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Theorem nmo 28121
Description: Negation of "at most one". (Contributed by Thierry Arnoux, 26-Feb-2017.)
Hypothesis
Ref Expression
nmo.1  |-  F/ y
ph
Assertion
Ref Expression
nmo  |-  ( -. 
E* x ph  <->  A. y E. x ( ph  /\  x  =/=  y ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem nmo
StepHypRef Expression
1 nmo.1 . . . 4  |-  F/ y
ph
21mo2 2308 . . 3  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
32notbii 298 . 2  |-  ( -. 
E* x ph  <->  -.  E. y A. x ( ph  ->  x  =  y ) )
4 alnex 1665 . 2  |-  ( A. y  -.  A. x (
ph  ->  x  =  y )  <->  -.  E. y A. x ( ph  ->  x  =  y ) )
5 exnal 1699 . . . 4  |-  ( E. x  -.  ( ph  ->  x  =  y )  <->  -.  A. x ( ph  ->  x  =  y ) )
6 pm4.61 428 . . . . . 6  |-  ( -.  ( ph  ->  x  =  y )  <->  ( ph  /\ 
-.  x  =  y ) )
7 biid 240 . . . . . . . 8  |-  ( x  =  y  <->  x  =  y )
87necon3bbii 2671 . . . . . . 7  |-  ( -.  x  =  y  <->  x  =/=  y )
98anbi2i 700 . . . . . 6  |-  ( (
ph  /\  -.  x  =  y )  <->  ( ph  /\  x  =/=  y ) )
106, 9bitri 253 . . . . 5  |-  ( -.  ( ph  ->  x  =  y )  <->  ( ph  /\  x  =/=  y ) )
1110exbii 1718 . . . 4  |-  ( E. x  -.  ( ph  ->  x  =  y )  <->  E. x ( ph  /\  x  =/=  y ) )
125, 11bitr3i 255 . . 3  |-  ( -. 
A. x ( ph  ->  x  =  y )  <->  E. x ( ph  /\  x  =/=  y ) )
1312albii 1691 . 2  |-  ( A. y  -.  A. x (
ph  ->  x  =  y )  <->  A. y E. x
( ph  /\  x  =/=  y ) )
143, 4, 133bitr2i 277 1  |-  ( -. 
E* x ph  <->  A. y E. x ( ph  /\  x  =/=  y ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371   A.wal 1442   E.wex 1663   F/wnf 1667   E*wmo 2300    =/= wne 2622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-nf 1668  df-eu 2303  df-mo 2304  df-ne 2624
This theorem is referenced by: (None)
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