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Theorem cbvopab2 4490
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 1772 . . . . . 6  |-  F/ z  w  =  <. x ,  y >.
2 cbvopab2.1 . . . . . 6  |-  F/ z
ph
31, 2nfan 2022 . . . . 5  |-  F/ z ( w  =  <. x ,  y >.  /\  ph )
4 nfv 1772 . . . . . 6  |-  F/ y  w  =  <. x ,  z >.
5 cbvopab2.2 . . . . . 6  |-  F/ y ps
64, 5nfan 2022 . . . . 5  |-  F/ y ( w  =  <. x ,  z >.  /\  ps )
7 opeq2 4181 . . . . . . 7  |-  ( y  =  z  ->  <. x ,  y >.  =  <. x ,  z >. )
87eqeq2d 2472 . . . . . 6  |-  ( y  =  z  ->  (
w  =  <. x ,  y >.  <->  w  =  <. x ,  z >.
) )
9 cbvopab2.3 . . . . . 6  |-  ( y  =  z  ->  ( ph 
<->  ps ) )
108, 9anbi12d 722 . . . . 5  |-  ( y  =  z  ->  (
( w  =  <. x ,  y >.  /\  ph ) 
<->  ( w  =  <. x ,  z >.  /\  ps ) ) )
113, 6, 10cbvex 2126 . . . 4  |-  ( E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. z
( w  =  <. x ,  z >.  /\  ps ) )
1211exbii 1729 . . 3  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph ) 
<->  E. x E. z
( w  =  <. x ,  z >.  /\  ps ) )
1312abbii 2578 . 2  |-  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }  =  {
w  |  E. x E. z ( w  = 
<. x ,  z >.  /\  ps ) }
14 df-opab 4478 . 2  |-  { <. x ,  y >.  |  ph }  =  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
15 df-opab 4478 . 2  |-  { <. x ,  z >.  |  ps }  =  { w  |  E. x E. z
( w  =  <. x ,  z >.  /\  ps ) }
1613, 14, 153eqtr4i 2494 1  |-  { <. x ,  y >.  |  ph }  =  { <. x ,  z >.  |  ps }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455   E.wex 1674   F/wnf 1678   {cab 2448   <.cop 3986   {copab 4476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-opab 4478
This theorem is referenced by:  cbvoprab3  6399
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