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Theorem nfeud2 2290
Description: Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.) (Proof shortened by Wolf Lammen, 4-Oct-2018.)
Hypotheses
Ref Expression
nfeud2.1  |-  F/ y
ph
nfeud2.2  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
Assertion
Ref Expression
nfeud2  |-  ( ph  ->  F/ x E! y ps )

Proof of Theorem nfeud2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-eu 2279 . 2  |-  ( E! y ps  <->  E. z A. y ( ps  <->  y  =  z ) )
2 nfv 1683 . . 3  |-  F/ z
ph
3 nfeud2.1 . . . 4  |-  F/ y
ph
4 nfeud2.2 . . . . 5  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
5 nfeqf1 2016 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  F/ x  y  =  z )
65adantl 466 . . . . 5  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x  y  =  z )
74, 6nfbid 1880 . . . 4  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ( ps  <->  y  =  z ) )
83, 7nfald2 2046 . . 3  |-  ( ph  ->  F/ x A. y
( ps  <->  y  =  z ) )
92, 8nfexd 1899 . 2  |-  ( ph  ->  F/ x E. z A. y ( ps  <->  y  =  z ) )
101, 9nfxfrd 1626 1  |-  ( ph  ->  F/ x E! y ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1377   E.wex 1596   F/wnf 1599   E!weu 2275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600  df-eu 2279
This theorem is referenced by:  nfmod2  2291  nfeud  2292  nfreud  3034
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