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Theorem mo3 2262
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 2260 . 2  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
31mo 2253 . 2  |-  ( E. y A. x (
ph  ->  x  =  y )  <->  A. x A. y
( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
42, 3bitri 241 1  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   E.wex 1547   F/wnf 1550   [wsb 1655   E*wmo 2232
This theorem is referenced by:  mo4f  2263  mopick  2293  rmo3  3184  isarep2  5466  mo5f  23809  rmo3f  23819  rmo4fOLD  23820  pm14.12  27283  bnj580  28615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236
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