ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sb8mo Unicode version

Theorem sb8mo 1989
Description: Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Hypothesis
Ref Expression
sb8eu.1  |-  F/ y
ph
Assertion
Ref Expression
sb8mo  |-  ( E* x ph  <->  E* y [ y  /  x ] ph )

Proof of Theorem sb8mo
StepHypRef Expression
1 sb8eu.1 . . . 4  |-  F/ y
ph
21sb8e 1811 . . 3  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
31sb8eu 1988 . . 3  |-  ( E! x ph  <->  E! y [ y  /  x ] ph )
42, 3imbi12i 238 . 2  |-  ( ( E. x ph  ->  E! x ph )  <->  ( E. y [ y  /  x ] ph  ->  E! y [ y  /  x ] ph ) )
5 df-mo 1979 . 2  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
6 df-mo 1979 . 2  |-  ( E* y [ y  /  x ] ph  <->  ( E. y [ y  /  x ] ph  ->  E! y [ y  /  x ] ph ) )
74, 5, 63bitr4i 211 1  |-  ( E* x ph  <->  E* y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   F/wnf 1419   E.wex 1451   [wsb 1718   E!weu 1975   E*wmo 1976
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator