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  653  pm4.8  707  oibabs  714  imorr  721  pm2.53  722  imorri  749  jao1i  796  pm2.64  801  pm2.82  812  condcOLD  854  pm5.12dc  910  pm5.14dc  911  peircedc  914  pm4.83dc  951  dedlem0a  968  oplem1  975  stdpc4  1773  sbequi  1837  sbidm  1849  eumo  2056  moimv  2090  euim  2092  alral  2520  r19.12  2581  r19.27av  2610  r19.37  2627  gencbval  2783  eqvinc  2858  eqvincg  2859  rr19.3v  2874  ralidm  3521  ralm  3525  class2seteq  4158  exmid0el  4199  sotritric  4318  elnnnn0b  9191  zltnle  9270  iccneg  9958  qltnle  10214  frec2uzlt2d  10372  hashfzp1  10770  algcvgblem  12014  bj-trst  14031  bj-findis  14271