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Theorem cbvopab1s 4475
Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003.)
Assertion
Ref Expression
cbvopab1s  |-  { <. x ,  y >.  |  ph }  =  { <. z ,  y >.  |  [
z  /  x ] ph }
Distinct variable groups:    x, y,
z    ph, z
Allowed substitution hints:    ph( x, y)

Proof of Theorem cbvopab1s
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 nfv 1761 . . . 4  |-  F/ z E. y ( w  =  <. x ,  y
>.  /\  ph )
2 nfv 1761 . . . . . 6  |-  F/ x  w  =  <. z ,  y >.
3 nfs1v 2266 . . . . . 6  |-  F/ x [ z  /  x ] ph
42, 3nfan 2011 . . . . 5  |-  F/ x
( w  =  <. z ,  y >.  /\  [
z  /  x ] ph )
54nfex 2031 . . . 4  |-  F/ x E. y ( w  = 
<. z ,  y >.  /\  [ z  /  x ] ph )
6 opeq1 4166 . . . . . . 7  |-  ( x  =  z  ->  <. x ,  y >.  =  <. z ,  y >. )
76eqeq2d 2461 . . . . . 6  |-  ( x  =  z  ->  (
w  =  <. x ,  y >.  <->  w  =  <. z ,  y >.
) )
8 sbequ12 2083 . . . . . 6  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
97, 8anbi12d 717 . . . . 5  |-  ( x  =  z  ->  (
( w  =  <. x ,  y >.  /\  ph ) 
<->  ( w  =  <. z ,  y >.  /\  [
z  /  x ] ph ) ) )
109exbidv 1768 . . . 4  |-  ( x  =  z  ->  ( E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. y
( w  =  <. z ,  y >.  /\  [
z  /  x ] ph ) ) )
111, 5, 10cbvex 2115 . . 3  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph ) 
<->  E. z E. y
( w  =  <. z ,  y >.  /\  [
z  /  x ] ph ) )
1211abbii 2567 . 2  |-  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }  =  {
w  |  E. z E. y ( w  = 
<. z ,  y >.  /\  [ z  /  x ] ph ) }
13 df-opab 4462 . 2  |-  { <. x ,  y >.  |  ph }  =  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
14 df-opab 4462 . 2  |-  { <. z ,  y >.  |  [
z  /  x ] ph }  =  { w  |  E. z E. y
( w  =  <. z ,  y >.  /\  [
z  /  x ] ph ) }
1512, 13, 143eqtr4i 2483 1  |-  { <. x ,  y >.  |  ph }  =  { <. z ,  y >.  |  [
z  /  x ] ph }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 371    = wceq 1444   E.wex 1663   [wsb 1797   {cab 2437   <.cop 3974   {copab 4460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-opab 4462
This theorem is referenced by: (None)
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