Metamath Proof Explorer


Table of Contents - 1.2.8. Mixed connectives

This section gathers theorems of propositional calculus which use (either in their statement or proof) mixed connectives (at least conjunction and disjunction).

As noted in the "note on definitions" in the section comment for logical equivalence, some theorem statements may contain for instance only conjunction or only disjunction, but both definitions are used in their proofs to make them shorter (this is exemplified in orim12d versus orim12dALT). These theorems are mostly grouped at the beginning of this section.

The family of theorems starting with animorl focus on the relation between conjunction and disjunction and can be seen as the starting point of mixed connectives in statements. This sectioning is not rigorously true, since for instance the section begins with jaao and related theorems.

  1. jaao
  2. jaoa
  3. jaoian
  4. jaodan
  5. mpjaodan
  6. pm3.44
  7. jao
  8. jaob
  9. pm4.77
  10. pm3.48
  11. orim12d
  12. orim1d
  13. orim2d
  14. orim2
  15. pm2.38
  16. pm2.36
  17. pm2.37
  18. pm2.81
  19. pm2.8
  20. pm2.73
  21. pm2.74
  22. pm2.82
  23. pm4.39
  24. animorl
  25. animorr
  26. animorlr
  27. animorrl
  28. ianor
  29. anor
  30. ioran
  31. pm4.52
  32. pm4.53
  33. pm4.54
  34. pm4.55
  35. pm4.56
  36. oran
  37. pm4.57
  38. pm3.1
  39. pm3.11
  40. pm3.12
  41. pm3.13
  42. pm3.14
  43. pm4.44
  44. pm4.45
  45. orabs
  46. oranabs
  47. pm5.61
  48. pm5.6
  49. orcanai
  50. pm4.79
  51. pm5.53
  52. ordi
  53. ordir
  54. andi
  55. andir
  56. orddi
  57. anddi
  58. pm5.17
  59. pm5.15
  60. pm5.16
  61. xor
  62. nbi2
  63. xordi
  64. pm5.54
  65. pm5.62
  66. pm5.63
  67. niabn
  68. ninba
  69. pm4.43
  70. pm4.82
  71. pm4.83
  72. pclem6
  73. bigolden
  74. pm5.71
  75. pm5.75
  76. ecase2d
  77. ecase2dOLD
  78. ecase3
  79. ecase
  80. ecase3d
  81. ecased
  82. ecase3ad
  83. ecase3adOLD
  84. ccase
  85. ccased
  86. ccase2
  87. 4cases
  88. 4casesdan
  89. cases
  90. dedlem0a
  91. dedlem0b
  92. dedlema
  93. dedlemb
  94. cases2
  95. cases2ALT
  96. dfbi3
  97. pm5.24
  98. 4exmid
  99. consensus
  100. pm4.42
  101. prlem1
  102. prlem2
  103. oplem1
  104. dn1
  105. bianir
  106. jaoi2
  107. jaoi3
  108. ornld