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Theorem nmo 23961
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 2309 . . 3  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
32notbii 288 . 2  |-  ( -. 
E* x ph  <->  -.  E. y A. x ( ph  ->  x  =  y ) )
4 alnex 1552 . 2  |-  ( A. y  -.  A. x (
ph  ->  x  =  y )  <->  -.  E. y A. x ( ph  ->  x  =  y ) )
5 exnal 1583 . . . 4  |-  ( E. x  -.  ( ph  ->  x  =  y )  <->  -.  A. x ( ph  ->  x  =  y ) )
6 pm4.61 416 . . . . . 6  |-  ( -.  ( ph  ->  x  =  y )  <->  ( ph  /\ 
-.  x  =  y ) )
7 biid 228 . . . . . . . 8  |-  ( x  =  y  <->  x  =  y )
87necon3bbii 2629 . . . . . . 7  |-  ( -.  x  =  y  <->  x  =/=  y )
98anbi2i 676 . . . . . 6  |-  ( (
ph  /\  -.  x  =  y )  <->  ( ph  /\  x  =/=  y ) )
106, 9bitri 241 . . . . 5  |-  ( -.  ( ph  ->  x  =  y )  <->  ( ph  /\  x  =/=  y ) )
1110exbii 1592 . . . 4  |-  ( E. x  -.  ( ph  ->  x  =  y )  <->  E. x ( ph  /\  x  =/=  y ) )
125, 11bitr3i 243 . . 3  |-  ( -. 
A. x ( ph  ->  x  =  y )  <->  E. x ( ph  /\  x  =/=  y ) )
1312albii 1575 . 2  |-  ( A. y  -.  A. x (
ph  ->  x  =  y )  <->  A. y E. x
( ph  /\  x  =/=  y ) )
143, 4, 133bitr2i 265 1  |-  ( -. 
E* x ph  <->  A. y E. x ( ph  /\  x  =/=  y ) )
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*wmo 2281    =/= wne 2598
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-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-ne 2600
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