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. ecase3
  78. ecase
  79. ecase3d
  80. ecased
  81. ecase3ad
  82. ccase
  83. ccased
  84. ccase2
  85. 4cases
  86. 4casesdan
  87. cases
  88. dedlem0a
  89. dedlem0b
  90. dedlema
  91. dedlemb
  92. cases2
  93. cases2ALT
  94. dfbi3
  95. pm5.24
  96. 4exmid
  97. consensus
  98. pm4.42
  99. prlem1
  100. prlem2
  101. oplem1
  102. dn1
  103. bianir
  104. jaoi2
  105. jaoi3
  106. ornld