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Theorem moim 2009
Description: "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.)
Assertion
Ref Expression
moim  |-  ( A. x ( ph  ->  ps )  ->  ( E* x ps  ->  E* x ph ) )

Proof of Theorem moim
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 nfa1 1477 . . 3  |-  F/ x A. x ( ph  ->  ps )
2 ax-4 1443 . . . . . 6  |-  ( A. x ( ph  ->  ps )  ->  ( ph  ->  ps ) )
3 spsbim 1768 . . . . . 6  |-  ( A. x ( ph  ->  ps )  ->  ( [
y  /  x ] ph  ->  [ y  /  x ] ps ) )
42, 3anim12d 328 . . . . 5  |-  ( A. x ( ph  ->  ps )  ->  ( ( ph  /\  [ y  /  x ] ph )  -> 
( ps  /\  [
y  /  x ] ps ) ) )
54imim1d 74 . . . 4  |-  ( A. x ( ph  ->  ps )  ->  ( (
( ps  /\  [
y  /  x ] ps )  ->  x  =  y )  ->  (
( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) ) )
65alimdv 1804 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( A. y ( ( ps 
/\  [ y  /  x ] ps )  ->  x  =  y )  ->  A. y ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y ) ) )
71, 6alimd 1457 . 2  |-  ( A. x ( ph  ->  ps )  ->  ( A. x A. y ( ( ps  /\  [ y  /  x ] ps )  ->  x  =  y )  ->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) ) )
8 ax-17 1462 . . 3  |-  ( ps 
->  A. y ps )
98mo3h 1998 . 2  |-  ( E* x ps  <->  A. x A. y ( ( ps 
/\  [ y  /  x ] ps )  ->  x  =  y )
)
10 ax-17 1462 . . 3  |-  ( ph  ->  A. y ph )
1110mo3h 1998 . 2  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
127, 9, 113imtr4g 203 1  |-  ( A. x ( ph  ->  ps )  ->  ( E* x ps  ->  E* x ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1285   [wsb 1689   E*wmo 1946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949
This theorem is referenced by:  moimi  2010  euimmo  2012  moexexdc  2029  euexex  2030  rmoim  2805  rmoimi2  2807  disjss1  3807  reusv1  4254  funmo  4996
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