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  655  pm4.8  709  oibabs  716  imorr  723  pm2.53  724  imorri  751  jao1i  798  pm2.64  803  pm2.82  814  condcOLD  856  pm5.12dc  912  pm5.14dc  913  peircedc  916  pm4.83dc  954  dedlem0a  971  oplem1  978  stdpc4  1798  sbequi  1862  sbidm  1874  eumo  2086  moimv  2120  euim  2122  alral  2551  r19.12  2612  r19.27av  2641  r19.37  2658  gencbval  2821  eqvinc  2896  eqvincg  2897  rr19.3v  2912  ralidm  3561  ralm  3564  class2seteq  4207  exmid0el  4248  sotritric  4371  elnnnn0b  9339  zltnle  9418  iccneg  10111  qltnle  10386  frec2uzlt2d  10549  hashfzp1  10969  algcvgblem  12371  bj-trst  15675  bj-findis  15915