MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sb8iota Structured version   Unicode version

Theorem sb8iota 5497
Description: Variable substitution in description binder. Compare sb8eu 2301. (Contributed by NM, 18-Mar-2013.)
Hypothesis
Ref Expression
sb8iota.1  |-  F/ y
ph
Assertion
Ref Expression
sb8iota  |-  ( iota
x ph )  =  ( iota y [ y  /  x ] ph )

Proof of Theorem sb8iota
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1674 . . . . . 6  |-  F/ w
( ph  <->  x  =  z
)
21sb8 2134 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  <->  A. w [ w  /  x ] ( ph  <->  x  =  z ) )
3 sbbi 2103 . . . . . . 7  |-  ( [ w  /  x ]
( ph  <->  x  =  z
)  <->  ( [ w  /  x ] ph  <->  [ w  /  x ] x  =  z ) )
4 sb8iota.1 . . . . . . . . 9  |-  F/ y
ph
54nfsb 2155 . . . . . . . 8  |-  F/ y [ w  /  x ] ph
6 equsb3 2146 . . . . . . . . 9  |-  ( [ w  /  x ]
x  =  z  <->  w  =  z )
7 nfv 1674 . . . . . . . . 9  |-  F/ y  w  =  z
86, 7nfxfr 1616 . . . . . . . 8  |-  F/ y [ w  /  x ] x  =  z
95, 8nfbi 1872 . . . . . . 7  |-  F/ y ( [ w  /  x ] ph  <->  [ w  /  x ] x  =  z )
103, 9nfxfr 1616 . . . . . 6  |-  F/ y [ w  /  x ] ( ph  <->  x  =  z )
11 nfv 1674 . . . . . 6  |-  F/ w [ y  /  x ] ( ph  <->  x  =  z )
12 sbequ 2077 . . . . . 6  |-  ( w  =  y  ->  ( [ w  /  x ] ( ph  <->  x  =  z )  <->  [ y  /  x ] ( ph  <->  x  =  z ) ) )
1310, 11, 12cbval 1981 . . . . 5  |-  ( A. w [ w  /  x ] ( ph  <->  x  =  z )  <->  A. y [ y  /  x ] ( ph  <->  x  =  z ) )
14 equsb3 2146 . . . . . . 7  |-  ( [ y  /  x ]
x  =  z  <->  y  =  z )
1514sblbis 2106 . . . . . 6  |-  ( [ y  /  x ]
( ph  <->  x  =  z
)  <->  ( [ y  /  x ] ph  <->  y  =  z ) )
1615albii 1611 . . . . 5  |-  ( A. y [ y  /  x ] ( ph  <->  x  =  z )  <->  A. y
( [ y  /  x ] ph  <->  y  =  z ) )
172, 13, 163bitri 271 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  <->  A. y
( [ y  /  x ] ph  <->  y  =  z ) )
1817abbii 2588 . . 3  |-  { z  |  A. x (
ph 
<->  x  =  z ) }  =  { z  |  A. y ( [ y  /  x ] ph  <->  y  =  z ) }
1918unieqi 4209 . 2  |-  U. {
z  |  A. x
( ph  <->  x  =  z
) }  =  U. { z  |  A. y ( [ y  /  x ] ph  <->  y  =  z ) }
20 dfiota2 5491 . 2  |-  ( iota
x ph )  =  U. { z  |  A. x ( ph  <->  x  =  z ) }
21 dfiota2 5491 . 2  |-  ( iota y [ y  /  x ] ph )  = 
U. { z  | 
A. y ( [ y  /  x ] ph 
<->  y  =  z ) }
2219, 20, 213eqtr4i 2493 1  |-  ( iota
x ph )  =  ( iota y [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   A.wal 1368    = wceq 1370   F/wnf 1590   [wsb 1702   {cab 2439   U.cuni 4200   iotacio 5488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2805  df-sn 3987  df-uni 4201  df-iota 5490
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator