Users' Mathboxes Mathbox for Andrew Salmon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dropab1 Unicode version

Theorem dropab1 27650
Description: Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
dropab1  |-  ( A. x  x  =  y  ->  { <. x ,  z
>.  |  ph }  =  { <. y ,  z
>.  |  ph } )

Proof of Theorem dropab1
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 opeq1 3796 . . . . . . . 8  |-  ( x  =  y  ->  <. x ,  z >.  =  <. y ,  z >. )
21sps 1739 . . . . . . 7  |-  ( A. x  x  =  y  -> 
<. x ,  z >.  =  <. y ,  z
>. )
32eqeq2d 2294 . . . . . 6  |-  ( A. x  x  =  y  ->  ( w  =  <. x ,  z >.  <->  w  =  <. y ,  z >.
) )
43anbi1d 685 . . . . 5  |-  ( A. x  x  =  y  ->  ( ( w  = 
<. x ,  z >.  /\  ph )  <->  ( w  =  <. y ,  z
>.  /\  ph ) ) )
54drex2 1908 . . . 4  |-  ( A. x  x  =  y  ->  ( E. z ( w  =  <. x ,  z >.  /\  ph ) 
<->  E. z ( w  =  <. y ,  z
>.  /\  ph ) ) )
65drex1 1907 . . 3  |-  ( A. x  x  =  y  ->  ( E. x E. z ( w  = 
<. x ,  z >.  /\  ph )  <->  E. y E. z ( w  = 
<. y ,  z >.  /\  ph ) ) )
76abbidv 2397 . 2  |-  ( A. x  x  =  y  ->  { w  |  E. x E. z ( w  =  <. x ,  z
>.  /\  ph ) }  =  { w  |  E. y E. z
( w  =  <. y ,  z >.  /\  ph ) } )
8 df-opab 4078 . 2  |-  { <. x ,  z >.  |  ph }  =  { w  |  E. x E. z
( w  =  <. x ,  z >.  /\  ph ) }
9 df-opab 4078 . 2  |-  { <. y ,  z >.  |  ph }  =  { w  |  E. y E. z
( w  =  <. y ,  z >.  /\  ph ) }
107, 8, 93eqtr4g 2340 1  |-  ( A. x  x  =  y  ->  { <. x ,  z
>.  |  ph }  =  { <. y ,  z
>.  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623   {cab 2269   <.cop 3643   {copab 4076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078
  Copyright terms: Public domain W3C validator