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  a1ddd  1456  stdpc4  1799  sbequi  1863  sbidm  1875  eumo  2087  moimv  2122  euim  2124  alral  2553  r19.12  2614  r19.27av  2643  r19.37  2660  gencbval  2826  eqvinc  2903  eqvincg  2904  rr19.3v  2919  ralidm  3569  ralm  3572  class2seteq  4223  exmid0el  4264  sotritric  4389  elnnnn0b  9374  zltnle  9453  iccneg  10146  qltnle  10423  frec2uzlt2d  10586  hashfzp1  11006  algcvgblem  12486  bj-trst  15875  bj-findis  16114