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Theorem cbvopab1s 3860
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 1437 . . . 4  |-  F/ z E. y ( w  =  <. x ,  y
>.  /\  ph )
2 nfv 1437 . . . . . 6  |-  F/ x  w  =  <. z ,  y >.
3 nfs1v 1831 . . . . . 6  |-  F/ x [ z  /  x ] ph
42, 3nfan 1473 . . . . 5  |-  F/ x
( w  =  <. z ,  y >.  /\  [
z  /  x ] ph )
54nfex 1544 . . . 4  |-  F/ x E. y ( w  = 
<. z ,  y >.  /\  [ z  /  x ] ph )
6 opeq1 3577 . . . . . . 7  |-  ( x  =  z  ->  <. x ,  y >.  =  <. z ,  y >. )
76eqeq2d 2067 . . . . . 6  |-  ( x  =  z  ->  (
w  =  <. x ,  y >.  <->  w  =  <. z ,  y >.
) )
8 sbequ12 1670 . . . . . 6  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
97, 8anbi12d 450 . . . . 5  |-  ( x  =  z  ->  (
( w  =  <. x ,  y >.  /\  ph ) 
<->  ( w  =  <. z ,  y >.  /\  [
z  /  x ] ph ) ) )
109exbidv 1722 . . . 4  |-  ( x  =  z  ->  ( E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. y
( w  =  <. z ,  y >.  /\  [
z  /  x ] ph ) ) )
111, 5, 10cbvex 1655 . . 3  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph ) 
<->  E. z E. y
( w  =  <. z ,  y >.  /\  [
z  /  x ] ph ) )
1211abbii 2169 . 2  |-  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }  =  {
w  |  E. z E. y ( w  = 
<. z ,  y >.  /\  [ z  /  x ] ph ) }
13 df-opab 3847 . 2  |-  { <. x ,  y >.  |  ph }  =  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
14 df-opab 3847 . 2  |-  { <. z ,  y >.  |  [
z  /  x ] ph }  =  { w  |  E. z E. y
( w  =  <. z ,  y >.  /\  [
z  /  x ] ph ) }
1512, 13, 143eqtr4i 2086 1  |-  { <. x ,  y >.  |  ph }  =  { <. z ,  y >.  |  [
z  /  x ] ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 101    = wceq 1259   E.wex 1397   [wsb 1661   {cab 2042   <.cop 3406   {copab 3845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-opab 3847
This theorem is referenced by: (None)
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