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Theorem eqer 6592
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 4770 . . . 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 2195 . . . . 5  |-  ( [_ z  /  x ]_ A  =  [_ w  /  x ]_ A  ->  [_ w  /  x ]_ A  = 
[_ z  /  x ]_ A )
6 eqer.1 . . . . . 6  |-  ( x  =  y  ->  A  =  B )
76, 1eqerlem 6591 . . . . 5  |-  ( z R w  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A
)
86, 1eqerlem 6591 . . . . 5  |-  ( w R z  <->  [_ w  /  x ]_ A  =  [_ z  /  x ]_ A
)
95, 7, 83imtr4i 201 . . . 4  |-  ( z R w  ->  w R z )
109adantl 277 . . 3  |-  ( ( T.  /\  z R w )  ->  w R z )
11 eqtr 2207 . . . . 5  |-  ( (
[_ z  /  x ]_ A  =  [_ w  /  x ]_ A  /\  [_ w  /  x ]_ A  =  [_ v  /  x ]_ A )  ->  [_ z  /  x ]_ A  =  [_ v  /  x ]_ A )
126, 1eqerlem 6591 . . . . . 6  |-  ( w R v  <->  [_ w  /  x ]_ A  =  [_ v  /  x ]_ A
)
137, 12anbi12i 460 . . . . 5  |-  ( ( z R w  /\  w R v )  <->  ( [_ z  /  x ]_ A  =  [_ w  /  x ]_ A  /\  [_ w  /  x ]_ A  = 
[_ v  /  x ]_ A ) )
146, 1eqerlem 6591 . . . . 5  |-  ( z R v  <->  [_ z  /  x ]_ A  =  [_ v  /  x ]_ A
)
1511, 13, 143imtr4i 201 . . . 4  |-  ( ( z R w  /\  w R v )  -> 
z R v )
1615adantl 277 . . 3  |-  ( ( T.  /\  ( z R w  /\  w R v ) )  ->  z R v )
17 vex 2755 . . . . 5  |-  z  e. 
_V
18 eqid 2189 . . . . . 6  |-  [_ z  /  x ]_ A  = 
[_ z  /  x ]_ A
196, 1eqerlem 6591 . . . . . 6  |-  ( z R z  <->  [_ z  /  x ]_ A  =  [_ z  /  x ]_ A
)
2018, 19mpbir 146 . . . . 5  |-  z R z
2117, 202th 174 . . . 4  |-  ( z  e.  _V  <->  z R
z )
2221a1i 9 . . 3  |-  ( T. 
->  ( z  e.  _V  <->  z R z ) )
233, 10, 16, 22iserd 6586 . 2  |-  ( T. 
->  R  Er  _V )
2423mptru 1373 1  |-  R  Er  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364   T. wtru 1365    e. wcel 2160   _Vcvv 2752   [_csb 3072   class class class wbr 4018   {copab 4078   Rel wrel 4649    Er wer 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-er 6560
This theorem is referenced by:  ider  6593
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