Natural Deduction
(p <=> q) <=> (-p <=> -q)
assume p <=> q. assume -p. assume q. q => p. p. F. therefore -q. therefore -p => -q. assume -q. assume p. p => q. q. F. therefore -p. therefore -q => -p. -p <=> -q. therefore (p <=> q) => (-p <=> -q). assume -p <=> -q. assume p. -q => -p. assume -q. -p. F. therefore --q. q. therefore p => q. assume q. -p => -q. -q => -p. assume -(--q + -p). assume --q. --q + -p. F. therefore ---q. -q. assume -p. F. therefore --p. -p. F. therefore --(--q + -p). --q + -p. assume --q. assume -p. -q. F. therefore --p. p. therefore --q => p. assume -p. assume -p. -q. F. therefore --p. p. therefore -p => p. p. therefore q => p. p <=> q. therefore (-p <=> -q) => (p <=> q). p <=> q <=> (-p <=> -q).
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Last Modified : 27-Feb-2023