MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reu6i Unicode version

Theorem reu6i 2969
Description: A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)
Assertion
Ref Expression
reu6i  |-  ( ( B  e.  A  /\  A. x  e.  A  (
ph 
<->  x  =  B ) )  ->  E! x  e.  A  ph )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    ph( x)

Proof of Theorem reu6i
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqeq2 2305 . . . . 5  |-  ( y  =  B  ->  (
x  =  y  <->  x  =  B ) )
21bibi2d 309 . . . 4  |-  ( y  =  B  ->  (
( ph  <->  x  =  y
)  <->  ( ph  <->  x  =  B ) ) )
32ralbidv 2576 . . 3  |-  ( y  =  B  ->  ( A. x  e.  A  ( ph  <->  x  =  y
)  <->  A. x  e.  A  ( ph  <->  x  =  B
) ) )
43rspcev 2897 . 2  |-  ( ( B  e.  A  /\  A. x  e.  A  (
ph 
<->  x  =  B ) )  ->  E. y  e.  A  A. x  e.  A  ( ph  <->  x  =  y ) )
5 reu6 2967 . 2  |-  ( E! x  e.  A  ph  <->  E. y  e.  A  A. x  e.  A  ( ph 
<->  x  =  y ) )
64, 5sylibr 203 1  |-  ( ( B  e.  A  /\  A. x  e.  A  (
ph 
<->  x  =  B ) )  ->  E! x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   E!wreu 2558
This theorem is referenced by:  eqreu  2970  riota5f  6345  negeu  9058  creur  9756  creui  9757  dfod2  14893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-reu 2563  df-v 2803
  Copyright terms: Public domain W3C validator