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Theorem nfeuv 2024
Description: Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 2025 but has the additional condition that  x and  y must be distinct. (Contributed by Jim Kingdon, 23-May-2018.)
Hypothesis
Ref Expression
nfeuv.1  |-  F/ x ph
Assertion
Ref Expression
nfeuv  |-  F/ x E! y ph
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem nfeuv
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nfeuv.1 . . . . 5  |-  F/ x ph
2 nfv 1508 . . . . 5  |-  F/ x  y  =  z
31, 2nfbi 1569 . . . 4  |-  F/ x
( ph  <->  y  =  z )
43nfal 1556 . . 3  |-  F/ x A. y ( ph  <->  y  =  z )
54nfex 1617 . 2  |-  F/ x E. z A. y (
ph 
<->  y  =  z )
6 df-eu 2009 . . 3  |-  ( E! y ph  <->  E. z A. y ( ph  <->  y  =  z ) )
76nfbii 1453 . 2  |-  ( F/ x E! y ph  <->  F/ x E. z A. y ( ph  <->  y  =  z ) )
85, 7mpbir 145 1  |-  F/ x E! y ph
Colors of variables: wff set class
Syntax hints:    <-> wb 104   A.wal 1333   F/wnf 1440   E.wex 1472   E!weu 2006
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-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-eu 2009
This theorem is referenced by:  nfeu  2025
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