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 ph and ps to the assertion of ph 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	|- ( ph -> ( ps -> ph ) )

Detailed syntax breakdown of Axiom ax-1

Step	Hyp	Ref	Expression
1		wph	. 2 wff ph
2		wps	. . 3 wff ps
3	2, 1	wi 4	. 2 wff ( ps -> ph )
4	1, 3	wi 4	1 wff ( ph -> ( ps -> ph ) )

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  1797  sbequi  1861  sbidm  1873  eumo  2085  moimv  2119  euim  2121  alral  2550  r19.12  2611  r19.27av  2640  r19.37  2657  gencbval  2820  eqvinc  2895  eqvincg  2896  rr19.3v  2911  ralidm  3560  ralm  3563  class2seteq  4206  exmid0el  4247  sotritric  4370  elnnnn0b  9338  zltnle  9417  iccneg  10110  qltnle  10384  frec2uzlt2d  10547  hashfzp1  10967  algcvgblem  12342  bj-trst  15637  bj-findis  15877