Users' Mathboxes Mathbox for ML < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  csboprabg Structured version   Visualization version   Unicode version

Theorem csboprabg 31801
Description: Move class substitution in and out of class abstractions of nested ordered pairs. (Contributed by ML, 25-Oct-2020.)
Assertion
Ref Expression
csboprabg  |-  ( A  e.  V  ->  [_ A  /  x ]_ { <. <.
y ,  z >. ,  d >.  |  ph }  =  { <. <. y ,  z >. ,  d
>.  |  [. A  /  x ]. ph } )
Distinct variable groups:    A, d    y, A    z, A    V, d    y, V    z, V    x, d    x, y    x, z
Allowed substitution hints:    ph( x, y, z, d)    A( x)    V( x)

Proof of Theorem csboprabg
Dummy variable  c is distinct from all other variables.
StepHypRef Expression
1 csbab 3801 . . 3  |-  [_ A  /  x ]_ { c  |  E. y E. z E. d ( c  =  <. <. y ,  z >. ,  d
>.  /\  ph ) }  =  { c  | 
[. A  /  x ]. E. y E. z E. d ( c  = 
<. <. y ,  z
>. ,  d >.  /\ 
ph ) }
2 sbcex2 3306 . . . . 5  |-  ( [. A  /  x ]. E. y E. z E. d
( c  =  <. <.
y ,  z >. ,  d >.  /\  ph ) 
<->  E. y [. A  /  x ]. E. z E. d ( c  = 
<. <. y ,  z
>. ,  d >.  /\ 
ph ) )
3 sbcex2 3306 . . . . . . 7  |-  ( [. A  /  x ]. E. z E. d ( c  =  <. <. y ,  z
>. ,  d >.  /\ 
ph )  <->  E. z [. A  /  x ]. E. d ( c  =  <. <. y ,  z
>. ,  d >.  /\ 
ph ) )
4 sbcex2 3306 . . . . . . . . 9  |-  ( [. A  /  x ]. E. d ( c  = 
<. <. y ,  z
>. ,  d >.  /\ 
ph )  <->  E. d [. A  /  x ]. ( c  =  <. <.
y ,  z >. ,  d >.  /\  ph ) )
5 sbcan 3298 . . . . . . . . . . 11  |-  ( [. A  /  x ]. (
c  =  <. <. y ,  z >. ,  d
>.  /\  ph )  <->  ( [. A  /  x ]. c  =  <. <. y ,  z
>. ,  d >.  /\ 
[. A  /  x ]. ph ) )
6 sbcg 3321 . . . . . . . . . . . 12  |-  ( A  e.  V  ->  ( [. A  /  x ]. c  =  <. <.
y ,  z >. ,  d >.  <->  c  =  <. <. y ,  z
>. ,  d >. ) )
76anbi1d 719 . . . . . . . . . . 11  |-  ( A  e.  V  ->  (
( [. A  /  x ]. c  =  <. <.
y ,  z >. ,  d >.  /\  [. A  /  x ]. ph )  <->  ( c  =  <. <. y ,  z >. ,  d
>.  /\  [. A  /  x ]. ph ) ) )
85, 7syl5bb 265 . . . . . . . . . 10  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( c  =  <. <.
y ,  z >. ,  d >.  /\  ph ) 
<->  ( c  =  <. <.
y ,  z >. ,  d >.  /\  [. A  /  x ]. ph )
) )
98exbidv 1776 . . . . . . . . 9  |-  ( A  e.  V  ->  ( E. d [. A  /  x ]. ( c  = 
<. <. y ,  z
>. ,  d >.  /\ 
ph )  <->  E. d
( c  =  <. <.
y ,  z >. ,  d >.  /\  [. A  /  x ]. ph )
) )
104, 9syl5bb 265 . . . . . . . 8  |-  ( A  e.  V  ->  ( [. A  /  x ]. E. d ( c  =  <. <. y ,  z
>. ,  d >.  /\ 
ph )  <->  E. d
( c  =  <. <.
y ,  z >. ,  d >.  /\  [. A  /  x ]. ph )
) )
1110exbidv 1776 . . . . . . 7  |-  ( A  e.  V  ->  ( E. z [. A  /  x ]. E. d ( c  =  <. <. y ,  z >. ,  d
>.  /\  ph )  <->  E. z E. d ( c  = 
<. <. y ,  z
>. ,  d >.  /\ 
[. A  /  x ]. ph ) ) )
123, 11syl5bb 265 . . . . . 6  |-  ( A  e.  V  ->  ( [. A  /  x ]. E. z E. d
( c  =  <. <.
y ,  z >. ,  d >.  /\  ph ) 
<->  E. z E. d
( c  =  <. <.
y ,  z >. ,  d >.  /\  [. A  /  x ]. ph )
) )
1312exbidv 1776 . . . . 5  |-  ( A  e.  V  ->  ( E. y [. A  /  x ]. E. z E. d ( c  = 
<. <. y ,  z
>. ,  d >.  /\ 
ph )  <->  E. y E. z E. d ( c  =  <. <. y ,  z >. ,  d
>.  /\  [. A  /  x ]. ph ) ) )
142, 13syl5bb 265 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. E. y E. z E. d ( c  = 
<. <. y ,  z
>. ,  d >.  /\ 
ph )  <->  E. y E. z E. d ( c  =  <. <. y ,  z >. ,  d
>.  /\  [. A  /  x ]. ph ) ) )
1514abbidv 2589 . . 3  |-  ( A  e.  V  ->  { c  |  [. A  /  x ]. E. y E. z E. d ( c  =  <. <. y ,  z >. ,  d
>.  /\  ph ) }  =  { c  |  E. y E. z E. d ( c  = 
<. <. y ,  z
>. ,  d >.  /\ 
[. A  /  x ]. ph ) } )
161, 15syl5eq 2517 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ { c  |  E. y E. z E. d ( c  =  <. <. y ,  z >. ,  d
>.  /\  ph ) }  =  { c  |  E. y E. z E. d ( c  = 
<. <. y ,  z
>. ,  d >.  /\ 
[. A  /  x ]. ph ) } )
17 df-oprab 6312 . . 3  |-  { <. <.
y ,  z >. ,  d >.  |  ph }  =  { c  |  E. y E. z E. d ( c  = 
<. <. y ,  z
>. ,  d >.  /\ 
ph ) }
1817csbeq2i 3786 . 2  |-  [_ A  /  x ]_ { <. <.
y ,  z >. ,  d >.  |  ph }  =  [_ A  /  x ]_ { c  |  E. y E. z E. d ( c  = 
<. <. y ,  z
>. ,  d >.  /\ 
ph ) }
19 df-oprab 6312 . 2  |-  { <. <.
y ,  z >. ,  d >.  |  [. A  /  x ]. ph }  =  { c  |  E. y E. z E. d
( c  =  <. <.
y ,  z >. ,  d >.  /\  [. A  /  x ]. ph ) }
2016, 18, 193eqtr4g 2530 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ { <. <.
y ,  z >. ,  d >.  |  ph }  =  { <. <. y ,  z >. ,  d
>.  |  [. A  /  x ]. ph } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   {cab 2457   [.wsbc 3255   [_csb 3349   <.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-or 377  df-an 378  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-in 3397  df-ss 3404  df-nul 3723  df-oprab 6312
This theorem is referenced by:  csbmpt22g  31802
  Copyright terms: Public domain W3C validator