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Theorem eqeu 3107
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 3039 . . . 4  |-  ( A  e.  B  ->  ( ps  ->  E. x ph )
)
32imp 420 . . 3  |-  ( ( A  e.  B  /\  ps )  ->  E. x ph )
433adant3 978 . 2  |-  ( ( A  e.  B  /\  ps  /\  A. x (
ph  ->  x  =  A ) )  ->  E. x ph )
5 eqeq2 2447 . . . . . . 7  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
65imbi2d 309 . . . . . 6  |-  ( y  =  A  ->  (
( ph  ->  x  =  y )  <->  ( ph  ->  x  =  A ) ) )
76albidv 1636 . . . . 5  |-  ( y  =  A  ->  ( A. x ( ph  ->  x  =  y )  <->  A. x
( ph  ->  x  =  A ) ) )
87spcegv 3039 . . . 4  |-  ( A  e.  B  ->  ( A. x ( ph  ->  x  =  A )  ->  E. y A. x (
ph  ->  x  =  y ) ) )
98imp 420 . . 3  |-  ( ( A  e.  B  /\  A. x ( ph  ->  x  =  A ) )  ->  E. y A. x
( ph  ->  x  =  y ) )
1093adant2 977 . 2  |-  ( ( A  e.  B  /\  ps  /\  A. x (
ph  ->  x  =  A ) )  ->  E. y A. x ( ph  ->  x  =  y ) )
11 nfv 1630 . . 3  |-  F/ y
ph
1211eu3 2309 . 2  |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
134, 10, 12sylanbrc 647 1  |-  ( ( A  e.  B  /\  ps  /\  A. x (
ph  ->  x  =  A ) )  ->  E! x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ w3a 937   A.wal 1550   E.wex 1551    = wceq 1653    e. wcel 1726   E!weu 2283
This theorem is referenced by:  neibastop3  26405  upixp  26445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960
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