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  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  1789  sbequi  1853  sbidm  1865  eumo  2077  moimv  2111  euim  2113  alral  2542  r19.12  2603  r19.27av  2632  r19.37  2649  gencbval  2812  eqvinc  2887  eqvincg  2888  rr19.3v  2903  ralidm  3552  ralm  3555  class2seteq  4197  exmid0el  4238  sotritric  4360  elnnnn0b  9310  zltnle  9389  iccneg  10081  qltnle  10350  frec2uzlt2d  10513  hashfzp1  10933  algcvgblem  12242  bj-trst  15469  bj-findis  15709