MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sb8eu Structured version   Unicode version

Theorem sb8eu 2319
Description: Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Aug-2019.)
Hypothesis
Ref Expression
sb8eu.1  |-  F/ y
ph
Assertion
Ref Expression
sb8eu  |-  ( E! x ph  <->  E! y [ y  /  x ] ph )

Proof of Theorem sb8eu
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1712 . . . . 5  |-  F/ w
( ph  <->  x  =  z
)
21sb8 2169 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  <->  A. w [ w  /  x ] ( ph  <->  x  =  z ) )
3 equsb3 2178 . . . . . 6  |-  ( [ w  /  x ]
x  =  z  <->  w  =  z )
43sblbis 2147 . . . . 5  |-  ( [ w  /  x ]
( ph  <->  x  =  z
)  <->  ( [ w  /  x ] ph  <->  w  =  z ) )
54albii 1645 . . . 4  |-  ( A. w [ w  /  x ] ( ph  <->  x  =  z )  <->  A. w
( [ w  /  x ] ph  <->  w  =  z ) )
6 sb8eu.1 . . . . . . 7  |-  F/ y
ph
76nfsb 2186 . . . . . 6  |-  F/ y [ w  /  x ] ph
8 nfv 1712 . . . . . 6  |-  F/ y  w  =  z
97, 8nfbi 1939 . . . . 5  |-  F/ y ( [ w  /  x ] ph  <->  w  =  z )
10 nfv 1712 . . . . 5  |-  F/ w
( [ y  /  x ] ph  <->  y  =  z )
11 sbequ 2119 . . . . . 6  |-  ( w  =  y  ->  ( [ w  /  x ] ph  <->  [ y  /  x ] ph ) )
12 equequ1 1803 . . . . . 6  |-  ( w  =  y  ->  (
w  =  z  <->  y  =  z ) )
1311, 12bibi12d 319 . . . . 5  |-  ( w  =  y  ->  (
( [ w  /  x ] ph  <->  w  =  z )  <->  ( [
y  /  x ] ph 
<->  y  =  z ) ) )
149, 10, 13cbval 2026 . . . 4  |-  ( A. w ( [ w  /  x ] ph  <->  w  =  z )  <->  A. y
( [ y  /  x ] ph  <->  y  =  z ) )
152, 5, 143bitri 271 . . 3  |-  ( A. x ( ph  <->  x  =  z )  <->  A. y
( [ y  /  x ] ph  <->  y  =  z ) )
1615exbii 1672 . 2  |-  ( E. z A. x (
ph 
<->  x  =  z )  <->  E. z A. y ( [ y  /  x ] ph  <->  y  =  z ) )
17 df-eu 2288 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
18 df-eu 2288 . 2  |-  ( E! y [ y  /  x ] ph  <->  E. z A. y ( [ y  /  x ] ph  <->  y  =  z ) )
1916, 17, 183bitr4i 277 1  |-  ( E! x ph  <->  E! y [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   A.wal 1396   E.wex 1617   F/wnf 1621   [wsb 1744   E!weu 2284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288
This theorem is referenced by:  sb8mo  2321  cbveu  2322  eu1  2328  cbvreu  3079
  Copyright terms: Public domain W3C validator