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Theorem cbvopab1s 4369
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 1673 . . . 4  |-  F/ z E. y ( w  =  <. x ,  y
>.  /\  ph )
2 nfv 1673 . . . . . 6  |-  F/ x  w  =  <. z ,  y >.
3 nfs1v 2142 . . . . . 6  |-  F/ x [ z  /  x ] ph
42, 3nfan 1861 . . . . 5  |-  F/ x
( w  =  <. z ,  y >.  /\  [
z  /  x ] ph )
54nfex 1874 . . . 4  |-  F/ x E. y ( w  = 
<. z ,  y >.  /\  [ z  /  x ] ph )
6 opeq1 4064 . . . . . . 7  |-  ( x  =  z  ->  <. x ,  y >.  =  <. z ,  y >. )
76eqeq2d 2454 . . . . . 6  |-  ( x  =  z  ->  (
w  =  <. x ,  y >.  <->  w  =  <. z ,  y >.
) )
8 sbequ12 1936 . . . . . 6  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
97, 8anbi12d 710 . . . . 5  |-  ( x  =  z  ->  (
( w  =  <. x ,  y >.  /\  ph ) 
<->  ( w  =  <. z ,  y >.  /\  [
z  /  x ] ph ) ) )
109exbidv 1680 . . . 4  |-  ( x  =  z  ->  ( E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. y
( w  =  <. z ,  y >.  /\  [
z  /  x ] ph ) ) )
111, 5, 10cbvex 1970 . . 3  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph ) 
<->  E. z E. y
( w  =  <. z ,  y >.  /\  [
z  /  x ] ph ) )
1211abbii 2560 . 2  |-  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }  =  {
w  |  E. z E. y ( w  = 
<. z ,  y >.  /\  [ z  /  x ] ph ) }
13 df-opab 4356 . 2  |-  { <. x ,  y >.  |  ph }  =  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
14 df-opab 4356 . 2  |-  { <. z ,  y >.  |  [
z  /  x ] ph }  =  { w  |  E. z E. y
( w  =  <. z ,  y >.  /\  [
z  /  x ] ph ) }
1512, 13, 143eqtr4i 2473 1  |-  { <. x ,  y >.  |  ph }  =  { <. z ,  y >.  |  [
z  /  x ] ph }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1369   E.wex 1586   [wsb 1700   {cab 2429   <.cop 3888   {copab 4354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-opab 4356
This theorem is referenced by: (None)
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