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Theorem dropab1 29701
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 4058 . . . . . . . 8  |-  ( x  =  y  ->  <. x ,  z >.  =  <. y ,  z >. )
21sps 1800 . . . . . . 7  |-  ( A. x  x  =  y  -> 
<. x ,  z >.  =  <. y ,  z
>. )
32eqeq2d 2453 . . . . . 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 2020 . . . 4  |-  ( A. x  x  =  y  ->  ( E. z ( w  =  <. x ,  z >.  /\  ph ) 
<->  E. z ( w  =  <. y ,  z
>.  /\  ph ) ) )
65drex1 2019 . . 3  |-  ( A. x  x  =  y  ->  ( E. x E. z ( w  = 
<. x ,  z >.  /\  ph )  <->  E. y E. z ( w  = 
<. y ,  z >.  /\  ph ) ) )
76abbidv 2556 . 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 4350 . 2  |-  { <. x ,  z >.  |  ph }  =  { w  |  E. x E. z
( w  =  <. x ,  z >.  /\  ph ) }
9 df-opab 4350 . 2  |-  { <. y ,  z >.  |  ph }  =  { w  |  E. y E. z
( w  =  <. y ,  z >.  /\  ph ) }
107, 8, 93eqtr4g 2499 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 1367    = wceq 1369   E.wex 1586   {cab 2428   <.cop 3882   {copab 4348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-opab 4350
This theorem is referenced by: (None)
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