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Theorem negsym1 32111
Description: In the paper "On Variable Functors of Propositional Arguments", Lukasiewicz introduced a system that can handle variable connectives. This was done by introducing a variable, marked with a lowercase delta, which takes a wff as input. In the system, "delta 𝜑 " means that "something is true of 𝜑." "delta 𝜑 " can be substituted with ¬ 𝜑, 𝜓𝜑, 𝑥𝜑, etc.

Later on, Meredith discovered a single axiom, in the form of ( delta delta ⊥ → delta 𝜑 ). This represents the shortest theorem in the extended propositional calculus that cannot be derived as an instance of a theorem in propositional calculus.

A symmetry with ¬. (Contributed by Anthony Hart, 4-Sep-2011.)

Ref Expression
negsym1 (¬ ¬ ⊥ → ¬ 𝜑)

Proof of Theorem negsym1
StepHypRef Expression
1 fal 1487 . 2 ¬ ⊥
21pm2.24i 146 1 (¬ ¬ ⊥ → ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wfal 1485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 197  df-tru 1483  df-fal 1486
This theorem is referenced by: (None)
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