Users' Mathboxes Mathbox for Andrew Salmon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dropab2 Unicode version

Theorem dropab2 27754
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
dropab2  |-  ( A. x  x  =  y  ->  { <. z ,  x >.  |  ph }  =  { <. z ,  y
>.  |  ph } )

Proof of Theorem dropab2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 opeq2 3813 . . . . . . . 8  |-  ( x  =  y  ->  <. z ,  x >.  =  <. z ,  y >. )
21sps 1751 . . . . . . 7  |-  ( A. x  x  =  y  -> 
<. z ,  x >.  = 
<. z ,  y >.
)
32eqeq2d 2307 . . . . . 6  |-  ( A. x  x  =  y  ->  ( w  =  <. z ,  x >.  <->  w  =  <. z ,  y >.
) )
43anbi1d 685 . . . . 5  |-  ( A. x  x  =  y  ->  ( ( w  = 
<. z ,  x >.  /\ 
ph )  <->  ( w  =  <. z ,  y
>.  /\  ph ) ) )
54drex1 1920 . . . 4  |-  ( A. x  x  =  y  ->  ( E. x ( w  =  <. z ,  x >.  /\  ph )  <->  E. y ( w  = 
<. z ,  y >.  /\  ph ) ) )
65drex2 1921 . . 3  |-  ( A. x  x  =  y  ->  ( E. z E. x ( w  = 
<. z ,  x >.  /\ 
ph )  <->  E. z E. y ( w  = 
<. z ,  y >.  /\  ph ) ) )
76abbidv 2410 . 2  |-  ( A. x  x  =  y  ->  { w  |  E. z E. x ( w  =  <. z ,  x >.  /\  ph ) }  =  { w  |  E. z E. y
( w  =  <. z ,  y >.  /\  ph ) } )
8 df-opab 4094 . 2  |-  { <. z ,  x >.  |  ph }  =  { w  |  E. z E. x
( w  =  <. z ,  x >.  /\  ph ) }
9 df-opab 4094 . 2  |-  { <. z ,  y >.  |  ph }  =  { w  |  E. z E. y
( w  =  <. z ,  y >.  /\  ph ) }
107, 8, 93eqtr4g 2353 1  |-  ( A. x  x  =  y  ->  { <. z ,  x >.  |  ph }  =  { <. z ,  y
>.  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632   {cab 2282   <.cop 3656   {copab 4092
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094
  Copyright terms: Public domain W3C validator