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Theorem moimv 2009
Description: Move antecedent outside of "at most one." (Contributed by NM, 28-Jul-1995.)
Assertion
Ref Expression
moimv  |-  ( E* x ( ph  ->  ps )  ->  ( ph  ->  E* x ps )
)
Distinct variable group:    ph, x
Allowed substitution hint:    ps( x)

Proof of Theorem moimv
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ax-1 5 . . . . . . 7  |-  ( ps 
->  ( ph  ->  ps ) )
21a1i 9 . . . . . 6  |-  ( ph  ->  ( ps  ->  ( ph  ->  ps ) ) )
32sbimi 1689 . . . . . . 7  |-  ( [ y  /  x ] ph  ->  [ y  /  x ] ( ps  ->  (
ph  ->  ps ) ) )
4 nfv 1462 . . . . . . . 8  |-  F/ x ph
54sbf 1702 . . . . . . 7  |-  ( [ y  /  x ] ph 
<-> 
ph )
6 sbim 1870 . . . . . . 7  |-  ( [ y  /  x ]
( ps  ->  ( ph  ->  ps ) )  <-> 
( [ y  /  x ] ps  ->  [ y  /  x ] (
ph  ->  ps ) ) )
73, 5, 63imtr3i 198 . . . . . 6  |-  ( ph  ->  ( [ y  /  x ] ps  ->  [ y  /  x ] (
ph  ->  ps ) ) )
82, 7anim12d 328 . . . . 5  |-  ( ph  ->  ( ( ps  /\  [ y  /  x ] ps )  ->  ( (
ph  ->  ps )  /\  [ y  /  x ]
( ph  ->  ps )
) ) )
98imim1d 74 . . . 4  |-  ( ph  ->  ( ( ( (
ph  ->  ps )  /\  [ y  /  x ]
( ph  ->  ps )
)  ->  x  =  y )  ->  (
( ps  /\  [
y  /  x ] ps )  ->  x  =  y ) ) )
1092alimdv 1804 . . 3  |-  ( ph  ->  ( A. x A. y ( ( (
ph  ->  ps )  /\  [ y  /  x ]
( ph  ->  ps )
)  ->  x  =  y )  ->  A. x A. y ( ( ps 
/\  [ y  /  x ] ps )  ->  x  =  y )
) )
11 ax-17 1460 . . . 4  |-  ( (
ph  ->  ps )  ->  A. y ( ph  ->  ps ) )
1211mo3h 1996 . . 3  |-  ( E* x ( ph  ->  ps )  <->  A. x A. y
( ( ( ph  ->  ps )  /\  [
y  /  x ]
( ph  ->  ps )
)  ->  x  =  y ) )
13 ax-17 1460 . . . 4  |-  ( ps 
->  A. y ps )
1413mo3h 1996 . . 3  |-  ( E* x ps  <->  A. x A. y ( ( ps 
/\  [ y  /  x ] ps )  ->  x  =  y )
)
1510, 12, 143imtr4g 203 . 2  |-  ( ph  ->  ( E* x (
ph  ->  ps )  ->  E* x ps ) )
1615com12 30 1  |-  ( E* x ( ph  ->  ps )  ->  ( ph  ->  E* x ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1283   [wsb 1687   E*wmo 1944
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947
This theorem is referenced by: (None)
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