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Theorem eqeu 3267
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 3192 . . . 4  |-  ( A  e.  B  ->  ( ps  ->  E. x ph )
)
32imp 427 . . 3  |-  ( ( A  e.  B  /\  ps )  ->  E. x ph )
433adant3 1014 . 2  |-  ( ( A  e.  B  /\  ps  /\  A. x (
ph  ->  x  =  A ) )  ->  E. x ph )
5 eqeq2 2469 . . . . . . 7  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
65imbi2d 314 . . . . . 6  |-  ( y  =  A  ->  (
( ph  ->  x  =  y )  <->  ( ph  ->  x  =  A ) ) )
76albidv 1718 . . . . 5  |-  ( y  =  A  ->  ( A. x ( ph  ->  x  =  y )  <->  A. x
( ph  ->  x  =  A ) ) )
87spcegv 3192 . . . 4  |-  ( A  e.  B  ->  ( A. x ( ph  ->  x  =  A )  ->  E. y A. x (
ph  ->  x  =  y ) ) )
98imp 427 . . 3  |-  ( ( A  e.  B  /\  A. x ( ph  ->  x  =  A ) )  ->  E. y A. x
( ph  ->  x  =  y ) )
1093adant2 1013 . 2  |-  ( ( A  e.  B  /\  ps  /\  A. x (
ph  ->  x  =  A ) )  ->  E. y A. x ( ph  ->  x  =  y ) )
11 eu3v 2314 . 2  |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
124, 10, 11sylanbrc 662 1  |-  ( ( A  e.  B  /\  ps  /\  A. x (
ph  ->  x  =  A ) )  ->  E! x ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 971   A.wal 1396    = wceq 1398   E.wex 1617    e. wcel 1823   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  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108
This theorem is referenced by:  rngurd  28013  neibastop3  30420  upixp  30460  zrinitorngc  33062  zrtermorngc  33063  zrtermoringc  33132
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