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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | pm1.2 901 | Axiom *1.2 of [WhiteheadRussell] p. 96, which they call "Taut". (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∨ 𝜑) → 𝜑) | ||
Theorem | oridm 902 | 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 | pm4.25 903 | Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) |
⊢ (𝜑 ↔ (𝜑 ∨ 𝜑)) | ||
Theorem | pm2.4 904 | Theorem *2.4 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∨ (𝜑 ∨ 𝜓)) → (𝜑 ∨ 𝜓)) | ||
Theorem | pm2.41 905 | Theorem *2.41 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜓 ∨ (𝜑 ∨ 𝜓)) → (𝜑 ∨ 𝜓)) | ||
Theorem | orim12i 906 | 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 907 | Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜑 ∨ 𝜒) → (𝜓 ∨ 𝜒)) | ||
Theorem | orim2i 908 | Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜒 ∨ 𝜑) → (𝜒 ∨ 𝜓)) | ||
Theorem | orim12dALT 909 | Alternate proof of orim12d 962 which does not depend on df-an 397. 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 910 | 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 911 | 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 912 | Infer the disjunction of two equivalences. (Contributed by NM, 3-Jan-1993.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜒 ↔ 𝜃) ⇒ ⊢ ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜃)) | ||
Theorem | orbi2d 913 | 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 914 | 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 915 | Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒))) | ||
Theorem | orbi12d 916 | 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 | pm1.5 917 | Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∨ (𝜓 ∨ 𝜒)) → (𝜓 ∨ (𝜑 ∨ 𝜒))) | ||
Theorem | or12 918 | 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 919 | 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 | pm2.31 920 | Theorem *2.31 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∨ (𝜓 ∨ 𝜒)) → ((𝜑 ∨ 𝜓) ∨ 𝜒)) | ||
Theorem | pm2.32 921 | Theorem *2.32 of [WhiteheadRussell] p. 105. (Contributed by NM, 3-Jan-2005.) |
⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) → (𝜑 ∨ (𝜓 ∨ 𝜒))) | ||
Theorem | pm2.3 922 | Theorem *2.3 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∨ (𝜓 ∨ 𝜒)) → (𝜑 ∨ (𝜒 ∨ 𝜓))) | ||
Theorem | or32 923 | 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 924 | Rearrangement of 4 disjuncts. (Contributed by NM, 12-Aug-1994.) |
⊢ (((𝜑 ∨ 𝜓) ∨ (𝜒 ∨ 𝜃)) ↔ ((𝜑 ∨ 𝜒) ∨ (𝜓 ∨ 𝜃))) | ||
Theorem | or42 925 | Rearrangement of 4 disjuncts. (Contributed by NM, 10-Jan-2005.) |
⊢ (((𝜑 ∨ 𝜓) ∨ (𝜒 ∨ 𝜃)) ↔ ((𝜑 ∨ 𝜒) ∨ (𝜃 ∨ 𝜓))) | ||
Theorem | orordi 926 | Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.) |
⊢ ((𝜑 ∨ (𝜓 ∨ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ∨ (𝜑 ∨ 𝜒))) | ||
Theorem | orordir 927 | Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.) |
⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ ((𝜑 ∨ 𝜒) ∨ (𝜓 ∨ 𝜒))) | ||
Theorem | orimdi 928 | Disjunction distributes over implication. (Contributed by Wolf Lammen, 5-Jan-2013.) |
⊢ ((𝜑 ∨ (𝜓 → 𝜒)) ↔ ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒))) | ||
Theorem | pm2.76 929 | Theorem *2.76 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∨ (𝜓 → 𝜒)) → ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒))) | ||
Theorem | pm2.85 930 | 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 | pm2.75 931 | 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 | pm4.78 932 | 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 933 | A disjunction with a true formula is equivalent to that true formula. (Contributed by NM, 23-May-1999.) |
⊢ (𝜑 → (𝜑 ↔ (𝜑 ∨ 𝜓))) | ||
Theorem | biorf 934 | 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 935 | A wff is equivalent to its negated disjunction with falsehood. (Contributed by NM, 9-Jul-2012.) |
⊢ (𝜑 → (𝜓 ↔ (¬ 𝜑 ∨ 𝜓))) | ||
Theorem | biorfi 936 | 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 | pm2.26 937 | 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 | pm2.63 938 | Theorem *2.63 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∨ 𝜓) → ((¬ 𝜑 ∨ 𝜓) → 𝜓)) | ||
Theorem | pm2.64 939 | Theorem *2.64 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∨ 𝜓) → ((𝜑 ∨ ¬ 𝜓) → 𝜑)) | ||
Theorem | pm2.42 940 | Theorem *2.42 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.) |
⊢ ((¬ 𝜑 ∨ (𝜑 → 𝜓)) → (𝜑 → 𝜓)) | ||
Theorem | pm5.11g 941 | A general instance of Theorem *5.11 of [WhiteheadRussell] p. 123. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 → 𝜓) ∨ (¬ 𝜑 → 𝜒)) | ||
Theorem | pm5.11 942 | Theorem *5.11 of [WhiteheadRussell] p. 123. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 → 𝜓) ∨ (¬ 𝜑 → 𝜓)) | ||
Theorem | pm5.12 943 | Theorem *5.12 of [WhiteheadRussell] p. 123. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 → 𝜓) ∨ (𝜑 → ¬ 𝜓)) | ||
Theorem | pm5.14 944 | Theorem *5.14 of [WhiteheadRussell] p. 123. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 → 𝜓) ∨ (𝜓 → 𝜒)) | ||
Theorem | pm5.13 945 | 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 | pm5.55 946 | 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 | pm4.72 947 | 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 948 | 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 949 | 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 950 | 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 | pm5.7 951 | Disjunction distributes over the biconditional. Theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbidi 950. (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 962 versus orim12dALT 909). These theorems are mostly grouped at the beginning of this section. The family of theorems starting with animorl 975 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 952 and related theorems. | ||
Theorem | jaao 952 | 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 953 | 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 954 | 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 955 | 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 956 | Eliminate a disjunction in a deduction. A translation of natural deduction rule ∨ E (∨ elimination), see natded 28767. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜑 ∧ 𝜃) → 𝜒) & ⊢ (𝜑 → (𝜓 ∨ 𝜃)) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | pm3.44 957 | 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 958 | 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 959 | 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 | pm4.77 960 | Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) |
⊢ (((𝜓 → 𝜑) ∧ (𝜒 → 𝜑)) ↔ ((𝜓 ∨ 𝜒) → 𝜑)) | ||
Theorem | pm3.48 961 | Theorem *3.48 of [WhiteheadRussell] p. 114. (Contributed by NM, 28-Jan-1997.) |
⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → ((𝜑 ∨ 𝜒) → (𝜓 ∨ 𝜃))) | ||
Theorem | orim12d 962 | Disjoin antecedents and consequents in a deduction. See orim12dALT 909 for a proof which does not depend on df-an 397. (Contributed by NM, 10-May-1994.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜃 → 𝜏)) ⇒ ⊢ (𝜑 → ((𝜓 ∨ 𝜃) → (𝜒 ∨ 𝜏))) | ||
Theorem | orim1d 963 | Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → ((𝜓 ∨ 𝜃) → (𝜒 ∨ 𝜃))) | ||
Theorem | orim2d 964 | Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → ((𝜃 ∨ 𝜓) → (𝜃 ∨ 𝜒))) | ||
Theorem | orim2 965 | Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜓 → 𝜒) → ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒))) | ||
Theorem | pm2.38 966 | Theorem *2.38 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.) |
⊢ ((𝜓 → 𝜒) → ((𝜓 ∨ 𝜑) → (𝜒 ∨ 𝜑))) | ||
Theorem | pm2.36 967 | Theorem *2.36 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.) |
⊢ ((𝜓 → 𝜒) → ((𝜑 ∨ 𝜓) → (𝜒 ∨ 𝜑))) | ||
Theorem | pm2.37 968 | Theorem *2.37 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.) |
⊢ ((𝜓 → 𝜒) → ((𝜓 ∨ 𝜑) → (𝜑 ∨ 𝜒))) | ||
Theorem | pm2.81 969 | Theorem *2.81 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜓 → (𝜒 → 𝜃)) → ((𝜑 ∨ 𝜓) → ((𝜑 ∨ 𝜒) → (𝜑 ∨ 𝜃)))) | ||
Theorem | pm2.8 970 | 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 | pm2.73 971 | Theorem *2.73 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 → 𝜓) → (((𝜑 ∨ 𝜓) ∨ 𝜒) → (𝜓 ∨ 𝜒))) | ||
Theorem | pm2.74 972 | 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 | pm2.82 973 | Theorem *2.82 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) |
⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) → (((𝜑 ∨ ¬ 𝜒) ∨ 𝜃) → ((𝜑 ∨ 𝜓) ∨ 𝜃))) | ||
Theorem | pm4.39 974 | Theorem *4.39 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.) |
⊢ (((𝜑 ↔ 𝜒) ∧ (𝜓 ↔ 𝜃)) → ((𝜑 ∨ 𝜓) ↔ (𝜒 ∨ 𝜃))) | ||
Theorem | animorl 975 | Conjunction implies disjunction with one common formula (1/4). (Contributed by BJ, 4-Oct-2019.) |
⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∨ 𝜒)) | ||
Theorem | animorr 976 | Conjunction implies disjunction with one common formula (2/4). (Contributed by BJ, 4-Oct-2019.) |
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∨ 𝜓)) | ||
Theorem | animorlr 977 | Conjunction implies disjunction with one common formula (3/4). (Contributed by BJ, 4-Oct-2019.) |
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∨ 𝜑)) | ||
Theorem | animorrl 978 | Conjunction implies disjunction with one common formula (4/4). (Contributed by BJ, 4-Oct-2019.) |
⊢ ((𝜑 ∧ 𝜓) → (𝜓 ∨ 𝜒)) | ||
Theorem | ianor 979 | 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 980 | 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 981 | 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 | pm4.52 982 | 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 | pm4.53 983 | Theorem *4.53 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) |
⊢ (¬ (𝜑 ∧ ¬ 𝜓) ↔ (¬ 𝜑 ∨ 𝜓)) | ||
Theorem | pm4.54 984 | 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 | pm4.55 985 | Theorem *4.55 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) |
⊢ (¬ (¬ 𝜑 ∧ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)) | ||
Theorem | pm4.56 986 | Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) |
⊢ ((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓)) | ||
Theorem | oran 987 | 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 | pm4.57 988 | Theorem *4.57 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) |
⊢ (¬ (¬ 𝜑 ∧ ¬ 𝜓) ↔ (𝜑 ∨ 𝜓)) | ||
Theorem | pm3.1 989 | Theorem *3.1 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∧ 𝜓) → ¬ (¬ 𝜑 ∨ ¬ 𝜓)) | ||
Theorem | pm3.11 990 | Theorem *3.11 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) |
⊢ (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓)) | ||
Theorem | pm3.12 991 | Theorem *3.12 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) |
⊢ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓)) | ||
Theorem | pm3.13 992 | Theorem *3.13 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) |
⊢ (¬ (𝜑 ∧ 𝜓) → (¬ 𝜑 ∨ ¬ 𝜓)) | ||
Theorem | pm3.14 993 | Theorem *3.14 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) |
⊢ ((¬ 𝜑 ∨ ¬ 𝜓) → ¬ (𝜑 ∧ 𝜓)) | ||
Theorem | pm4.44 994 | Theorem *4.44 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.) |
⊢ (𝜑 ↔ (𝜑 ∨ (𝜑 ∧ 𝜓))) | ||
Theorem | pm4.45 995 | Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.) |
⊢ (𝜑 ↔ (𝜑 ∧ (𝜑 ∨ 𝜓))) | ||
Theorem | orabs 996 | 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 997 | 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 | pm5.61 998 | 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)) | ||
Theorem | pm5.6 999 | Conjunction in antecedent versus disjunction in consequent. Theorem *5.6 of [WhiteheadRussell] p. 125. (Contributed by NM, 8-Jun-1994.) |
⊢ (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 ∨ 𝜒))) | ||
Theorem | orcanai 1000 | Change disjunction in consequent to conjunction in antecedent. (Contributed by NM, 8-Jun-1994.) |
⊢ (𝜑 → (𝜓 ∨ 𝜒)) ⇒ ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜒) |
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