MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvoprab1 Structured version   Unicode version

Theorem cbvoprab1 6157
Description: Rule used to change first bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 5-Dec-2016.)
Hypotheses
Ref Expression
cbvoprab1.1  |-  F/ w ph
cbvoprab1.2  |-  F/ x ps
cbvoprab1.3  |-  ( x  =  w  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvoprab1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. w ,  y >. ,  z
>.  |  ps }
Distinct variable group:    x, y, z, w
Allowed substitution hints:    ph( x, y, z, w)    ps( x, y, z, w)

Proof of Theorem cbvoprab1
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 nfv 1673 . . . . . 6  |-  F/ w  v  =  <. x ,  y >.
2 cbvoprab1.1 . . . . . 6  |-  F/ w ph
31, 2nfan 1861 . . . . 5  |-  F/ w
( v  =  <. x ,  y >.  /\  ph )
43nfex 1874 . . . 4  |-  F/ w E. y ( v  = 
<. x ,  y >.  /\  ph )
5 nfv 1673 . . . . . 6  |-  F/ x  v  =  <. w ,  y >.
6 cbvoprab1.2 . . . . . 6  |-  F/ x ps
75, 6nfan 1861 . . . . 5  |-  F/ x
( v  =  <. w ,  y >.  /\  ps )
87nfex 1874 . . . 4  |-  F/ x E. y ( v  = 
<. w ,  y >.  /\  ps )
9 opeq1 4058 . . . . . . 7  |-  ( x  =  w  ->  <. x ,  y >.  =  <. w ,  y >. )
109eqeq2d 2453 . . . . . 6  |-  ( x  =  w  ->  (
v  =  <. x ,  y >.  <->  v  =  <. w ,  y >.
) )
11 cbvoprab1.3 . . . . . 6  |-  ( x  =  w  ->  ( ph 
<->  ps ) )
1210, 11anbi12d 710 . . . . 5  |-  ( x  =  w  ->  (
( v  =  <. x ,  y >.  /\  ph ) 
<->  ( v  =  <. w ,  y >.  /\  ps ) ) )
1312exbidv 1680 . . . 4  |-  ( x  =  w  ->  ( E. y ( v  = 
<. x ,  y >.  /\  ph )  <->  E. y
( v  =  <. w ,  y >.  /\  ps ) ) )
144, 8, 13cbvex 1970 . . 3  |-  ( E. x E. y ( v  =  <. x ,  y >.  /\  ph ) 
<->  E. w E. y
( v  =  <. w ,  y >.  /\  ps ) )
1514opabbii 4355 . 2  |-  { <. v ,  z >.  |  E. x E. y ( v  =  <. x ,  y
>.  /\  ph ) }  =  { <. v ,  z >.  |  E. w E. y ( v  =  <. w ,  y
>.  /\  ps ) }
16 dfoprab2 6132 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. v ,  z >.  |  E. x E. y ( v  =  <. x ,  y
>.  /\  ph ) }
17 dfoprab2 6132 . 2  |-  { <. <.
w ,  y >. ,  z >.  |  ps }  =  { <. v ,  z >.  |  E. w E. y ( v  =  <. w ,  y
>.  /\  ps ) }
1815, 16, 173eqtr4i 2472 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. w ,  y >. ,  z
>.  |  ps }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586   F/wnf 1589   <.cop 3882   {copab 4348   {coprab 6091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-opab 4350  df-oprab 6094
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator