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Theorem eu1 2134
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.)
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 2066 . . 3  |-  F/ x [ y  /  x ] ph
21euf 2120 . 2  |-  ( E! y [ y  /  x ] ph  <->  E. x A. y ( [ y  /  x ] ph  <->  y  =  x ) )
3 eu1.1 . . 3  |-  F/ y
ph
43sb8eu 2132 . 2  |-  ( E! x ph  <->  E! y [ y  /  x ] ph )
5 equcom 1824 . . . . . . 7  |-  ( x  =  y  <->  y  =  x )
65imbi2i 305 . . . . . 6  |-  ( ( [ y  /  x ] ph  ->  x  =  y )  <->  ( [
y  /  x ] ph  ->  y  =  x ) )
76albii 1554 . . . . 5  |-  ( A. y ( [ y  /  x ] ph  ->  x  =  y )  <->  A. y ( [ y  /  x ] ph  ->  y  =  x ) )
83sb6rf 1985 . . . . 5  |-  ( ph  <->  A. y ( y  =  x  ->  [ y  /  x ] ph )
)
97, 8anbi12i 681 . . . 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 439 . . . 4  |-  ( (
ph  /\  A. y
( [ y  /  x ] ph  ->  x  =  y ) )  <-> 
( A. y ( [ y  /  x ] ph  ->  x  =  y )  /\  ph ) )
11 albiim 1612 . . . 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 270 . . 3  |-  ( (
ph  /\  A. y
( [ y  /  x ] ph  ->  x  =  y ) )  <->  A. y ( [ y  /  x ] ph  <->  y  =  x ) )
1312exbii 1580 . 2  |-  ( E. x ( ph  /\  A. y ( [ y  /  x ] ph  ->  x  =  y ) )  <->  E. x A. y
( [ y  /  x ] ph  <->  y  =  x ) )
142, 4, 133bitr4i 270 1  |-  ( E! x ph  <->  E. x
( ph  /\  A. y
( [ y  /  x ] ph  ->  x  =  y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1532   E.wex 1537   F/wnf 1539    = wceq 1619   [wsb 1882   E!weu 2114
This theorem is referenced by:  euex  2136  eu2  2138  kmlem15  7674
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118
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