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Theorem dropab1 27607
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 3976 . . . . . . . 8  |-  ( x  =  y  ->  <. x ,  z >.  =  <. y ,  z >. )
21sps 1770 . . . . . . 7  |-  ( A. x  x  =  y  -> 
<. x ,  z >.  =  <. y ,  z
>. )
32eqeq2d 2446 . . . . . 6  |-  ( A. x  x  =  y  ->  ( w  =  <. x ,  z >.  <->  w  =  <. y ,  z >.
) )
43anbi1d 686 . . . . 5  |-  ( A. x  x  =  y  ->  ( ( w  = 
<. x ,  z >.  /\  ph )  <->  ( w  =  <. y ,  z
>.  /\  ph ) ) )
54drex2 2056 . . . 4  |-  ( A. x  x  =  y  ->  ( E. z ( w  =  <. x ,  z >.  /\  ph ) 
<->  E. z ( w  =  <. y ,  z
>.  /\  ph ) ) )
65drex1 2055 . . 3  |-  ( A. x  x  =  y  ->  ( E. x E. z ( w  = 
<. x ,  z >.  /\  ph )  <->  E. y E. z ( w  = 
<. y ,  z >.  /\  ph ) ) )
76abbidv 2549 . 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 4259 . 2  |-  { <. x ,  z >.  |  ph }  =  { w  |  E. x E. z
( w  =  <. x ,  z >.  /\  ph ) }
9 df-opab 4259 . 2  |-  { <. y ,  z >.  |  ph }  =  { w  |  E. y E. z
( w  =  <. y ,  z >.  /\  ph ) }
107, 8, 93eqtr4g 2492 1  |-  ( A. x  x  =  y  ->  { <. x ,  z
>.  |  ph }  =  { <. y ,  z
>.  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1549   E.wex 1550    = wceq 1652   {cab 2421   <.cop 3809   {copab 4257
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 2416
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-opab 4259
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