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Theorem mo3h 2089
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.) (New usage is discouraged.)
Hypothesis
Ref Expression
mo3h.1  |-  ( ph  ->  A. y ph )
Assertion
Ref Expression
mo3h  |-  ( 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 mo3h
StepHypRef Expression
1 mo3h.1 . . . . . . 7  |-  ( ph  ->  A. y ph )
21nfi 1472 . . . . . 6  |-  F/ y
ph
32eu2 2080 . . . . 5  |-  ( E! x ph  <->  ( E. x ph  /\  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) ) )
43imbi2i 226 . . . 4  |-  ( ( E. x ph  ->  E! x ph )  <->  ( E. x ph  ->  ( E. x ph  /\  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) ) ) )
5 df-mo 2040 . . . 4  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
6 anclb 319 . . . 4  |-  ( ( E. x ph  ->  A. x A. y ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y ) )  <->  ( E. x ph  ->  ( E. x ph  /\  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) ) ) )
74, 5, 63bitr4i 212 . . 3  |-  ( E* x ph  <->  ( E. x ph  ->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) ) )
8 19.38 1686 . . . . 5  |-  ( ( E. x ph  ->  A. x A. y ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y ) )  ->  A. x ( ph  ->  A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) ) )
9219.21 1593 . . . . . 6  |-  ( A. y ( ph  ->  ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y ) )  <->  ( ph  ->  A. y ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y ) ) )
109albii 1480 . . . . 5  |-  ( A. x A. y ( ph  ->  ( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )  <->  A. x
( ph  ->  A. y
( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) ) )
118, 10sylibr 134 . . . 4  |-  ( ( E. x ph  ->  A. x A. y ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y ) )  ->  A. x A. y (
ph  ->  ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) ) )
12 anabs5 573 . . . . . 6  |-  ( (
ph  /\  ( ph  /\ 
[ y  /  x ] ph ) )  <->  ( ph  /\ 
[ y  /  x ] ph ) )
13 pm3.31 262 . . . . . 6  |-  ( (
ph  ->  ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )  ->  ( ( ph  /\  ( ph  /\  [
y  /  x ] ph ) )  ->  x  =  y ) )
1412, 13biimtrrid 153 . . . . 5  |-  ( (
ph  ->  ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )  ->  ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
15142alimi 1466 . . . 4  |-  ( A. x A. y ( ph  ->  ( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )  ->  A. x A. y ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y ) )
1611, 15syl 14 . . 3  |-  ( ( E. x ph  ->  A. x A. y ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y ) )  ->  A. x A. y ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y ) )
177, 16sylbi 121 . 2  |-  ( E* x ph  ->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
183simplbi2com 1454 . . 3  |-  ( A. x A. y ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  ( E. x ph  ->  E! x ph ) )
1918, 5sylibr 134 . 2  |-  ( A. x A. y ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  E* x ph )
2017, 19impbii 126 1  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1361   E.wex 1502   [wsb 1772   E!weu 2036   E*wmo 2037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040
This theorem is referenced by:  mo3  2090  mo2dc  2091  mo4f  2096  moim  2100  moimv  2102  moanim  2110  mopick  2114
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