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Axiom ax-3 719
 Description: Axiom Transp. Axiom A3 of [Margaris] p. 49. We take this as an additional axiom which transforms intuitionistic logic to classical logic, but there are others which would have the same effect, including exmid 1885, peirce 1760, or notnot2 725. This axiom swaps or "transposes" the order of the consequents when negation is removed. An informal example is that the statement "if there are no clouds in the sky, it is not raining" implies the statement "if it is raining, there are clouds in the sky." This axiom is called Transp or "the principle of transposition" in Principia Mathematica (Theorem *2.17 of [WhiteheadRussell] p. 103). We will also use the term "contraposition" for this principle, although the reader is advised that in the field of philosophical logic, "contraposition" has a different technical meaning. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
Assertion
Ref Expression
ax-3 ((¬ φ → ¬ ψ) → (ψφ))

Detailed syntax breakdown of Axiom ax-3
StepHypRef Expression
1 wph . . . 4 wff φ
21wn 3 . . 3 wff ¬ φ
3 wps . . . 4 wff ψ
43wn 3 . . 3 wff ¬ ψ
52, 4wi 4 . 2 wff φ → ¬ ψ)
63, 1wi 4 . 2 wff (ψφ)
75, 6wi 4 1 wff ((¬ φ → ¬ ψ) → (ψφ))
 Colors of variables: wff set class This axiom is referenced by:  con4d  721  con34b  753
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