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Theorem eu3h 1961
Description: An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.)
Hypothesis
Ref Expression
eu3h.1  |-  ( ph  ->  A. y ph )
Assertion
Ref Expression
eu3h  |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem eu3h
StepHypRef Expression
1 euex 1946 . . 3  |-  ( E! x ph  ->  E. x ph )
2 eu3h.1 . . . 4  |-  ( ph  ->  A. y ph )
32eumo0 1947 . . 3  |-  ( E! x ph  ->  E. y A. x ( ph  ->  x  =  y ) )
41, 3jca 294 . 2  |-  ( E! x ph  ->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
52nfi 1367 . . . . 5  |-  F/ y
ph
65mo23 1957 . . . 4  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
76anim2i 328 . . 3  |-  ( ( E. x ph  /\  E. y A. x (
ph  ->  x  =  y ) )  ->  ( E. x ph  /\  A. x A. y ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y ) ) )
85eu2 1960 . . 3  |-  ( E! x ph  <->  ( E. x ph  /\  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) ) )
97, 8sylibr 141 . 2  |-  ( ( E. x ph  /\  E. y A. x (
ph  ->  x  =  y ) )  ->  E! x ph )
104, 9impbii 121 1  |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102   A.wal 1257   E.wex 1397   [wsb 1661   E!weu 1916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-eu 1919
This theorem is referenced by:  eu3  1962  mo2r  1968  2eu4  2009
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