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  659  pm4.8  715  oibabs  722  imorr  729  pm2.53  730  imorri  757  jao1i  804  pm2.64  809  pm2.82  820  condcOLD  862  pm5.12dc  918  pm5.14dc  919  peircedc  922  pm4.83dc  960  dedlem0a  977  oplem1  984  a1ddd  1481  stdpc4  1824  sbequi  1888  sbidm  1900  eumo  2114  moimv  2149  euim  2151  alral  2589  r19.12  2651  r19.27av  2680  r19.37  2697  gencbval  2865  eqvinc  2942  eqvincg  2943  rr19.3v  2958  ralidm  3612  ralm  3615  class2seteq  4278  exmid0el  4319  sotritric  4447  elnnnn0b  9542  zltnle  9625  iccneg  10325  qltnle  10607  frec2uzlt2d  10770  hashfzp1  11193  algcvgblem  12750  clwwlknonex2lem2  16450  bj-trst  16528  bj-findis  16766