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Theorem nfeud2 2298
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 2288 . 2  |-  ( E! y ps  <->  E. z A. y ( ps  <->  y  =  z ) )
2 nfv 1712 . . 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 2047 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  F/ x  y  =  z )
65adantl 464 . . . . 5  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x  y  =  z )
74, 6nfbid 1938 . . . 4  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ( ps  <->  y  =  z ) )
83, 7nfald2 2077 . . 3  |-  ( ph  ->  F/ x A. y
( ps  <->  y  =  z ) )
92, 8nfexd 1957 . 2  |-  ( ph  ->  F/ x E. z A. y ( ps  <->  y  =  z ) )
101, 9nfxfrd 1651 1  |-  ( ph  ->  F/ x E! y ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1396   E.wex 1617   F/wnf 1621   E!weu 2284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-nf 1622  df-eu 2288
This theorem is referenced by:  nfmod2  2299  nfeud  2300  nfreud  3027
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