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Theorem cbvopab1s 4280
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 1629 . . . 4  |-  F/ z E. y ( w  =  <. x ,  y
>.  /\  ph )
2 nfv 1629 . . . . . 6  |-  F/ x  w  =  <. z ,  y >.
3 nfs1v 2182 . . . . . 6  |-  F/ x [ z  /  x ] ph
42, 3nfan 1846 . . . . 5  |-  F/ x
( w  =  <. z ,  y >.  /\  [
z  /  x ] ph )
54nfex 1865 . . . 4  |-  F/ x E. y ( w  = 
<. z ,  y >.  /\  [ z  /  x ] ph )
6 opeq1 3984 . . . . . . 7  |-  ( x  =  z  ->  <. x ,  y >.  =  <. z ,  y >. )
76eqeq2d 2447 . . . . . 6  |-  ( x  =  z  ->  (
w  =  <. x ,  y >.  <->  w  =  <. z ,  y >.
) )
8 sbequ12 1944 . . . . . 6  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
97, 8anbi12d 692 . . . . 5  |-  ( x  =  z  ->  (
( w  =  <. x ,  y >.  /\  ph ) 
<->  ( w  =  <. z ,  y >.  /\  [
z  /  x ] ph ) ) )
109exbidv 1636 . . . 4  |-  ( x  =  z  ->  ( E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. y
( w  =  <. z ,  y >.  /\  [
z  /  x ] ph ) ) )
111, 5, 10cbvex 1983 . . 3  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph ) 
<->  E. z E. y
( w  =  <. z ,  y >.  /\  [
z  /  x ] ph ) )
1211abbii 2548 . 2  |-  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }  =  {
w  |  E. z E. y ( w  = 
<. z ,  y >.  /\  [ z  /  x ] ph ) }
13 df-opab 4267 . 2  |-  { <. x ,  y >.  |  ph }  =  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
14 df-opab 4267 . 2  |-  { <. z ,  y >.  |  [
z  /  x ] ph }  =  { w  |  E. z E. y
( w  =  <. z ,  y >.  /\  [
z  /  x ] ph ) }
1512, 13, 143eqtr4i 2466 1  |-  { <. x ,  y >.  |  ph }  =  { <. z ,  y >.  |  [
z  /  x ] ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1550    = wceq 1652   [wsb 1658   {cab 2422   <.cop 3817   {copab 4265
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-opab 4267
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