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Theorem cbvopab2 4009
Description: Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.)
Hypotheses
Ref Expression
cbvopab2.1  |-  F/ z
ph
cbvopab2.2  |-  F/ y ps
cbvopab2.3  |-  ( y  =  z  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvopab2  |-  { <. x ,  y >.  |  ph }  =  { <. x ,  z >.  |  ps }
Distinct variable group:    x, y, z
Allowed substitution hints:    ph( x, y, z)    ps( x, y, z)

Proof of Theorem cbvopab2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 nfv 1509 . . . . . 6  |-  F/ z  w  =  <. x ,  y >.
2 cbvopab2.1 . . . . . 6  |-  F/ z
ph
31, 2nfan 1545 . . . . 5  |-  F/ z ( w  =  <. x ,  y >.  /\  ph )
4 nfv 1509 . . . . . 6  |-  F/ y  w  =  <. x ,  z >.
5 cbvopab2.2 . . . . . 6  |-  F/ y ps
64, 5nfan 1545 . . . . 5  |-  F/ y ( w  =  <. x ,  z >.  /\  ps )
7 opeq2 3713 . . . . . . 7  |-  ( y  =  z  ->  <. x ,  y >.  =  <. x ,  z >. )
87eqeq2d 2152 . . . . . 6  |-  ( y  =  z  ->  (
w  =  <. x ,  y >.  <->  w  =  <. x ,  z >.
) )
9 cbvopab2.3 . . . . . 6  |-  ( y  =  z  ->  ( ph 
<->  ps ) )
108, 9anbi12d 465 . . . . 5  |-  ( y  =  z  ->  (
( w  =  <. x ,  y >.  /\  ph ) 
<->  ( w  =  <. x ,  z >.  /\  ps ) ) )
113, 6, 10cbvex 1730 . . . 4  |-  ( E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. z
( w  =  <. x ,  z >.  /\  ps ) )
1211exbii 1585 . . 3  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph ) 
<->  E. x E. z
( w  =  <. x ,  z >.  /\  ps ) )
1312abbii 2256 . 2  |-  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }  =  {
w  |  E. x E. z ( w  = 
<. x ,  z >.  /\  ps ) }
14 df-opab 3997 . 2  |-  { <. x ,  y >.  |  ph }  =  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
15 df-opab 3997 . 2  |-  { <. x ,  z >.  |  ps }  =  { w  |  E. x E. z
( w  =  <. x ,  z >.  /\  ps ) }
1613, 14, 153eqtr4i 2171 1  |-  { <. x ,  y >.  |  ph }  =  { <. x ,  z >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1332   F/wnf 1437   E.wex 1469   {cab 2126   <.cop 3534   {copab 3995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3079  df-sn 3537  df-pr 3538  df-op 3540  df-opab 3997
This theorem is referenced by:  cbvoprab3  5854
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