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Theorem nfeud2 2293
Description: Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.)
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 2285 . 2  |-  ( E! y ps  <->  E. z A. y ( ps  <->  y  =  z ) )
2 nfv 1629 . . 3  |-  F/ z
ph
3 nfeud2.1 . . . . 5  |-  F/ y
ph
4 nfnae 2044 . . . . 5  |-  F/ y  -.  A. x  x  =  z
53, 4nfan 1846 . . . 4  |-  F/ y ( ph  /\  -.  A. x  x  =  z )
6 nfeud2.2 . . . . . 6  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
76adantlr 696 . . . . 5  |-  ( ( ( ph  /\  -.  A. x  x  =  z )  /\  -.  A. x  x  =  y
)  ->  F/ x ps )
8 nfeqf 2009 . . . . . . 7  |-  ( ( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )  ->  F/ x  y  =  z )
98ancoms 440 . . . . . 6  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  y )  ->  F/ x  y  =  z )
109adantll 695 . . . . 5  |-  ( ( ( ph  /\  -.  A. x  x  =  z )  /\  -.  A. x  x  =  y
)  ->  F/ x  y  =  z )
117, 10nfbid 1854 . . . 4  |-  ( ( ( ph  /\  -.  A. x  x  =  z )  /\  -.  A. x  x  =  y
)  ->  F/ x
( ps  <->  y  =  z ) )
125, 11nfald2 2064 . . 3  |-  ( (
ph  /\  -.  A. x  x  =  z )  ->  F/ x A. y
( ps  <->  y  =  z ) )
132, 12nfexd2 2065 . 2  |-  ( ph  ->  F/ x E. z A. y ( ps  <->  y  =  z ) )
141, 13nfxfrd 1580 1  |-  ( ph  ->  F/ x E! y ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549   E.wex 1550   F/wnf 1553   E!weu 2281
This theorem is referenced by:  nfmod2  2294  nfeud  2295  nfreud  2880
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-eu 2285
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