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Theorem oprabbid 6363
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 718 . . . . . 6  |-  ( ph  ->  ( ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ps )  <->  ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ch ) ) )
63, 5exbid 1984 . . . . 5  |-  ( ph  ->  ( E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ps )  <->  E. z
( w  =  <. <.
x ,  y >. ,  z >.  /\  ch ) ) )
72, 6exbid 1984 . . . 4  |-  ( ph  ->  ( E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ps )  <->  E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ch ) ) )
81, 7exbid 1984 . . 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 2589 . 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 6312 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ps }  =  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ps ) }
11 df-oprab 6312 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ch }  =  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ch ) }
129, 10, 113eqtr4g 2530 1  |-  ( ph  ->  { <. <. x ,  y
>. ,  z >.  |  ps }  =  { <. <. x ,  y
>. ,  z >.  |  ch } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452   E.wex 1671   F/wnf 1675   {cab 2457   <.cop 3965   {coprab 6309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-oprab 6312
This theorem is referenced by:  oprabbidv  6364  mpt2eq123  6369
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