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Axiom ax-1 5
Description: Axiom Simp. Axiom A1 of [Margaris] p. 49. One of the 3 axioms of propositional calculus. The 3 axioms are also given as Definition 2.1 of [Hamilton] p. 28. This axiom is called Simp or "the principle of simplification" in Principia Mathematica (Theorem *2.02 of [WhiteheadRussell] p. 100) because "it enables us to pass from the joint assertion of φ and ψ to the assertion of φ simply."

General remarks: Propositional calculus (axioms ax-1 5 through ax-3 719 and rule ax-mp 7) can be thought of as asserting formulas that are universally "true" when their variables are replaced by any combination of "true" and "false." Propositional calculus was first formalized by Frege in 1879, using as his axioms (in addition to rule ax-mp 7) the wffs ax-1 5, ax-2 6, pm2.04 74, con3 549, notnot2 725, and notnot1 540. Around 1930, Lukasiewicz simplified the system by eliminating the third (which follows from the first two, as you can see by looking at the proof of pm2.04 74) and replacing the last three with our ax-3 719. (Thanks to Ted Ulrich for this information.)

The theorems of propositional calculus are also called tautologies. Tautologies can be proved very simply using truth tables, based on the true/false interpretation of propositional calculus. To do this, we assign all possible combinations of true and false to the wff variables and verify that the result (using the rules described in wi 4 and wn 3) always evaluates to true. This is called the semantic approach. Our approach is called the syntactic approach, in which everything is derived from axioms. A metatheorem called the Completeness Theorem for Propositional Calculus shows that the two approaches are equivalent and even provides an algorithm for automatically generating syntactic proofs from a truth table. Those proofs, however, tend to be long, since truth tables grow exponentially with the number of variables, and the much shorter proofs that we show here were found manually. (Contributed by NM, 5-Aug-1993.)

Ref Expression
ax-1 (φ → (ψφ))

Detailed syntax breakdown of Axiom ax-1
StepHypRef Expression
1 wph . 2 wff φ
2 wps . . 3 wff ψ
32, 1wi 4 . 2 wff (ψφ)
41, 3wi 4 1 wff (φ → (ψφ))
Colors of variables: wff set class
This axiom is referenced by:  a1i  9  id  17  id1  18  a1d  20  a1dd  40  jarr  89  pm2.86i  90  pm2.86d  91  pm5.1im  160  biimt  228  pm5.4  236  pm4.45im  315  pm2.51  559  pm4.8  601  pm2.53  616  imorri  643  pm2.64  688  pm2.82  699  biort  713  condc  720  oibabs  795  pm5.12dc  804  pm5.14dc  805  oplem1  850  stdpc4  1537  sbequi  1595  sbidm  1605  moimv  1809  euim  1811  ax11  1918  ax11f  1919  ax11eq  1920  ax11el  1921  ax11indi  1923  ax11indalem  1924  ax11inda2ALT  1925  ax11inda2  1926
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