Metamath Proof Explorer


Table of Contents - 1.2.3. Logical implication

The results in this section are based on implication only, and avoid ax-3, so are intuitionistic. The system { ax-mp, ax-1, ax-2 } axiomatizes what is sometimes called "intuitionistic implicational calculus" or "minimal implicational calculus".

In an implication, the wff before the arrow is called the "antecedent" and the wff after the arrow is called the "consequent".

  1. mp2
  2. mp2b
  3. a1i
  4. 2a1i
  5. mp1i
  6. a2i
  7. mpd
  8. imim2i
  9. syl
  10. 3syl
  11. 4syl
  12. mpi
  13. mpisyl
  14. id
  15. idALT
  16. idd
  17. a1d
  18. 2a1d
  19. a1i13
  20. 2a1
  21. a2d
  22. sylcom
  23. syl5com
  24. com12
  25. syl11
  26. syl5
  27. syl6
  28. syl56
  29. syl6com
  30. mpcom
  31. syli
  32. syl2im
  33. syl2imc
  34. pm2.27
  35. mpdd
  36. mpid
  37. mpdi
  38. mpii
  39. syld
  40. syldc
  41. mp2d
  42. a1dd
  43. 2a1dd
  44. pm2.43i
  45. pm2.43d
  46. pm2.43a
  47. pm2.43b
  48. pm2.43
  49. imim2d
  50. imim2
  51. embantd
  52. 3syld
  53. sylsyld
  54. imim12i
  55. imim1i
  56. imim3i
  57. sylc
  58. syl3c
  59. syl6mpi
  60. mpsyl
  61. mpsylsyld
  62. syl6c
  63. syl6ci
  64. syldd
  65. syl5d
  66. syl7
  67. syl6d
  68. syl8
  69. syl9
  70. syl9r
  71. syl10
  72. a1ddd
  73. imim12d
  74. imim1d
  75. imim1
  76. pm2.83
  77. peirceroll
  78. com23
  79. com3r
  80. com13
  81. com3l
  82. pm2.04
  83. com34
  84. com4l
  85. com4t
  86. com4r
  87. com24
  88. com14
  89. com45
  90. com35
  91. com25
  92. com5l
  93. com15
  94. com52l
  95. com52r
  96. com5r
  97. imim12
  98. jarr
  99. jarri
  100. pm2.86d
  101. pm2.86
  102. pm2.86i
  103. loolin
  104. loowoz