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Theorem dropab1 27319
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 3927 . . . . . . . 8  |-  ( x  =  y  ->  <. x ,  z >.  =  <. y ,  z >. )
21sps 1762 . . . . . . 7  |-  ( A. x  x  =  y  -> 
<. x ,  z >.  =  <. y ,  z
>. )
32eqeq2d 2399 . . . . . 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 2009 . . . 4  |-  ( A. x  x  =  y  ->  ( E. z ( w  =  <. x ,  z >.  /\  ph ) 
<->  E. z ( w  =  <. y ,  z
>.  /\  ph ) ) )
65drex1 2008 . . 3  |-  ( A. x  x  =  y  ->  ( E. x E. z ( w  = 
<. x ,  z >.  /\  ph )  <->  E. y E. z ( w  = 
<. y ,  z >.  /\  ph ) ) )
76abbidv 2502 . 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 4209 . 2  |-  { <. x ,  z >.  |  ph }  =  { w  |  E. x E. z
( w  =  <. x ,  z >.  /\  ph ) }
9 df-opab 4209 . 2  |-  { <. y ,  z >.  |  ph }  =  { w  |  E. y E. z
( w  =  <. y ,  z >.  /\  ph ) }
107, 8, 93eqtr4g 2445 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 1546   E.wex 1547    = wceq 1649   {cab 2374   <.cop 3761   {copab 4207
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-opab 4209
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