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Theorem mo3 2336
Description: Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that  y not occur in  ph in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) (Proof shortened by Wolf Lammen, 18-Aug-2019.)
Hypothesis
Ref Expression
mo3.1  |-  F/ y
ph
Assertion
Ref Expression
mo3  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem mo3
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nfmo1 2310 . . 3  |-  F/ x E* x ph
2 mo3.1 . . . . 5  |-  F/ y
ph
32nfmo 2316 . . . 4  |-  F/ y E* x ph
4 mo2v 2306 . . . . 5  |-  ( E* x ph  <->  E. z A. x ( ph  ->  x  =  z ) )
5 sp 1937 . . . . . . . 8  |-  ( A. x ( ph  ->  x  =  z )  -> 
( ph  ->  x  =  z ) )
6 spsbim 2223 . . . . . . . . 9  |-  ( A. x ( ph  ->  x  =  z )  -> 
( [ y  /  x ] ph  ->  [ y  /  x ] x  =  z ) )
7 equsb3 2261 . . . . . . . . 9  |-  ( [ y  /  x ]
x  =  z  <->  y  =  z )
86, 7syl6ib 230 . . . . . . . 8  |-  ( A. x ( ph  ->  x  =  z )  -> 
( [ y  /  x ] ph  ->  y  =  z ) )
95, 8anim12d 566 . . . . . . 7  |-  ( A. x ( ph  ->  x  =  z )  -> 
( ( ph  /\  [ y  /  x ] ph )  ->  ( x  =  z  /\  y  =  z ) ) )
10 equtr2 1869 . . . . . . 7  |-  ( ( x  =  z  /\  y  =  z )  ->  x  =  y )
119, 10syl6 34 . . . . . 6  |-  ( A. x ( ph  ->  x  =  z )  -> 
( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
1211exlimiv 1776 . . . . 5  |-  ( E. z A. x (
ph  ->  x  =  z )  ->  ( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y )
)
134, 12sylbi 199 . . . 4  |-  ( E* x ph  ->  (
( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
143, 13alrimi 1955 . . 3  |-  ( E* x ph  ->  A. y
( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
151, 14alrimi 1955 . 2  |-  ( E* x ph  ->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
16 nfs1v 2266 . . . . . . . 8  |-  F/ x [ y  /  x ] ph
17 pm3.21 450 . . . . . . . . 9  |-  ( [ y  /  x ] ph  ->  ( ph  ->  (
ph  /\  [ y  /  x ] ph )
) )
1817imim1d 78 . . . . . . . 8  |-  ( [ y  /  x ] ph  ->  ( ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  ( ph  ->  x  =  y ) ) )
1916, 18alimd 1954 . . . . . . 7  |-  ( [ y  /  x ] ph  ->  ( A. x
( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  A. x
( ph  ->  x  =  y ) ) )
2019com12 32 . . . . . 6  |-  ( A. x ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y )  -> 
( [ y  /  x ] ph  ->  A. x
( ph  ->  x  =  y ) ) )
2120aleximi 1704 . . . . 5  |-  ( A. y A. x ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  ( E. y [ y  /  x ] ph  ->  E. y A. x ( ph  ->  x  =  y ) ) )
222sb8e 2254 . . . . 5  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
232mo2 2308 . . . . 5  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
2421, 22, 233imtr4g 274 . . . 4  |-  ( A. y A. x ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  ( E. x ph  ->  E* x ph ) )
25 moabs 2330 . . . 4  |-  ( E* x ph  <->  ( E. x ph  ->  E* x ph ) )
2624, 25sylibr 216 . . 3  |-  ( A. y A. x ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  E* x ph )
2726alcoms 1921 . 2  |-  ( A. x A. y ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  E* x ph )
2815, 27impbii 191 1  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371   A.wal 1442   E.wex 1663   F/wnf 1667   [wsb 1797   E*wmo 2300
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-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304
This theorem is referenced by:  mo  2337  eu2  2338  mo4f  2345  2mo  2380  rmo3  3358  isarep2  5663  mo5f  28120  rmo3f  28131  rmo4fOLD  28132  bnj580  29724  pm14.12  36772
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