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Theorem mo5f 23974
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 2314 . 2  |-  ( E* x ph  <->  A. x A. j ( ( ph  /\ 
[ j  /  x ] ph )  ->  x  =  j ) )
3 mo5f.1 . . . . . 6  |-  F/ i
ph
43nfsb 2187 . . . . . 6  |-  F/ i [ j  /  x ] ph
53, 4nfan 1847 . . . . 5  |-  F/ i ( ph  /\  [
j  /  x ] ph )
6 nfv 1630 . . . . 5  |-  F/ i  x  =  j
75, 6nfim 1833 . . . 4  |-  F/ i ( ( ph  /\  [ j  /  x ] ph )  ->  x  =  j )
87nfal 1865 . . 3  |-  F/ i A. j ( (
ph  /\  [ j  /  x ] ph )  ->  x  =  j )
98sb8 2170 . 2  |-  ( A. x A. j ( (
ph  /\  [ j  /  x ] ph )  ->  x  =  j )  <->  A. i [ i  /  x ] A. j ( ( ph  /\  [
j  /  x ] ph )  ->  x  =  j ) )
10 sbim 2137 . . . . 5  |-  ( [ i  /  x ]
( ( ph  /\  [ j  /  x ] ph )  ->  x  =  j )  <->  ( [
i  /  x ]
( ph  /\  [ j  /  x ] ph )  ->  [ i  /  x ] x  =  j ) )
11 sban 2141 . . . . . . 7  |-  ( [ i  /  x ]
( ph  /\  [ j  /  x ] ph ) 
<->  ( [ i  /  x ] ph  /\  [
i  /  x ] [ j  /  x ] ph ) )
12 nfs1v 2184 . . . . . . . . . 10  |-  F/ x [ j  /  x ] ph
1312sbf 2119 . . . . . . . . 9  |-  ( [ i  /  x ] [ j  /  x ] ph  <->  [ j  /  x ] ph )
1413bicomi 195 . . . . . . . 8  |-  ( [ j  /  x ] ph 
<->  [ i  /  x ] [ j  /  x ] ph )
1514anbi2i 677 . . . . . . 7  |-  ( ( [ i  /  x ] ph  /\  [ j  /  x ] ph ) 
<->  ( [ i  /  x ] ph  /\  [
i  /  x ] [ j  /  x ] ph ) )
1611, 15bitr4i 245 . . . . . 6  |-  ( [ i  /  x ]
( ph  /\  [ j  /  x ] ph ) 
<->  ( [ i  /  x ] ph  /\  [
j  /  x ] ph ) )
17 equsb3 2180 . . . . . 6  |-  ( [ i  /  x ]
x  =  j  <->  i  =  j )
1816, 17imbi12i 318 . . . . 5  |-  ( ( [ i  /  x ] ( ph  /\  [ j  /  x ] ph )  ->  [ i  /  x ] x  =  j )  <->  ( ( [ i  /  x ] ph  /\  [ j  /  x ] ph )  ->  i  =  j ) )
1910, 18bitri 242 . . . 4  |-  ( [ i  /  x ]
( ( ph  /\  [ j  /  x ] ph )  ->  x  =  j )  <->  ( ( [ i  /  x ] ph  /\  [ j  /  x ] ph )  ->  i  =  j ) )
2019sbalv 2208 . . 3  |-  ( [ i  /  x ] A. j ( ( ph  /\ 
[ j  /  x ] ph )  ->  x  =  j )  <->  A. j
( ( [ i  /  x ] ph  /\ 
[ j  /  x ] ph )  ->  i  =  j ) )
2120albii 1576 . 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 ) )
222, 9, 213bitri 264 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 178    /\ wa 360   A.wal 1550   F/wnf 1554   [wsb 1659   E*wmo 2284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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
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