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Theorem eu1 2164
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 2045 . . 3  |-  F/ x [ y  /  x ] ph
21euf 2149 . 2  |-  ( E! y [ y  /  x ] ph  <->  E. x A. y ( [ y  /  x ] ph  <->  y  =  x ) )
3 eu1.1 . . 3  |-  F/ y
ph
43sb8eu 2161 . 2  |-  ( E! x ph  <->  E! y [ y  /  x ] ph )
5 equcom 1647 . . . . . . 7  |-  ( x  =  y  <->  y  =  x )
65imbi2i 303 . . . . . 6  |-  ( ( [ y  /  x ] ph  ->  x  =  y )  <->  ( [
y  /  x ] ph  ->  y  =  x ) )
76albii 1553 . . . . 5  |-  ( A. y ( [ y  /  x ] ph  ->  x  =  y )  <->  A. y ( [ y  /  x ] ph  ->  y  =  x ) )
83sb6rf 2031 . . . . 5  |-  ( ph  <->  A. y ( y  =  x  ->  [ y  /  x ] ph )
)
97, 8anbi12i 678 . . . 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 437 . . . 4  |-  ( (
ph  /\  A. y
( [ y  /  x ] ph  ->  x  =  y ) )  <-> 
( A. y ( [ y  /  x ] ph  ->  x  =  y )  /\  ph ) )
11 albiim 1598 . . . 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 268 . . 3  |-  ( (
ph  /\  A. y
( [ y  /  x ] ph  ->  x  =  y ) )  <->  A. y ( [ y  /  x ] ph  <->  y  =  x ) )
1312exbii 1569 . 2  |-  ( E. x ( ph  /\  A. y ( [ y  /  x ] ph  ->  x  =  y ) )  <->  E. x A. y
( [ y  /  x ] ph  <->  y  =  x ) )
142, 4, 133bitr4i 268 1  |-  ( E! x ph  <->  E. x
( ph  /\  A. y
( [ y  /  x ] ph  ->  x  =  y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528   F/wnf 1531    = wceq 1623   [wsb 1629   E!weu 2143
This theorem is referenced by:  euex  2166  eu2  2168  kmlem15  7790
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147
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