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Theorem eu1 1967
Description: An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.)
Hypothesis
Ref Expression
eu1.1  |-  ( ph  ->  A. 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 hbs1 1856 . . 3  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
21euf 1947 . 2  |-  ( E! y [ y  /  x ] ph  <->  E. x A. y ( [ y  /  x ] ph  <->  y  =  x ) )
3 eu1.1 . . 3  |-  ( ph  ->  A. y ph )
43sb8euh 1965 . 2  |-  ( E! x ph  <->  E! y [ y  /  x ] ph )
5 equcom 1634 . . . . . . 7  |-  ( x  =  y  <->  y  =  x )
65imbi2i 224 . . . . . 6  |-  ( ( [ y  /  x ] ph  ->  x  =  y )  <->  ( [
y  /  x ] ph  ->  y  =  x ) )
76albii 1400 . . . . 5  |-  ( A. y ( [ y  /  x ] ph  ->  x  =  y )  <->  A. y ( [ y  /  x ] ph  ->  y  =  x ) )
83sb6rf 1775 . . . . 5  |-  ( ph  <->  A. y ( y  =  x  ->  [ y  /  x ] ph )
)
97, 8anbi12i 448 . . . 4  |-  ( ( A. y ( [ y  /  x ] ph  ->  x  =  y )  /\  ph )  <->  ( A. y ( [ y  /  x ] ph  ->  y  =  x )  /\  A. y
( y  =  x  ->  [ y  /  x ] ph ) ) )
10 ancom 262 . . . 4  |-  ( (
ph  /\  A. y
( [ y  /  x ] ph  ->  x  =  y ) )  <-> 
( A. y ( [ y  /  x ] ph  ->  x  =  y )  /\  ph ) )
11 albiim 1417 . . . 4  |-  ( A. y ( [ y  /  x ] ph  <->  y  =  x )  <->  ( A. y ( [ y  /  x ] ph  ->  y  =  x )  /\  A. y ( y  =  x  ->  [ y  /  x ] ph ) ) )
129, 10, 113bitr4i 210 . . 3  |-  ( (
ph  /\  A. y
( [ y  /  x ] ph  ->  x  =  y ) )  <->  A. y ( [ y  /  x ] ph  <->  y  =  x ) )
1312exbii 1537 . 2  |-  ( E. x ( ph  /\  A. y ( [ y  /  x ] ph  ->  x  =  y ) )  <->  E. x A. y
( [ y  /  x ] ph  <->  y  =  x ) )
142, 4, 133bitr4i 210 1  |-  ( E! x ph  <->  E. x
( ph  /\  A. y
( [ y  /  x ] ph  ->  x  =  y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283   E.wex 1422   [wsb 1686   E!weu 1942
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-eu 1945
This theorem is referenced by:  euex  1972  eu2  1986
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