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Theorem nmo 23978
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 2312 . . 3  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
32notbii 289 . 2  |-  ( -. 
E* x ph  <->  -.  E. y A. x ( ph  ->  x  =  y ) )
4 alnex 1553 . 2  |-  ( A. y  -.  A. x (
ph  ->  x  =  y )  <->  -.  E. y A. x ( ph  ->  x  =  y ) )
5 exnal 1584 . . . 4  |-  ( E. x  -.  ( ph  ->  x  =  y )  <->  -.  A. x ( ph  ->  x  =  y ) )
6 pm4.61 417 . . . . . 6  |-  ( -.  ( ph  ->  x  =  y )  <->  ( ph  /\ 
-.  x  =  y ) )
7 biid 229 . . . . . . . 8  |-  ( x  =  y  <->  x  =  y )
87necon3bbii 2634 . . . . . . 7  |-  ( -.  x  =  y  <->  x  =/=  y )
98anbi2i 677 . . . . . 6  |-  ( (
ph  /\  -.  x  =  y )  <->  ( ph  /\  x  =/=  y ) )
106, 9bitri 242 . . . . 5  |-  ( -.  ( ph  ->  x  =  y )  <->  ( ph  /\  x  =/=  y ) )
1110exbii 1593 . . . 4  |-  ( E. x  -.  ( ph  ->  x  =  y )  <->  E. x ( ph  /\  x  =/=  y ) )
125, 11bitr3i 244 . . 3  |-  ( -. 
A. x ( ph  ->  x  =  y )  <->  E. x ( ph  /\  x  =/=  y ) )
1312albii 1576 . 2  |-  ( A. y  -.  A. x (
ph  ->  x  =  y )  <->  A. y E. x
( ph  /\  x  =/=  y ) )
143, 4, 133bitr2i 266 1  |-  ( -. 
E* x ph  <->  A. y E. x ( ph  /\  x  =/=  y ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550   E.wex 1551   F/wnf 1554   E*wmo 2284    =/= wne 2601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-ne 2603
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