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Theorem mo3 2175
Description: Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that  y not occur in  ph in place of our hypothesis. (Contributed by NM, 8-Mar-1995.)
Hypothesis
Ref Expression
mo3.1  |-  F/ y
ph
Assertion
Ref Expression
mo3  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem mo3
StepHypRef Expression
1 mo3.1 . . 3  |-  F/ y
ph
21mo2 2173 . 2  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
31mo 2166 . 2  |-  ( E. y A. x (
ph  ->  x  =  y )  <->  A. x A. y
( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
42, 3bitri 242 1  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1532   E.wex 1533   F/wnf 1536   [wsb 1635   E*wmo 2145
This theorem is referenced by:  mo4f  2176  mopick  2206  rmo3  3079  isarep2  5297  isconcl5a  25500  isconcl5ab  25501  pm14.12  27020  bnj580  28212
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149
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