MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvoprab2 Structured version   Visualization version   Unicode version

Theorem cbvoprab2 6383
Description: Change the second bound variable in an operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypotheses
Ref Expression
cbvoprab2.1  |-  F/ w ph
cbvoprab2.2  |-  F/ y ps
cbvoprab2.3  |-  ( y  =  w  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvoprab2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. x ,  w >. ,  z >.  |  ps }
Distinct variable group:    x, w, y, z
Allowed substitution hints:    ph( x, y, z, w)    ps( x, y, z, w)

Proof of Theorem cbvoprab2
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 nfv 1769 . . . . . . 7  |-  F/ w  v  =  <. <. x ,  y >. ,  z
>.
2 cbvoprab2.1 . . . . . . 7  |-  F/ w ph
31, 2nfan 2031 . . . . . 6  |-  F/ w
( v  =  <. <.
x ,  y >. ,  z >.  /\  ph )
43nfex 2050 . . . . 5  |-  F/ w E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )
5 nfv 1769 . . . . . . 7  |-  F/ y  v  =  <. <. x ,  w >. ,  z >.
6 cbvoprab2.2 . . . . . . 7  |-  F/ y ps
75, 6nfan 2031 . . . . . 6  |-  F/ y ( v  =  <. <.
x ,  w >. ,  z >.  /\  ps )
87nfex 2050 . . . . 5  |-  F/ y E. z ( v  =  <. <. x ,  w >. ,  z >.  /\  ps )
9 opeq2 4159 . . . . . . . . 9  |-  ( y  =  w  ->  <. x ,  y >.  =  <. x ,  w >. )
109opeq1d 4164 . . . . . . . 8  |-  ( y  =  w  ->  <. <. x ,  y >. ,  z
>.  =  <. <. x ,  w >. ,  z >.
)
1110eqeq2d 2481 . . . . . . 7  |-  ( y  =  w  ->  (
v  =  <. <. x ,  y >. ,  z
>. 
<->  v  =  <. <. x ,  w >. ,  z >.
) )
12 cbvoprab2.3 . . . . . . 7  |-  ( y  =  w  ->  ( ph 
<->  ps ) )
1311, 12anbi12d 725 . . . . . 6  |-  ( y  =  w  ->  (
( v  =  <. <.
x ,  y >. ,  z >.  /\  ph ) 
<->  ( v  =  <. <.
x ,  w >. ,  z >.  /\  ps )
) )
1413exbidv 1776 . . . . 5  |-  ( y  =  w  ->  ( E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. z
( v  =  <. <.
x ,  w >. ,  z >.  /\  ps )
) )
154, 8, 14cbvex 2128 . . . 4  |-  ( E. y E. z ( v  =  <. <. x ,  y >. ,  z
>.  /\  ph )  <->  E. w E. z ( v  = 
<. <. x ,  w >. ,  z >.  /\  ps ) )
1615exbii 1726 . . 3  |-  ( E. x E. y E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. x E. w E. z ( v  =  <. <. x ,  w >. ,  z >.  /\  ps ) )
1716abbii 2587 . 2  |-  { v  |  E. x E. y E. z ( v  =  <. <. x ,  y
>. ,  z >.  /\ 
ph ) }  =  { v  |  E. x E. w E. z
( v  =  <. <.
x ,  w >. ,  z >.  /\  ps ) }
18 df-oprab 6312 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { v  |  E. x E. y E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
19 df-oprab 6312 . 2  |-  { <. <.
x ,  w >. ,  z >.  |  ps }  =  { v  |  E. x E. w E. z ( v  = 
<. <. x ,  w >. ,  z >.  /\  ps ) }
2017, 18, 193eqtr4i 2503 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. x ,  w >. ,  z >.  |  ps }
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-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-oprab 6312
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator