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Theorem moimv 1966
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 1647 . . . . . . 7  |-  ( [ y  /  x ] ph  ->  [ y  /  x ] ( ps  ->  (
ph  ->  ps ) ) )
4 nfv 1421 . . . . . . . 8  |-  F/ x ph
54sbf 1660 . . . . . . 7  |-  ( [ y  /  x ] ph 
<-> 
ph )
6 sbim 1827 . . . . . . 7  |-  ( [ y  /  x ]
( ps  ->  ( ph  ->  ps ) )  <-> 
( [ y  /  x ] ps  ->  [ y  /  x ] (
ph  ->  ps ) ) )
73, 5, 63imtr3i 189 . . . . . 6  |-  ( ph  ->  ( [ y  /  x ] ps  ->  [ y  /  x ] (
ph  ->  ps ) ) )
82, 7anim12d 318 . . . . 5  |-  ( ph  ->  ( ( ps  /\  [ y  /  x ] ps )  ->  ( (
ph  ->  ps )  /\  [ y  /  x ]
( ph  ->  ps )
) ) )
98imim1d 69 . . . 4  |-  ( ph  ->  ( ( ( (
ph  ->  ps )  /\  [ y  /  x ]
( ph  ->  ps )
)  ->  x  =  y )  ->  (
( ps  /\  [
y  /  x ] ps )  ->  x  =  y ) ) )
1092alimdv 1761 . . 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 1419 . . . 4  |-  ( (
ph  ->  ps )  ->  A. y ( ph  ->  ps ) )
1211mo3h 1953 . . 3  |-  ( E* x ( ph  ->  ps )  <->  A. x A. y
( ( ( ph  ->  ps )  /\  [
y  /  x ]
( ph  ->  ps )
)  ->  x  =  y ) )
13 ax-17 1419 . . . 4  |-  ( ps 
->  A. y ps )
1413mo3h 1953 . . 3  |-  ( E* x ps  <->  A. x A. y ( ( ps 
/\  [ y  /  x ] ps )  ->  x  =  y )
)
1510, 12, 143imtr4g 194 . 2  |-  ( ph  ->  ( E* x (
ph  ->  ps )  ->  E* x ps ) )
1615com12 27 1  |-  ( E* x ( ph  ->  ps )  ->  ( ph  ->  E* x ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97   A.wal 1241   [wsb 1645   E*wmo 1901
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904
This theorem is referenced by: (None)
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