  x ]_ A  ->  [_ w  /  x ]_ A  = 
[_ z  /  x ]_ A )
6 eqer.1 . . . . . 6  |-  ( x  =  y  ->  A  =  B )
76, 1eqerlem 6390 . . . . 5  |-  ( z R w  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A
)
86, 1eqerlem 6390 . . . . 5  |-  ( w R z  <->  [_ w  /  x ]_ A  =  [_ z  /  x ]_ A
)
95, 7, 83imtr4i 200 . . . 4  |-  ( z R w  ->  w R z )
109adantl 273 . . 3  |-  ( ( T.  /\  z R w )  ->  w R z )
11 eqtr 2117 . . . . 5  |-  ( (
[_ z  /  x ]_ A  =  [_ w  /  x ]_ A  /\  [_ w  /  x ]_ A  =  [_ v  /  x ]_ A )  ->  [_ z  /  x ]_ A  =  [_ v  /  x ]_ A )
126, 1eqerlem 6390 . . . . . 6  |-  ( w R v  <->  [_ w  /  x ]_ A  =  [_ v  /  x ]_ A
)
137, 12anbi12i 451 . . . . 5  |-  ( ( z R w  /\  w R v )  <->  ( [_ z  /  x ]_ A  =  [_ w  /  x ]_ A  /\  [_ w  /  x ]_ A  = 
[_ v  /  x ]_ A ) )
146, 1eqerlem 6390 . . . . 5  |-  ( z R v  <->  [_ z  /  x ]_ A  =  [_ v  /  x ]_ A
)
1511, 13, 143imtr4i 200 . . . 4  |-  ( ( z R w  /\  w R v )  -> 
z R v )
1615adantl 273 . . 3  |-  ( ( T.  /\  ( z R w  /\  w R v ) )  ->  z R v )
17 vex 2644 . . . . 5  |-  z  e. 
_V
18 eqid 2100 . . . . . 6  |-  [_ z  /  x ]_ A  = 
[_ z  /  x ]_ A
196, 1eqerlem 6390 . . . . . 6  |-  ( z R z  <->  [_ z  /  x ]_ A  =  [_ z  /  x ]_ A
)
2018, 19mpbir 145 . . . . 5  |-  z R z
2117, 202th 173 . . . 4  |-  ( z  e.  _V  <->  z R
z )
2221a1i 9 . . 3  |-  ( T. 
->  ( z  e.  _V  <->  z R z ) )
233, 10, 16, 22iserd 6385 . 2  |-  ( T. 
->  R  Er  _V )
2423mptru 1308 1  |-  R  Er  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1299   T. wtru 1300    e. wcel 1448   _Vcvv 2641   [_csb 2955   class class class wbr 3875   {copab 3928   Rel wrel 4482    Er wer 6356
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-csb 2956  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-er 6359
This theorem is referenced by:  ider  6392
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