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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dfor2 901 | Logical 'or' expressed in terms of implication only. Theorem *5.25 of [WhiteheadRussell] p. 124. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Wolf Lammen, 20-Oct-2012.) |
⊢ ((𝜑 ∨ 𝜓) ↔ ((𝜑 → 𝜓) → 𝜓)) | ||
Theorem | pm2.07 902 | Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.) |
⊢ (𝜑 → (𝜑 ∨ 𝜑)) | ||
Theorem | pm1.2 903 | Axiom *1.2 of [WhiteheadRussell] p. 96, which they call "Taut". (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∨ 𝜑) → 𝜑) | ||
Theorem | oridm 904 | Idempotent law for disjunction. Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 11-May-1993.) (Proof shortened by Andrew Salmon, 16-Apr-2011.) (Proof shortened by Wolf Lammen, 10-Mar-2013.) |
⊢ ((𝜑 ∨ 𝜑) ↔ 𝜑) | ||
Theorem | pm4.25 905 | Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) |
⊢ (𝜑 ↔ (𝜑 ∨ 𝜑)) | ||
Theorem | pm2.4 906 | Theorem *2.4 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∨ (𝜑 ∨ 𝜓)) → (𝜑 ∨ 𝜓)) | ||
Theorem | pm2.41 907 | Theorem *2.41 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜓 ∨ (𝜑 ∨ 𝜓)) → (𝜑 ∨ 𝜓)) | ||
Theorem | orim12i 908 | Disjoin antecedents and consequents of two premises. (Contributed by NM, 6-Jun-1994.) (Proof shortened by Wolf Lammen, 25-Jul-2012.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜒 → 𝜃) ⇒ ⊢ ((𝜑 ∨ 𝜒) → (𝜓 ∨ 𝜃)) | ||
Theorem | orim1i 909 | Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜑 ∨ 𝜒) → (𝜓 ∨ 𝜒)) | ||
Theorem | orim2i 910 | Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜒 ∨ 𝜑) → (𝜒 ∨ 𝜓)) | ||
Theorem | orim12dALT 911 | Alternate proof of orim12d 964 which does not depend on df-an 400. This is an illustration of the conservativity of definitions (definitions do not permit to prove additional theorems whose statements do not contain the defined symbol). (Contributed by Wolf Lammen, 8-Aug-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜃 → 𝜏)) ⇒ ⊢ (𝜑 → ((𝜓 ∨ 𝜃) → (𝜒 ∨ 𝜏))) | ||
Theorem | orbi2i 912 | Inference adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 12-Dec-2012.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓)) | ||
Theorem | orbi1i 913 | Inference adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒)) | ||
Theorem | orbi12i 914 | Infer the disjunction of two equivalences. (Contributed by NM, 3-Jan-1993.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜒 ↔ 𝜃) ⇒ ⊢ ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜃)) | ||
Theorem | orbi2d 915 | Deduction adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ((𝜃 ∨ 𝜓) ↔ (𝜃 ∨ 𝜒))) | ||
Theorem | orbi1d 916 | Deduction adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ((𝜓 ∨ 𝜃) ↔ (𝜒 ∨ 𝜃))) | ||
Theorem | orbi1 917 | Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒))) | ||
Theorem | orbi12d 918 | Deduction joining two equivalences to form equivalence of disjunctions. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → (𝜃 ↔ 𝜏)) ⇒ ⊢ (𝜑 → ((𝜓 ∨ 𝜃) ↔ (𝜒 ∨ 𝜏))) | ||
Theorem | pm1.5 919 | Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∨ (𝜓 ∨ 𝜒)) → (𝜓 ∨ (𝜑 ∨ 𝜒))) | ||
Theorem | or12 920 | Swap two disjuncts. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2012.) |
⊢ ((𝜑 ∨ (𝜓 ∨ 𝜒)) ↔ (𝜓 ∨ (𝜑 ∨ 𝜒))) | ||
Theorem | orass 921 | Associative law for disjunction. Theorem *4.33 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (𝜑 ∨ (𝜓 ∨ 𝜒))) | ||
Theorem | pm2.31 922 | Theorem *2.31 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∨ (𝜓 ∨ 𝜒)) → ((𝜑 ∨ 𝜓) ∨ 𝜒)) | ||
Theorem | pm2.32 923 | Theorem *2.32 of [WhiteheadRussell] p. 105. (Contributed by NM, 3-Jan-2005.) |
⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) → (𝜑 ∨ (𝜓 ∨ 𝜒))) | ||
Theorem | pm2.3 924 | Theorem *2.3 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∨ (𝜓 ∨ 𝜒)) → (𝜑 ∨ (𝜒 ∨ 𝜓))) | ||
Theorem | or32 925 | A rearrangement of disjuncts. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ ((𝜑 ∨ 𝜒) ∨ 𝜓)) | ||
Theorem | or4 926 | Rearrangement of 4 disjuncts. (Contributed by NM, 12-Aug-1994.) |
⊢ (((𝜑 ∨ 𝜓) ∨ (𝜒 ∨ 𝜃)) ↔ ((𝜑 ∨ 𝜒) ∨ (𝜓 ∨ 𝜃))) | ||
Theorem | or42 927 | Rearrangement of 4 disjuncts. (Contributed by NM, 10-Jan-2005.) |
⊢ (((𝜑 ∨ 𝜓) ∨ (𝜒 ∨ 𝜃)) ↔ ((𝜑 ∨ 𝜒) ∨ (𝜃 ∨ 𝜓))) | ||
Theorem | orordi 928 | Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.) |
⊢ ((𝜑 ∨ (𝜓 ∨ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ∨ (𝜑 ∨ 𝜒))) | ||
Theorem | orordir 929 | Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.) |
⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ ((𝜑 ∨ 𝜒) ∨ (𝜓 ∨ 𝜒))) | ||
Theorem | orimdi 930 | Disjunction distributes over implication. (Contributed by Wolf Lammen, 5-Jan-2013.) |
⊢ ((𝜑 ∨ (𝜓 → 𝜒)) ↔ ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒))) | ||
Theorem | pm2.76 931 | Theorem *2.76 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∨ (𝜓 → 𝜒)) → ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒))) | ||
Theorem | pm2.85 932 | Theorem *2.85 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Jan-2013.) |
⊢ (((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) → (𝜑 ∨ (𝜓 → 𝜒))) | ||
Theorem | pm2.75 933 | Theorem *2.75 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 4-Jan-2013.) |
⊢ ((𝜑 ∨ 𝜓) → ((𝜑 ∨ (𝜓 → 𝜒)) → (𝜑 ∨ 𝜒))) | ||
Theorem | pm4.78 934 | Implication distributes over disjunction. Theorem *4.78 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2012.) |
⊢ (((𝜑 → 𝜓) ∨ (𝜑 → 𝜒)) ↔ (𝜑 → (𝜓 ∨ 𝜒))) | ||
Theorem | biort 935 | A disjunction with a true formula is equivalent to that true formula. (Contributed by NM, 23-May-1999.) |
⊢ (𝜑 → (𝜑 ↔ (𝜑 ∨ 𝜓))) | ||
Theorem | biorf 936 | A wff is equivalent to its disjunction with falsehood. Theorem *4.74 of [WhiteheadRussell] p. 121. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2012.) |
⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓))) | ||
Theorem | biortn 937 | A wff is equivalent to its negated disjunction with falsehood. (Contributed by NM, 9-Jul-2012.) |
⊢ (𝜑 → (𝜓 ↔ (¬ 𝜑 ∨ 𝜓))) | ||
Theorem | biorfi 938 | A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 16-Jul-2021.) |
⊢ ¬ 𝜑 ⇒ ⊢ (𝜓 ↔ (𝜓 ∨ 𝜑)) | ||
Theorem | pm2.26 939 | Theorem *2.26 of [WhiteheadRussell] p. 104. See pm2.27 42. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Nov-2012.) |
⊢ (¬ 𝜑 ∨ ((𝜑 → 𝜓) → 𝜓)) | ||
Theorem | pm2.63 940 | Theorem *2.63 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∨ 𝜓) → ((¬ 𝜑 ∨ 𝜓) → 𝜓)) | ||
Theorem | pm2.64 941 | Theorem *2.64 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∨ 𝜓) → ((𝜑 ∨ ¬ 𝜓) → 𝜑)) | ||
Theorem | pm2.42 942 | Theorem *2.42 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.) |
⊢ ((¬ 𝜑 ∨ (𝜑 → 𝜓)) → (𝜑 → 𝜓)) | ||
Theorem | pm5.11g 943 | A general instance of Theorem *5.11 of [WhiteheadRussell] p. 123. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 → 𝜓) ∨ (¬ 𝜑 → 𝜒)) | ||
Theorem | pm5.11 944 | Theorem *5.11 of [WhiteheadRussell] p. 123. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 → 𝜓) ∨ (¬ 𝜑 → 𝜓)) | ||
Theorem | pm5.12 945 | Theorem *5.12 of [WhiteheadRussell] p. 123. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 → 𝜓) ∨ (𝜑 → ¬ 𝜓)) | ||
Theorem | pm5.14 946 | Theorem *5.14 of [WhiteheadRussell] p. 123. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 → 𝜓) ∨ (𝜓 → 𝜒)) | ||
Theorem | pm5.13 947 | Theorem *5.13 of [WhiteheadRussell] p. 123. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 14-Nov-2012.) |
⊢ ((𝜑 → 𝜓) ∨ (𝜓 → 𝜑)) | ||
Theorem | pm5.55 948 | Theorem *5.55 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 20-Jan-2013.) |
⊢ (((𝜑 ∨ 𝜓) ↔ 𝜑) ∨ ((𝜑 ∨ 𝜓) ↔ 𝜓)) | ||
Theorem | pm4.72 949 | Implication in terms of biconditional and disjunction. Theorem *4.72 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Jan-2013.) |
⊢ ((𝜑 → 𝜓) ↔ (𝜓 ↔ (𝜑 ∨ 𝜓))) | ||
Theorem | imimorb 950 | Simplify an implication between implications. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Wolf Lammen, 3-Apr-2013.) |
⊢ (((𝜓 → 𝜒) → (𝜑 → 𝜒)) ↔ (𝜑 → (𝜓 ∨ 𝜒))) | ||
Theorem | oibabs 951 | Absorption of disjunction into equivalence. (Contributed by NM, 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.) |
⊢ (((𝜑 ∨ 𝜓) → (𝜑 ↔ 𝜓)) ↔ (𝜑 ↔ 𝜓)) | ||
Theorem | orbidi 952 | Disjunction distributes over the biconditional. An axiom of system DS in Vladimir Lifschitz, "On calculational proofs" (1998), http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.25.3384. (Contributed by NM, 8-Jan-2005.) (Proof shortened by Wolf Lammen, 4-Feb-2013.) |
⊢ ((𝜑 ∨ (𝜓 ↔ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ↔ (𝜑 ∨ 𝜒))) | ||
Theorem | pm5.7 953 | Disjunction distributes over the biconditional. Theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbidi 952. (Contributed by Roy F. Longton, 21-Jun-2005.) |
⊢ (((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒)) ↔ (𝜒 ∨ (𝜑 ↔ 𝜓))) | ||
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 964 versus orim12dALT 911). These theorems are mostly grouped at the beginning of this section. The family of theorems starting with animorl 977 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 954 and related theorems. | ||
Theorem | jaao 954 | Inference conjoining and disjoining the antecedents of two implications. (Contributed by NM, 30-Sep-1999.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜃 → (𝜏 → 𝜒)) ⇒ ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∨ 𝜏) → 𝜒)) | ||
Theorem | jaoa 955 | Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜃 → (𝜏 → 𝜒)) ⇒ ⊢ ((𝜑 ∨ 𝜃) → ((𝜓 ∧ 𝜏) → 𝜒)) | ||
Theorem | jaoian 956 | Inference disjoining the antecedents of two implications. (Contributed by NM, 23-Oct-2005.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜃 ∧ 𝜓) → 𝜒) ⇒ ⊢ (((𝜑 ∨ 𝜃) ∧ 𝜓) → 𝜒) | ||
Theorem | jaodan 957 | Deduction disjoining the antecedents of two implications. (Contributed by NM, 14-Oct-2005.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜑 ∧ 𝜃) → 𝜒) ⇒ ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜃)) → 𝜒) | ||
Theorem | mpjaodan 958 | Eliminate a disjunction in a deduction. A translation of natural deduction rule ∨ E (∨ elimination), see natded 28332. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜑 ∧ 𝜃) → 𝜒) & ⊢ (𝜑 → (𝜓 ∨ 𝜃)) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | pm3.44 959 | Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Oct-2013.) |
⊢ (((𝜓 → 𝜑) ∧ (𝜒 → 𝜑)) → ((𝜓 ∨ 𝜒) → 𝜑)) | ||
Theorem | jao 960 | Disjunction of antecedents. Compare Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM, 5-Apr-1994.) (Proof shortened by Wolf Lammen, 4-Apr-2013.) |
⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜓) → ((𝜑 ∨ 𝜒) → 𝜓))) | ||
Theorem | jaob 961 | Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-May-1994.) (Proof shortened by Wolf Lammen, 9-Dec-2012.) |
⊢ (((𝜑 ∨ 𝜒) → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓))) | ||
Theorem | pm4.77 962 | Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) |
⊢ (((𝜓 → 𝜑) ∧ (𝜒 → 𝜑)) ↔ ((𝜓 ∨ 𝜒) → 𝜑)) | ||
Theorem | pm3.48 963 | Theorem *3.48 of [WhiteheadRussell] p. 114. (Contributed by NM, 28-Jan-1997.) |
⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → ((𝜑 ∨ 𝜒) → (𝜓 ∨ 𝜃))) | ||
Theorem | orim12d 964 | Disjoin antecedents and consequents in a deduction. See orim12dALT 911 for a proof which does not depend on df-an 400. (Contributed by NM, 10-May-1994.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜃 → 𝜏)) ⇒ ⊢ (𝜑 → ((𝜓 ∨ 𝜃) → (𝜒 ∨ 𝜏))) | ||
Theorem | orim1d 965 | Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → ((𝜓 ∨ 𝜃) → (𝜒 ∨ 𝜃))) | ||
Theorem | orim2d 966 | Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → ((𝜃 ∨ 𝜓) → (𝜃 ∨ 𝜒))) | ||
Theorem | orim2 967 | Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜓 → 𝜒) → ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒))) | ||
Theorem | pm2.38 968 | Theorem *2.38 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.) |
⊢ ((𝜓 → 𝜒) → ((𝜓 ∨ 𝜑) → (𝜒 ∨ 𝜑))) | ||
Theorem | pm2.36 969 | Theorem *2.36 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.) |
⊢ ((𝜓 → 𝜒) → ((𝜑 ∨ 𝜓) → (𝜒 ∨ 𝜑))) | ||
Theorem | pm2.37 970 | Theorem *2.37 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.) |
⊢ ((𝜓 → 𝜒) → ((𝜓 ∨ 𝜑) → (𝜑 ∨ 𝜒))) | ||
Theorem | pm2.81 971 | Theorem *2.81 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜓 → (𝜒 → 𝜃)) → ((𝜑 ∨ 𝜓) → ((𝜑 ∨ 𝜒) → (𝜑 ∨ 𝜃)))) | ||
Theorem | pm2.8 972 | Theorem *2.8 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Jan-2013.) |
⊢ ((𝜑 ∨ 𝜓) → ((¬ 𝜓 ∨ 𝜒) → (𝜑 ∨ 𝜒))) | ||
Theorem | pm2.73 973 | Theorem *2.73 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 → 𝜓) → (((𝜑 ∨ 𝜓) ∨ 𝜒) → (𝜓 ∨ 𝜒))) | ||
Theorem | pm2.74 974 | Theorem *2.74 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
⊢ ((𝜓 → 𝜑) → (((𝜑 ∨ 𝜓) ∨ 𝜒) → (𝜑 ∨ 𝜒))) | ||
Theorem | pm2.82 975 | Theorem *2.82 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) |
⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) → (((𝜑 ∨ ¬ 𝜒) ∨ 𝜃) → ((𝜑 ∨ 𝜓) ∨ 𝜃))) | ||
Theorem | pm4.39 976 | Theorem *4.39 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.) |
⊢ (((𝜑 ↔ 𝜒) ∧ (𝜓 ↔ 𝜃)) → ((𝜑 ∨ 𝜓) ↔ (𝜒 ∨ 𝜃))) | ||
Theorem | animorl 977 | Conjunction implies disjunction with one common formula (1/4). (Contributed by BJ, 4-Oct-2019.) |
⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∨ 𝜒)) | ||
Theorem | animorr 978 | Conjunction implies disjunction with one common formula (2/4). (Contributed by BJ, 4-Oct-2019.) |
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∨ 𝜓)) | ||
Theorem | animorlr 979 | Conjunction implies disjunction with one common formula (3/4). (Contributed by BJ, 4-Oct-2019.) |
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∨ 𝜑)) | ||
Theorem | animorrl 980 | Conjunction implies disjunction with one common formula (4/4). (Contributed by BJ, 4-Oct-2019.) |
⊢ ((𝜑 ∧ 𝜓) → (𝜓 ∨ 𝜒)) | ||
Theorem | ianor 981 | Negated conjunction in terms of disjunction (De Morgan's law). Theorem *4.51 of [WhiteheadRussell] p. 120. (Contributed by NM, 14-May-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓)) | ||
Theorem | anor 982 | Conjunction in terms of disjunction (De Morgan's law). Theorem *4.5 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 3-Nov-2012.) |
⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓)) | ||
Theorem | ioran 983 | Negated disjunction in terms of conjunction (De Morgan's law). Compare Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓)) | ||
Theorem | pm4.52 984 | Theorem *4.52 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Nov-2012.) |
⊢ ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (¬ 𝜑 ∨ 𝜓)) | ||
Theorem | pm4.53 985 | Theorem *4.53 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) |
⊢ (¬ (𝜑 ∧ ¬ 𝜓) ↔ (¬ 𝜑 ∨ 𝜓)) | ||
Theorem | pm4.54 986 | Theorem *4.54 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Nov-2012.) |
⊢ ((¬ 𝜑 ∧ 𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓)) | ||
Theorem | pm4.55 987 | Theorem *4.55 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) |
⊢ (¬ (¬ 𝜑 ∧ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)) | ||
Theorem | pm4.56 988 | Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) |
⊢ ((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓)) | ||
Theorem | oran 989 | Disjunction in terms of conjunction (De Morgan's law). Compare Theorem *4.57 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
⊢ ((𝜑 ∨ 𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓)) | ||
Theorem | pm4.57 990 | Theorem *4.57 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) |
⊢ (¬ (¬ 𝜑 ∧ ¬ 𝜓) ↔ (𝜑 ∨ 𝜓)) | ||
Theorem | pm3.1 991 | Theorem *3.1 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∧ 𝜓) → ¬ (¬ 𝜑 ∨ ¬ 𝜓)) | ||
Theorem | pm3.11 992 | Theorem *3.11 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) |
⊢ (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓)) | ||
Theorem | pm3.12 993 | Theorem *3.12 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) |
⊢ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓)) | ||
Theorem | pm3.13 994 | Theorem *3.13 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) |
⊢ (¬ (𝜑 ∧ 𝜓) → (¬ 𝜑 ∨ ¬ 𝜓)) | ||
Theorem | pm3.14 995 | Theorem *3.14 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) |
⊢ ((¬ 𝜑 ∨ ¬ 𝜓) → ¬ (𝜑 ∧ 𝜓)) | ||
Theorem | pm4.44 996 | Theorem *4.44 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.) |
⊢ (𝜑 ↔ (𝜑 ∨ (𝜑 ∧ 𝜓))) | ||
Theorem | pm4.45 997 | Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.) |
⊢ (𝜑 ↔ (𝜑 ∧ (𝜑 ∨ 𝜓))) | ||
Theorem | orabs 998 | Absorption of redundant internal disjunct. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 28-Feb-2014.) |
⊢ (𝜑 ↔ ((𝜑 ∨ 𝜓) ∧ 𝜑)) | ||
Theorem | oranabs 999 | Absorb a disjunct into a conjunct. (Contributed by Roy F. Longton, 23-Jun-2005.) (Proof shortened by Wolf Lammen, 10-Nov-2013.) |
⊢ (((𝜑 ∨ ¬ 𝜓) ∧ 𝜓) ↔ (𝜑 ∧ 𝜓)) | ||
Theorem | pm5.61 1000 | Theorem *5.61 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 30-Jun-2013.) |
⊢ (((𝜑 ∨ 𝜓) ∧ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓)) |
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