Axiom ax-1 6

Description: Axiom Simp. Axiom A1 of [Margaris] p. 49. One of the axioms of propositional calculus. 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."

The theorems of propositional calculus are also called tautologies. Although classical propositional logic tautologies can be proved using truth tables, there is no similarly simple system for intuitionistic propositional logic, so proving tautologies from axioms is the preferred approach. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
ax-1	⊢ (𝜑 → (𝜓 → 𝜑))

Detailed syntax breakdown of Axiom ax-1

Step	Hyp	Ref	Expression
1		wph	. 2 wff 𝜑
2		wps	. . 3 wff 𝜓
3	2, 1	wi 4	. 2 wff (𝜓 → 𝜑)
4	1, 3	wi 4	1 wff (𝜑 → (𝜓 → 𝜑))

This axiom is referenced by:  a1i  9  id  19  idALT  20  a1d  22  a1dd  48  jarr  97  jarri  98  pm2.86i  99  pm2.86d  100  pm5.1im  173  biimt  241  pm5.4  249  pm4.45im  334  conax1  654  pm4.8  708  oibabs  715  imorr  722  pm2.53  723  imorri  750  jao1i  797  pm2.64  802  pm2.82  813  condcOLD  855  pm5.12dc  911  pm5.14dc  912  peircedc  915  pm4.83dc  953  dedlem0a  970  oplem1  977  stdpc4  1786  sbequi  1850  sbidm  1862  eumo  2070  moimv  2104  euim  2106  alral  2535  r19.12  2596  r19.27av  2625  r19.37  2642  gencbval  2800  eqvinc  2875  eqvincg  2876  rr19.3v  2891  ralidm  3538  ralm  3542  class2seteq  4181  exmid0el  4222  sotritric  4342  elnnnn0b  9251  zltnle  9330  iccneg  10021  qltnle  10278  frec2uzlt2d  10437  hashfzp1  10839  algcvgblem  12084  bj-trst  14969  bj-findis  15209