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Theorem nfeud2 2311
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 2303 . 2  |-  ( E! y ps  <->  E. z A. y ( ps  <->  y  =  z ) )
2 nfv 1764 . . 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 2137 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  F/ x  y  =  z )
65adantl 472 . . . . 5  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x  y  =  z )
74, 6nfbid 2020 . . . 4  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ( ps  <->  y  =  z ) )
83, 7nfald2 2165 . . 3  |-  ( ph  ->  F/ x A. y
( ps  <->  y  =  z ) )
92, 8nfexd 2039 . 2  |-  ( ph  ->  F/ x E. z A. y ( ps  <->  y  =  z ) )
101, 9nfxfrd 1700 1  |-  ( ph  ->  F/ x E! y ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375   A.wal 1445   E.wex 1666   F/wnf 1670   E!weu 2299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1667  df-nf 1671  df-eu 2303
This theorem is referenced by:  nfmod2  2312  nfeud  2313  nfreud  2930
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