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Theorem cbvopab2v 3907
Description: Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999.)
Hypothesis
Ref Expression
cbvopab2v.1  |-  ( y  =  z  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvopab2v  |-  { <. x ,  y >.  |  ph }  =  { <. x ,  z >.  |  ps }
Distinct variable groups:    x, y, z    ph, z    ps, y
Allowed substitution hints:    ph( x, y)    ps( x, z)

Proof of Theorem cbvopab2v
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 opeq2 3618 . . . . . . 7  |-  ( y  =  z  ->  <. x ,  y >.  =  <. x ,  z >. )
21eqeq2d 2099 . . . . . 6  |-  ( y  =  z  ->  (
w  =  <. x ,  y >.  <->  w  =  <. x ,  z >.
) )
3 cbvopab2v.1 . . . . . 6  |-  ( y  =  z  ->  ( ph 
<->  ps ) )
42, 3anbi12d 457 . . . . 5  |-  ( y  =  z  ->  (
( w  =  <. x ,  y >.  /\  ph ) 
<->  ( w  =  <. x ,  z >.  /\  ps ) ) )
54cbvexv 1843 . . . 4  |-  ( E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. z
( w  =  <. x ,  z >.  /\  ps ) )
65exbii 1541 . . 3  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph ) 
<->  E. x E. z
( w  =  <. x ,  z >.  /\  ps ) )
76abbii 2203 . 2  |-  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }  =  {
w  |  E. x E. z ( w  = 
<. x ,  z >.  /\  ps ) }
8 df-opab 3892 . 2  |-  { <. x ,  y >.  |  ph }  =  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
9 df-opab 3892 . 2  |-  { <. x ,  z >.  |  ps }  =  { w  |  E. x E. z
( w  =  <. x ,  z >.  /\  ps ) }
107, 8, 93eqtr4i 2118 1  |-  { <. x ,  y >.  |  ph }  =  { <. x ,  z >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426   {cab 2074   <.cop 3444   {copab 3890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-op 3450  df-opab 3892
This theorem is referenced by: (None)
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