Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  3imp GIF version

Theorem 3imp 1133
 Description: Importation inference. (Contributed by NM, 8-Apr-1994.)
Hypothesis
Ref Expression
3imp.1 (𝜑 → (𝜓 → (𝜒𝜃)))
Assertion
Ref Expression
3imp ((𝜑𝜓𝜒) → 𝜃)

Proof of Theorem 3imp
StepHypRef Expression
1 df-3an 922 . 2 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
2 3imp.1 . . 3 (𝜑 → (𝜓 → (𝜒𝜃)))
32imp31 252 . 2 (((𝜑𝜓) ∧ 𝜒) → 𝜃)
41, 3sylbi 119 1 ((𝜑𝜓𝜒) → 𝜃)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   ∧ w3a 920 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105 This theorem depends on definitions:  df-bi 115  df-3an 922 This theorem is referenced by:  3impa  1134  3impb  1135  3impia  1136  3impib  1137  3com23  1145  3an1rs  1151  3imp1  1152  3impd  1153  syl3an2  1204  syl3an3  1205  3jao  1233  biimp3ar  1278  poxp  5904  tfrlemibxssdm  5996  tfr1onlembxssdm  6012  tfrcllembxssdm  6025  nndi  6150  nnmass  6151  pr2nelem  6571  difelfzle  9274  fzo1fzo0n0  9321  elfzo0z  9322  fzofzim  9326  elfzodifsumelfzo  9339  mulexp  9664  expadd  9667  expmul  9670  bernneq  9742  facdiv  9814  addmodlteqALT  10467  ltoddhalfle  10500  halfleoddlt  10501  dfgcd2  10610  cncongr1  10692  oddprmgt2  10722  prmfac1  10738
 Copyright terms: Public domain W3C validator