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Theorem cbvopab2v 3863
Description: Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999.)
Hypothesis
Ref Expression
cbvopab2v.1  |-  ( y  =  z  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvopab2v  |-  { <. x ,  y >.  |  ph }  =  { <. x ,  z >.  |  ps }
Distinct variable groups:    x, y, z    ph, z    ps, y
Allowed substitution hints:    ph( x, y)    ps( x, z)

Proof of Theorem cbvopab2v
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 opeq2 3579 . . . . . . 7  |-  ( y  =  z  ->  <. x ,  y >.  =  <. x ,  z >. )
21eqeq2d 2093 . . . . . 6  |-  ( y  =  z  ->  (
w  =  <. x ,  y >.  <->  w  =  <. x ,  z >.
) )
3 cbvopab2v.1 . . . . . 6  |-  ( y  =  z  ->  ( ph 
<->  ps ) )
42, 3anbi12d 457 . . . . 5  |-  ( y  =  z  ->  (
( w  =  <. x ,  y >.  /\  ph ) 
<->  ( w  =  <. x ,  z >.  /\  ps ) ) )
54cbvexv 1837 . . . 4  |-  ( E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. z
( w  =  <. x ,  z >.  /\  ps ) )
65exbii 1537 . . 3  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph ) 
<->  E. x E. z
( w  =  <. x ,  z >.  /\  ps ) )
76abbii 2195 . 2  |-  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }  =  {
w  |  E. x E. z ( w  = 
<. x ,  z >.  /\  ps ) }
8 df-opab 3848 . 2  |-  { <. x ,  y >.  |  ph }  =  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
9 df-opab 3848 . 2  |-  { <. x ,  z >.  |  ps }  =  { w  |  E. x E. z
( w  =  <. x ,  z >.  /\  ps ) }
107, 8, 93eqtr4i 2112 1  |-  { <. x ,  y >.  |  ph }  =  { <. x ,  z >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   E.wex 1422   {cab 2068   <.cop 3409   {copab 3846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-opab 3848
This theorem is referenced by: (None)
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