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Theorem moim 2327
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 imim1 72 . . . 4  |-  ( (
ph  ->  ps )  -> 
( ( ps  ->  x  =  y )  -> 
( ph  ->  x  =  y ) ) )
21al2imi 1570 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( A. x ( ps  ->  x  =  y )  ->  A. x ( ph  ->  x  =  y ) ) )
32eximdv 1632 . 2  |-  ( A. x ( ph  ->  ps )  ->  ( E. y A. x ( ps 
->  x  =  y
)  ->  E. y A. x ( ph  ->  x  =  y ) ) )
4 nfv 1629 . . 3  |-  F/ y ps
54mo2 2310 . 2  |-  ( E* x ps  <->  E. y A. x ( ps  ->  x  =  y ) )
6 nfv 1629 . . 3  |-  F/ y
ph
76mo2 2310 . 2  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
83, 5, 73imtr4g 262 1  |-  ( A. x ( ph  ->  ps )  ->  ( E* x ps  ->  E* x ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549   E.wex 1550   E*wmo 2282
This theorem is referenced by:  moimi  2328  euimmo  2330  moexex  2350  rmoim  3133  rmoimi2  3135  disjss1  4188  disjss3  4211  reusv1  4723  reusv2lem1  4724  funmo  5470  brdom6disj  8410  uptx  17657  taylf  20277  moimd  23974  ssrmo  23981  funressnfv  27968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286
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