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Theorem eu1 2359
Description: An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 29-Oct-2018.)
Hypothesis
Ref Expression
eu1.1  |-  F/ y
ph
Assertion
Ref Expression
eu1  |-  ( E! x ph  <->  E. x
( ph  /\  A. y
( [ y  /  x ] ph  ->  x  =  y ) ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem eu1
StepHypRef Expression
1 nfs1v 2286 . . 3  |-  F/ x [ y  /  x ] ph
21euf 2327 . 2  |-  ( E! y [ y  /  x ] ph  <->  E. x A. y ( [ y  /  x ] ph  <->  y  =  x ) )
3 eu1.1 . . 3  |-  F/ y
ph
43sb8eu 2352 . 2  |-  ( E! x ph  <->  E! y [ y  /  x ] ph )
53sb6rf 2272 . . . . 5  |-  ( ph  <->  A. y ( y  =  x  ->  [ y  /  x ] ph )
)
6 equcom 1870 . . . . . . 7  |-  ( x  =  y  <->  y  =  x )
76imbi2i 319 . . . . . 6  |-  ( ( [ y  /  x ] ph  ->  x  =  y )  <->  ( [
y  /  x ] ph  ->  y  =  x ) )
87albii 1699 . . . . 5  |-  ( A. y ( [ y  /  x ] ph  ->  x  =  y )  <->  A. y ( [ y  /  x ] ph  ->  y  =  x ) )
95, 8anbi12ci 712 . . . 4  |-  ( (
ph  /\  A. y
( [ y  /  x ] ph  ->  x  =  y ) )  <-> 
( A. y ( [ y  /  x ] ph  ->  y  =  x )  /\  A. y ( y  =  x  ->  [ y  /  x ] ph )
) )
10 albiim 1760 . . . 4  |-  ( A. y ( [ y  /  x ] ph  <->  y  =  x )  <->  ( A. y ( [ y  /  x ] ph  ->  y  =  x )  /\  A. y ( y  =  x  ->  [ y  /  x ] ph ) ) )
119, 10bitr4i 260 . . 3  |-  ( (
ph  /\  A. y
( [ y  /  x ] ph  ->  x  =  y ) )  <->  A. y ( [ y  /  x ] ph  <->  y  =  x ) )
1211exbii 1726 . 2  |-  ( E. x ( ph  /\  A. y ( [ y  /  x ] ph  ->  x  =  y ) )  <->  E. x A. y
( [ y  /  x ] ph  <->  y  =  x ) )
132, 4, 123bitr4i 285 1  |-  ( E! x ph  <->  E. x
( ph  /\  A. y
( [ y  /  x ] ph  ->  x  =  y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376   A.wal 1450   E.wex 1671   F/wnf 1675   [wsb 1805   E!weu 2319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323
This theorem is referenced by:  euexALT  2360  kmlem15  8612
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