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Theorem cbvopab2 4271
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 1629 . . . . . 6  |-  F/ z  w  =  <. x ,  y >.
2 cbvopab2.1 . . . . . 6  |-  F/ z
ph
31, 2nfan 1846 . . . . 5  |-  F/ z ( w  =  <. x ,  y >.  /\  ph )
4 nfv 1629 . . . . . 6  |-  F/ y  w  =  <. x ,  z >.
5 cbvopab2.2 . . . . . 6  |-  F/ y ps
64, 5nfan 1846 . . . . 5  |-  F/ y ( w  =  <. x ,  z >.  /\  ps )
7 opeq2 3977 . . . . . . 7  |-  ( y  =  z  ->  <. x ,  y >.  =  <. x ,  z >. )
87eqeq2d 2446 . . . . . 6  |-  ( y  =  z  ->  (
w  =  <. x ,  y >.  <->  w  =  <. x ,  z >.
) )
9 cbvopab2.3 . . . . . 6  |-  ( y  =  z  ->  ( ph 
<->  ps ) )
108, 9anbi12d 692 . . . . 5  |-  ( y  =  z  ->  (
( w  =  <. x ,  y >.  /\  ph ) 
<->  ( w  =  <. x ,  z >.  /\  ps ) ) )
113, 6, 10cbvex 1983 . . . 4  |-  ( E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. z
( w  =  <. x ,  z >.  /\  ps ) )
1211exbii 1592 . . 3  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph ) 
<->  E. x E. z
( w  =  <. x ,  z >.  /\  ps ) )
1312abbii 2547 . 2  |-  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }  =  {
w  |  E. x E. z ( w  = 
<. x ,  z >.  /\  ps ) }
14 df-opab 4259 . 2  |-  { <. x ,  y >.  |  ph }  =  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
15 df-opab 4259 . 2  |-  { <. x ,  z >.  |  ps }  =  { w  |  E. x E. z
( w  =  <. x ,  z >.  /\  ps ) }
1613, 14, 153eqtr4i 2465 1  |-  { <. x ,  y >.  |  ph }  =  { <. x ,  z >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550   F/wnf 1553    = wceq 1652   {cab 2421   <.cop 3809   {copab 4257
This theorem is referenced by:  cbvoprab3  6140
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
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