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Theorem eu3h 2042
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 2027 . . 3  |-  ( E! x ph  ->  E. x ph )
2 eu3h.1 . . . 4  |-  ( ph  ->  A. y ph )
32eumo0 2028 . . 3  |-  ( E! x ph  ->  E. y A. x ( ph  ->  x  =  y ) )
41, 3jca 304 . 2  |-  ( E! x ph  ->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
52nfi 1438 . . . . 5  |-  F/ y
ph
65mo23 2038 . . . 4  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
76anim2i 339 . . 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 2041 . . 3  |-  ( E! x ph  <->  ( E. x ph  /\  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) ) )
97, 8sylibr 133 . 2  |-  ( ( E. x ph  /\  E. y A. x (
ph  ->  x  =  y ) )  ->  E! x ph )
104, 9impbii 125 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 103    <-> wb 104   A.wal 1329   E.wex 1468   [wsb 1735   E!weu 1997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2000
This theorem is referenced by:  eu3  2043  mo2r  2049  2eu4  2090
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