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Axiom ax-3 8
 Description: Axiom Transp. Axiom A3 of [Margaris] p. 49. One of the 3 axioms of propositional calculus. It 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, 30-Sep-1992.) Use its alias con4 111 instead. (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 setvar class This axiom is referenced by:  con4  111  dfbi1ALT  203
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