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Theorem nfeuv 1960
Description: Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 1961 but has the additional constraint 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 1462 . . . . 5  |-  F/ x  y  =  z
31, 2nfbi 1522 . . . 4  |-  F/ x
( ph  <->  y  =  z )
43nfal 1509 . . 3  |-  F/ x A. y ( ph  <->  y  =  z )
54nfex 1569 . 2  |-  F/ x E. z A. y (
ph 
<->  y  =  z )
6 df-eu 1945 . . 3  |-  ( E! y ph  <->  E. z A. y ( ph  <->  y  =  z ) )
76nfbii 1403 . 2  |-  ( F/ x E! y ph  <->  F/ x E. z A. y ( ph  <->  y  =  z ) )
85, 7mpbir 144 1  |-  F/ x E! y ph
Colors of variables: wff set class
Syntax hints:    <-> wb 103   A.wal 1283   F/wnf 1390   E.wex 1422   E!weu 1942
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-eu 1945
This theorem is referenced by:  nfeu  1961
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