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Axiom ax-3 9
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, 5-Aug-1993.)
Assertion
Ref Expression
ax-3  |-  ( ( -.  ph  ->  -.  ps )  ->  ( ps  ->  ph ) )

Detailed syntax breakdown of Axiom ax-3
StepHypRef Expression
1 wph . . . 4  wff  ph
21wn 5 . . 3  wff  -.  ph
3 wps . . . 4  wff  ps
43wn 5 . . 3  wff  -.  ps
52, 4wi 6 . 2  wff  ( -. 
ph  ->  -.  ps )
63, 1wi 6 . 2  wff  ( ps 
->  ph )
75, 6wi 6 1  wff  ( ( -.  ph  ->  -.  ps )  ->  ( ps  ->  ph ) )
Colors of variables: wff set class
This axiom is referenced by:  con4d  99  dfbi1gb  187  con34b  285  meredith  1400  notnot2ALTVD  27381  con3ALTVD  27382  notnot2ALT2  27393
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