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Theorem nfeud2 2168
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 2160 . 2  |-  ( E! y ps  <->  E. z A. y ( ps  <->  y  =  z ) )
2 nfv 1609 . . 3  |-  F/ z
ph
3 nfeud2.1 . . . . 5  |-  F/ y
ph
4 nfnae 1909 . . . . 5  |-  F/ y  -.  A. x  x  =  z
53, 4nfan 1783 . . . 4  |-  F/ y ( ph  /\  -.  A. x  x  =  z )
6 nfeud2.2 . . . . . 6  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
76adantlr 695 . . . . 5  |-  ( ( ( ph  /\  -.  A. x  x  =  z )  /\  -.  A. x  x  =  y
)  ->  F/ x ps )
8 nfeqf 1911 . . . . . . 7  |-  ( ( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )  ->  F/ x  y  =  z )
98ancoms 439 . . . . . 6  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  y )  ->  F/ x  y  =  z )
109adantll 694 . . . . 5  |-  ( ( ( ph  /\  -.  A. x  x  =  z )  /\  -.  A. x  x  =  y
)  ->  F/ x  y  =  z )
117, 10nfbid 1774 . . . 4  |-  ( ( ( ph  /\  -.  A. x  x  =  z )  /\  -.  A. x  x  =  y
)  ->  F/ x
( ps  <->  y  =  z ) )
125, 11nfald2 1925 . . 3  |-  ( (
ph  /\  -.  A. x  x  =  z )  ->  F/ x A. y
( ps  <->  y  =  z ) )
132, 12nfexd2 1926 . 2  |-  ( ph  ->  F/ x E. z A. y ( ps  <->  y  =  z ) )
141, 13nfxfrd 1561 1  |-  ( ph  ->  F/ x E! y ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531   F/wnf 1534    = wceq 1632   E!weu 2156
This theorem is referenced by:  nfmod2  2169  nfeud  2170  nfreud  2725
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-eu 2160
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