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Theorem eqeu 3065
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 2997 . . . 4  |-  ( A  e.  B  ->  ( ps  ->  E. x ph )
)
32imp 419 . . 3  |-  ( ( A  e.  B  /\  ps )  ->  E. x ph )
433adant3 977 . 2  |-  ( ( A  e.  B  /\  ps  /\  A. x (
ph  ->  x  =  A ) )  ->  E. x ph )
5 eqeq2 2413 . . . . . . 7  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
65imbi2d 308 . . . . . 6  |-  ( y  =  A  ->  (
( ph  ->  x  =  y )  <->  ( ph  ->  x  =  A ) ) )
76albidv 1632 . . . . 5  |-  ( y  =  A  ->  ( A. x ( ph  ->  x  =  y )  <->  A. x
( ph  ->  x  =  A ) ) )
87spcegv 2997 . . . 4  |-  ( A  e.  B  ->  ( A. x ( ph  ->  x  =  A )  ->  E. y A. x (
ph  ->  x  =  y ) ) )
98imp 419 . . 3  |-  ( ( A  e.  B  /\  A. x ( ph  ->  x  =  A ) )  ->  E. y A. x
( ph  ->  x  =  y ) )
1093adant2 976 . 2  |-  ( ( A  e.  B  /\  ps  /\  A. x (
ph  ->  x  =  A ) )  ->  E. y A. x ( ph  ->  x  =  y ) )
11 nfv 1626 . . 3  |-  F/ y
ph
1211eu3 2280 . 2  |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
134, 10, 12sylanbrc 646 1  |-  ( ( A  e.  B  /\  ps  /\  A. x (
ph  ->  x  =  A ) )  ->  E! x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1721   E!weu 2254
This theorem is referenced by:  neibastop3  26281  upixp  26321
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918
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