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

Theorem eqeu 2936
Description: A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.)
Hypothesis
Ref Expression
eqeu.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
eqeu  |-  ( ( A  e.  B  /\  ps  /\  A. x (
ph  ->  x  =  A ) )  ->  E! x ph )
Distinct variable groups:    ps, x    x, A
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem eqeu
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqeu.1 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
21spcegv 2869 . . . 4  |-  ( A  e.  B  ->  ( ps  ->  E. x ph )
)
32imp 418 . . 3  |-  ( ( A  e.  B  /\  ps )  ->  E. x ph )
433adant3 975 . 2  |-  ( ( A  e.  B  /\  ps  /\  A. x (
ph  ->  x  =  A ) )  ->  E. x ph )
5 eqeq2 2292 . . . . . . 7  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
65imbi2d 307 . . . . . 6  |-  ( y  =  A  ->  (
( ph  ->  x  =  y )  <->  ( ph  ->  x  =  A ) ) )
76albidv 1611 . . . . 5  |-  ( y  =  A  ->  ( A. x ( ph  ->  x  =  y )  <->  A. x
( ph  ->  x  =  A ) ) )
87spcegv 2869 . . . 4  |-  ( A  e.  B  ->  ( A. x ( ph  ->  x  =  A )  ->  E. y A. x (
ph  ->  x  =  y ) ) )
98imp 418 . . 3  |-  ( ( A  e.  B  /\  A. x ( ph  ->  x  =  A ) )  ->  E. y A. x
( ph  ->  x  =  y ) )
1093adant2 974 . 2  |-  ( ( A  e.  B  /\  ps  /\  A. x (
ph  ->  x  =  A ) )  ->  E. y A. x ( ph  ->  x  =  y ) )
11 nfv 1605 . . 3  |-  F/ y
ph
1211eu3 2169 . 2  |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
134, 10, 12sylanbrc 645 1  |-  ( ( A  e.  B  /\  ps  /\  A. x (
ph  ->  x  =  A ) )  ->  E! x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   E!weu 2143
This theorem is referenced by:  eqeuOLD  26223  neibastop3  26311  upixp  26403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790
  Copyright terms: Public domain W3C validator