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Theorem moim 2090
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 1541 . . 3  |-  F/ x A. x ( ph  ->  ps )
2 ax-4 1510 . . . . . 6  |-  ( A. x ( ph  ->  ps )  ->  ( ph  ->  ps ) )
3 spsbim 1843 . . . . . 6  |-  ( A. x ( ph  ->  ps )  ->  ( [
y  /  x ] ph  ->  [ y  /  x ] ps ) )
42, 3anim12d 335 . . . . 5  |-  ( A. x ( ph  ->  ps )  ->  ( ( ph  /\  [ y  /  x ] ph )  -> 
( ps  /\  [
y  /  x ] ps ) ) )
54imim1d 75 . . . 4  |-  ( A. x ( ph  ->  ps )  ->  ( (
( ps  /\  [
y  /  x ] ps )  ->  x  =  y )  ->  (
( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) ) )
65alimdv 1879 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( A. y ( ( ps 
/\  [ y  /  x ] ps )  ->  x  =  y )  ->  A. y ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y ) ) )
71, 6alimd 1521 . 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 1526 . . 3  |-  ( ps 
->  A. y ps )
98mo3h 2079 . 2  |-  ( E* x ps  <->  A. x A. y ( ( ps 
/\  [ y  /  x ] ps )  ->  x  =  y )
)
10 ax-17 1526 . . 3  |-  ( ph  ->  A. y ph )
1110mo3h 2079 . 2  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
127, 9, 113imtr4g 205 1  |-  ( A. x ( ph  ->  ps )  ->  ( E* x ps  ->  E* x ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   A.wal 1351   [wsb 1762   E*wmo 2027
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030
This theorem is referenced by:  moimi  2091  euimmo  2093  moexexdc  2110  euexex  2111  rmoim  2938  rmoimi2  2940  ssrmof  3218  disjss1  3983  reusv1  4454  funmo  5226  uptx  13407
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