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Theorem dropab1 31518
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 4219 . . . . . . . 8  |-  ( x  =  y  ->  <. x ,  z >.  =  <. y ,  z >. )
21sps 1866 . . . . . . 7  |-  ( A. x  x  =  y  -> 
<. x ,  z >.  =  <. y ,  z
>. )
32eqeq2d 2471 . . . . . 6  |-  ( A. x  x  =  y  ->  ( w  =  <. x ,  z >.  <->  w  =  <. y ,  z >.
) )
43anbi1d 704 . . . . 5  |-  ( A. x  x  =  y  ->  ( ( w  = 
<. x ,  z >.  /\  ph )  <->  ( w  =  <. y ,  z
>.  /\  ph ) ) )
54drex2 2071 . . . 4  |-  ( A. x  x  =  y  ->  ( E. z ( w  =  <. x ,  z >.  /\  ph ) 
<->  E. z ( w  =  <. y ,  z
>.  /\  ph ) ) )
65drex1 2070 . . 3  |-  ( A. x  x  =  y  ->  ( E. x E. z ( w  = 
<. x ,  z >.  /\  ph )  <->  E. y E. z ( w  = 
<. y ,  z >.  /\  ph ) ) )
76abbidv 2593 . 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 4516 . 2  |-  { <. x ,  z >.  |  ph }  =  { w  |  E. x E. z
( w  =  <. x ,  z >.  /\  ph ) }
9 df-opab 4516 . 2  |-  { <. y ,  z >.  |  ph }  =  { w  |  E. y E. z
( w  =  <. y ,  z >.  /\  ph ) }
107, 8, 93eqtr4g 2523 1  |-  ( A. x  x  =  y  ->  { <. x ,  z
>.  |  ph }  =  { <. y ,  z
>.  |  ph } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1393    = wceq 1395   E.wex 1613   {cab 2442   <.cop 4038   {copab 4514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-opab 4516
This theorem is referenced by: (None)
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