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Theorem oprabbid 6067
Description: Equivalent wff's yield equal operation class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
oprabbid.1  |-  F/ x ph
oprabbid.2  |-  F/ y
ph
oprabbid.3  |-  F/ z
ph
oprabbid.4  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
oprabbid  |-  ( ph  ->  { <. <. x ,  y
>. ,  z >.  |  ps }  =  { <. <. x ,  y
>. ,  z >.  |  ch } )
Distinct variable groups:    x, z    y, z
Allowed substitution hints:    ph( x, y, z)    ps( x, y, z)    ch( x, y, z)

Proof of Theorem oprabbid
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 oprabbid.1 . . . 4  |-  F/ x ph
2 oprabbid.2 . . . . 5  |-  F/ y
ph
3 oprabbid.3 . . . . . 6  |-  F/ z
ph
4 oprabbid.4 . . . . . . 7  |-  ( ph  ->  ( ps  <->  ch )
)
54anbi2d 685 . . . . . 6  |-  ( ph  ->  ( ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ps )  <->  ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ch ) ) )
63, 5exbid 1781 . . . . 5  |-  ( ph  ->  ( E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ps )  <->  E. z
( w  =  <. <.
x ,  y >. ,  z >.  /\  ch ) ) )
72, 6exbid 1781 . . . 4  |-  ( ph  ->  ( E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ps )  <->  E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ch ) ) )
81, 7exbid 1781 . . 3  |-  ( ph  ->  ( E. x E. y E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ps )  <->  E. x E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ch ) ) )
98abbidv 2502 . 2  |-  ( ph  ->  { w  |  E. x E. y E. z
( w  =  <. <.
x ,  y >. ,  z >.  /\  ps ) }  =  {
w  |  E. x E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ch ) } )
10 df-oprab 6025 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ps }  =  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ps ) }
11 df-oprab 6025 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ch }  =  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ch ) }
129, 10, 113eqtr4g 2445 1  |-  ( ph  ->  { <. <. x ,  y
>. ,  z >.  |  ps }  =  { <. <. x ,  y
>. ,  z >.  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547   F/wnf 1550    = wceq 1649   {cab 2374   <.cop 3761   {coprab 6022
This theorem is referenced by:  oprabbidv  6068  mpt2eq123  6073
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-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-oprab 6025
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