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Theorem dropab2 27629
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 3987 . . . . . . . 8  |-  ( x  =  y  ->  <. z ,  x >.  =  <. z ,  y >. )
21sps 1771 . . . . . . 7  |-  ( A. x  x  =  y  -> 
<. z ,  x >.  = 
<. z ,  y >.
)
32eqeq2d 2449 . . . . . 6  |-  ( A. x  x  =  y  ->  ( w  =  <. z ,  x >.  <->  w  =  <. z ,  y >.
) )
43anbi1d 687 . . . . 5  |-  ( A. x  x  =  y  ->  ( ( w  = 
<. z ,  x >.  /\ 
ph )  <->  ( w  =  <. z ,  y
>.  /\  ph ) ) )
54drex1 2060 . . . 4  |-  ( A. x  x  =  y  ->  ( E. x ( w  =  <. z ,  x >.  /\  ph )  <->  E. y ( w  = 
<. z ,  y >.  /\  ph ) ) )
65drex2 2061 . . 3  |-  ( A. x  x  =  y  ->  ( E. z E. x ( w  = 
<. z ,  x >.  /\ 
ph )  <->  E. z E. y ( w  = 
<. z ,  y >.  /\  ph ) ) )
76abbidv 2552 . 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 4269 . 2  |-  { <. z ,  x >.  |  ph }  =  { w  |  E. z E. x
( w  =  <. z ,  x >.  /\  ph ) }
9 df-opab 4269 . 2  |-  { <. z ,  y >.  |  ph }  =  { w  |  E. z E. y
( w  =  <. z ,  y >.  /\  ph ) }
107, 8, 93eqtr4g 2495 1  |-  ( A. x  x  =  y  ->  { <. z ,  x >.  |  ph }  =  { <. z ,  y
>.  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   A.wal 1550   E.wex 1551    = wceq 1653   {cab 2424   <.cop 3819   {copab 4267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4269
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