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Theorem mo5f 23159
Description: Alternate definition of "at most one." (Contributed by Thierry Arnoux, 1-Mar-2017.)
Hypotheses
Ref Expression
mo5f.1  |-  F/ i
ph
mo5f.2  |-  F/ j
ph
Assertion
Ref Expression
mo5f  |-  ( E* x ph  <->  A. i A. j ( ( [ i  /  x ] ph  /\  [ j  /  x ] ph )  -> 
i  =  j ) )
Distinct variable group:    i, j, x
Allowed substitution hints:    ph( x, i, j)

Proof of Theorem mo5f
StepHypRef Expression
1 mo5f.2 . . 3  |-  F/ j
ph
21mo3 2187 . 2  |-  ( E* x ph  <->  A. x A. j ( ( ph  /\ 
[ j  /  x ] ph )  ->  x  =  j ) )
3 mo5f.1 . . . . . 6  |-  F/ i
ph
43nfsb 2061 . . . . . 6  |-  F/ i [ j  /  x ] ph
53, 4nfan 1783 . . . . 5  |-  F/ i ( ph  /\  [
j  /  x ] ph )
6 nfv 1609 . . . . 5  |-  F/ i  x  =  j
75, 6nfim 1781 . . . 4  |-  F/ i ( ( ph  /\  [ j  /  x ] ph )  ->  x  =  j )
87nfal 1778 . . 3  |-  F/ i A. j ( (
ph  /\  [ j  /  x ] ph )  ->  x  =  j )
98sb8 2045 . 2  |-  ( A. x A. j ( (
ph  /\  [ j  /  x ] ph )  ->  x  =  j )  <->  A. i [ i  /  x ] A. j ( ( ph  /\  [
j  /  x ] ph )  ->  x  =  j ) )
10 sbal 2079 . . . 4  |-  ( [ i  /  x ] A. j ( ( ph  /\ 
[ j  /  x ] ph )  ->  x  =  j )  <->  A. j [ i  /  x ] ( ( ph  /\ 
[ j  /  x ] ph )  ->  x  =  j ) )
11 sbim 2018 . . . . . 6  |-  ( [ i  /  x ]
( ( ph  /\  [ j  /  x ] ph )  ->  x  =  j )  <->  ( [
i  /  x ]
( ph  /\  [ j  /  x ] ph )  ->  [ i  /  x ] x  =  j ) )
12 sban 2022 . . . . . . . 8  |-  ( [ i  /  x ]
( ph  /\  [ j  /  x ] ph ) 
<->  ( [ i  /  x ] ph  /\  [
i  /  x ] [ j  /  x ] ph ) )
131nfs1 1997 . . . . . . . . . . 11  |-  F/ x [ j  /  x ] ph
1413sbf 1979 . . . . . . . . . 10  |-  ( [ i  /  x ] [ j  /  x ] ph  <->  [ j  /  x ] ph )
1514bicomi 193 . . . . . . . . 9  |-  ( [ j  /  x ] ph 
<->  [ i  /  x ] [ j  /  x ] ph )
1615anbi2i 675 . . . . . . . 8  |-  ( ( [ i  /  x ] ph  /\  [ j  /  x ] ph ) 
<->  ( [ i  /  x ] ph  /\  [
i  /  x ] [ j  /  x ] ph ) )
1712, 16bitr4i 243 . . . . . . 7  |-  ( [ i  /  x ]
( ph  /\  [ j  /  x ] ph ) 
<->  ( [ i  /  x ] ph  /\  [
j  /  x ] ph ) )
18 equsb3 2054 . . . . . . 7  |-  ( [ i  /  x ]
x  =  j  <->  i  =  j )
1917, 18imbi12i 316 . . . . . 6  |-  ( ( [ i  /  x ] ( ph  /\  [ j  /  x ] ph )  ->  [ i  /  x ] x  =  j )  <->  ( ( [ i  /  x ] ph  /\  [ j  /  x ] ph )  ->  i  =  j ) )
2011, 19bitri 240 . . . . 5  |-  ( [ i  /  x ]
( ( ph  /\  [ j  /  x ] ph )  ->  x  =  j )  <->  ( ( [ i  /  x ] ph  /\  [ j  /  x ] ph )  ->  i  =  j ) )
2120albii 1556 . . . 4  |-  ( A. j [ i  /  x ] ( ( ph  /\ 
[ j  /  x ] ph )  ->  x  =  j )  <->  A. j
( ( [ i  /  x ] ph  /\ 
[ j  /  x ] ph )  ->  i  =  j ) )
2210, 21bitri 240 . . 3  |-  ( [ i  /  x ] A. j ( ( ph  /\ 
[ j  /  x ] ph )  ->  x  =  j )  <->  A. j
( ( [ i  /  x ] ph  /\ 
[ j  /  x ] ph )  ->  i  =  j ) )
2322albii 1556 . 2  |-  ( A. i [ i  /  x ] A. j ( (
ph  /\  [ j  /  x ] ph )  ->  x  =  j )  <->  A. i A. j ( ( [ i  /  x ] ph  /\  [
j  /  x ] ph )  ->  i  =  j ) )
242, 9, 233bitri 262 1  |-  ( E* x ph  <->  A. i A. j ( ( [ i  /  x ] ph  /\  [ j  /  x ] ph )  -> 
i  =  j ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   F/wnf 1534    = wceq 1632   [wsb 1638   E*wmo 2157
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-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161
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