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Theorem List for Metamath Proof Explorer - 901-1000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempm4.25 901 Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
(𝜑 ↔ (𝜑𝜑))

Theorempm2.4 902 Theorem *2.4 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∨ (𝜑𝜓)) → (𝜑𝜓))

Theorempm2.41 903 Theorem *2.41 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
((𝜓 ∨ (𝜑𝜓)) → (𝜑𝜓))

Theoremorim12i 904 Disjoin antecedents and consequents of two premises. (Contributed by NM, 6-Jun-1994.) (Proof shortened by Wolf Lammen, 25-Jul-2012.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) → (𝜓𝜃))

Theoremorim1i 905 Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
(𝜑𝜓)       ((𝜑𝜒) → (𝜓𝜒))

Theoremorim2i 906 Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
(𝜑𝜓)       ((𝜒𝜑) → (𝜒𝜓))

Theoremorim12dALT 907 Alternate proof of orim12d 960 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 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) → (𝜒𝜏)))

Theoremorbi2i 908 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.)
(𝜑𝜓)       ((𝜒𝜑) ↔ (𝜒𝜓))

Theoremorbi1i 909 Inference adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.)
(𝜑𝜓)       ((𝜑𝜒) ↔ (𝜓𝜒))

Theoremorbi12i 910 Infer the disjunction of two equivalences. (Contributed by NM, 3-Jan-1993.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))

Theoremorbi2d 911 Deduction adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 21-Jun-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) ↔ (𝜃𝜒)))

Theoremorbi1d 912 Deduction adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 21-Jun-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))

Theoremorbi1 913 Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))

Theoremorbi12d 914 Deduction joining two equivalences to form equivalence of disjunctions. (Contributed by NM, 21-Jun-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))

Theorempm1.5 915 Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∨ (𝜓𝜒)) → (𝜓 ∨ (𝜑𝜒)))

Theoremor12 916 Swap two disjuncts. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
((𝜑 ∨ (𝜓𝜒)) ↔ (𝜓 ∨ (𝜑𝜒)))

Theoremorass 917 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.)
(((𝜑𝜓) ∨ 𝜒) ↔ (𝜑 ∨ (𝜓𝜒)))

Theorempm2.31 918 Theorem *2.31 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∨ (𝜓𝜒)) → ((𝜑𝜓) ∨ 𝜒))

Theorempm2.32 919 Theorem *2.32 of [WhiteheadRussell] p. 105. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∨ 𝜒) → (𝜑 ∨ (𝜓𝜒)))

Theorempm2.3 920 Theorem *2.3 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∨ (𝜓𝜒)) → (𝜑 ∨ (𝜒𝜓)))

Theoremor32 921 A rearrangement of disjuncts. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(((𝜑𝜓) ∨ 𝜒) ↔ ((𝜑𝜒) ∨ 𝜓))

Theoremor4 922 Rearrangement of 4 disjuncts. (Contributed by NM, 12-Aug-1994.)
(((𝜑𝜓) ∨ (𝜒𝜃)) ↔ ((𝜑𝜒) ∨ (𝜓𝜃)))

Theoremor42 923 Rearrangement of 4 disjuncts. (Contributed by NM, 10-Jan-2005.)
(((𝜑𝜓) ∨ (𝜒𝜃)) ↔ ((𝜑𝜒) ∨ (𝜃𝜓)))

Theoremorordi 924 Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ∨ (𝜑𝜒)))

Theoremorordir 925 Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
(((𝜑𝜓) ∨ 𝜒) ↔ ((𝜑𝜒) ∨ (𝜓𝜒)))

Theoremorimdi 926 Disjunction distributes over implication. (Contributed by Wolf Lammen, 5-Jan-2013.)
((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒)))

Theorempm2.76 927 Theorem *2.76 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∨ (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))

Theorempm2.85 928 Theorem *2.85 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Jan-2013.)
(((𝜑𝜓) → (𝜑𝜒)) → (𝜑 ∨ (𝜓𝜒)))

Theorempm2.75 929 Theorem *2.75 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 4-Jan-2013.)
((𝜑𝜓) → ((𝜑 ∨ (𝜓𝜒)) → (𝜑𝜒)))

Theorempm4.78 930 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.)
(((𝜑𝜓) ∨ (𝜑𝜒)) ↔ (𝜑 → (𝜓𝜒)))

Theorembiort 931 A wff disjoined with truth is true. (Contributed by NM, 23-May-1999.)
(𝜑 → (𝜑 ↔ (𝜑𝜓)))

Theorembiorf 932 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 ↔ (𝜑𝜓)))

Theorembiortn 933 A wff is equivalent to its negated disjunction with falsehood. (Contributed by NM, 9-Jul-2012.)
(𝜑 → (𝜓 ↔ (¬ 𝜑𝜓)))

Theorembiorfi 934 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 ↔ (𝜓𝜑))

Theorempm2.26 935 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 ∨ ((𝜑𝜓) → 𝜓))

Theorempm2.63 936 Theorem *2.63 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((¬ 𝜑𝜓) → 𝜓))

Theorempm2.64 937 Theorem *2.64 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑 ∨ ¬ 𝜓) → 𝜑))

Theorempm2.42 938 Theorem *2.42 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑 ∨ (𝜑𝜓)) → (𝜑𝜓))

Theorempm5.11g 939 A general instance of Theorem *5.11 of [WhiteheadRussell] p. 123. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) ∨ (¬ 𝜑𝜒))

Theorempm5.11 940 Theorem *5.11 of [WhiteheadRussell] p. 123. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) ∨ (¬ 𝜑𝜓))

Theorempm5.12 941 Theorem *5.12 of [WhiteheadRussell] p. 123. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) ∨ (𝜑 → ¬ 𝜓))

Theorempm5.14 942 Theorem *5.14 of [WhiteheadRussell] p. 123. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) ∨ (𝜓𝜒))

Theorempm5.13 943 Theorem *5.13 of [WhiteheadRussell] p. 123. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
((𝜑𝜓) ∨ (𝜓𝜑))

Theorempm5.55 944 Theorem *5.55 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 20-Jan-2013.)
(((𝜑𝜓) ↔ 𝜑) ∨ ((𝜑𝜓) ↔ 𝜓))

Theorempm4.72 945 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.)
((𝜑𝜓) ↔ (𝜓 ↔ (𝜑𝜓)))

Theoremimimorb 946 Simplify an implication between implications. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Wolf Lammen, 3-Apr-2013.)
(((𝜓𝜒) → (𝜑𝜒)) ↔ (𝜑 → (𝜓𝜒)))

Theoremoibabs 947 Absorption of disjunction into equivalence. (Contributed by NM, 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.)
(((𝜑𝜓) → (𝜑𝜓)) ↔ (𝜑𝜓))

Theoremorbidi 948 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 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))

Theorempm5.7 949 Disjunction distributes over the biconditional. Theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbidi 948. (Contributed by Roy F. Longton, 21-Jun-2005.)
(((𝜑𝜒) ↔ (𝜓𝜒)) ↔ (𝜒 ∨ (𝜑𝜓)))

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 960 versus orim12dALT 907). These theorems are mostly grouped at the beginning of this section.

The family of theorems starting with animorl 973 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 950 and related theorems.

Theoremjaao 950 Inference conjoining and disjoining the antecedents of two implications. (Contributed by NM, 30-Sep-1999.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜒))       ((𝜑𝜃) → ((𝜓𝜏) → 𝜒))

Theoremjaoa 951 Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜒))       ((𝜑𝜃) → ((𝜓𝜏) → 𝜒))

Theoremjaoian 952 Inference disjoining the antecedents of two implications. (Contributed by NM, 23-Oct-2005.)
((𝜑𝜓) → 𝜒)    &   ((𝜃𝜓) → 𝜒)       (((𝜑𝜃) ∧ 𝜓) → 𝜒)

Theoremjaodan 953 Deduction disjoining the antecedents of two implications. (Contributed by NM, 14-Oct-2005.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜃) → 𝜒)       ((𝜑 ∧ (𝜓𝜃)) → 𝜒)

Theoremmpjaodan 954 Eliminate a disjunction in a deduction. A translation of natural deduction rule E ( elimination), see natded 28096. (Contributed by Mario Carneiro, 29-May-2016.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜃) → 𝜒)    &   (𝜑 → (𝜓𝜃))       (𝜑𝜒)

Theorempm3.44 955 Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
(((𝜓𝜑) ∧ (𝜒𝜑)) → ((𝜓𝜒) → 𝜑))

Theoremjao 956 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.)
((𝜑𝜓) → ((𝜒𝜓) → ((𝜑𝜒) → 𝜓)))

Theoremjaob 957 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.)
(((𝜑𝜒) → 𝜓) ↔ ((𝜑𝜓) ∧ (𝜒𝜓)))

Theorempm4.77 958 Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
(((𝜓𝜑) ∧ (𝜒𝜑)) ↔ ((𝜓𝜒) → 𝜑))

Theorempm3.48 959 Theorem *3.48 of [WhiteheadRussell] p. 114. (Contributed by NM, 28-Jan-1997.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜒) → (𝜓𝜃)))

Theoremorim12d 960 Disjoin antecedents and consequents in a deduction. See orim12dALT 907 for a proof which does not depend on df-an 397. (Contributed by NM, 10-May-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) → (𝜒𝜏)))

Theoremorim1d 961 Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) → (𝜒𝜃)))

Theoremorim2d 962 Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) → (𝜃𝜒)))

Theoremorim2 963 Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97. (Contributed by NM, 3-Jan-2005.)
((𝜓𝜒) → ((𝜑𝜓) → (𝜑𝜒)))

Theorempm2.38 964 Theorem *2.38 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
((𝜓𝜒) → ((𝜓𝜑) → (𝜒𝜑)))

Theorempm2.36 965 Theorem *2.36 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
((𝜓𝜒) → ((𝜑𝜓) → (𝜒𝜑)))

Theorempm2.37 966 Theorem *2.37 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
((𝜓𝜒) → ((𝜓𝜑) → (𝜑𝜒)))

Theorempm2.81 967 Theorem *2.81 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
((𝜓 → (𝜒𝜃)) → ((𝜑𝜓) → ((𝜑𝜒) → (𝜑𝜃))))

Theorempm2.8 968 Theorem *2.8 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Jan-2013.)
((𝜑𝜓) → ((¬ 𝜓𝜒) → (𝜑𝜒)))

Theorempm2.73 969 Theorem *2.73 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → (((𝜑𝜓) ∨ 𝜒) → (𝜓𝜒)))

Theorempm2.74 970 Theorem *2.74 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.)
((𝜓𝜑) → (((𝜑𝜓) ∨ 𝜒) → (𝜑𝜒)))

Theorempm2.82 971 Theorem *2.82 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∨ 𝜒) → (((𝜑 ∨ ¬ 𝜒) ∨ 𝜃) → ((𝜑𝜓) ∨ 𝜃)))

Theorempm4.39 972 Theorem *4.39 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜒) ∧ (𝜓𝜃)) → ((𝜑𝜓) ↔ (𝜒𝜃)))

Theoremanimorl 973 Conjunction implies disjunction with one common formula (1/4). (Contributed by BJ, 4-Oct-2019.)
((𝜑𝜓) → (𝜑𝜒))

Theoremanimorr 974 Conjunction implies disjunction with one common formula (2/4). (Contributed by BJ, 4-Oct-2019.)
((𝜑𝜓) → (𝜒𝜓))

Theoremanimorlr 975 Conjunction implies disjunction with one common formula (3/4). (Contributed by BJ, 4-Oct-2019.)
((𝜑𝜓) → (𝜒𝜑))

Theoremanimorrl 976 Conjunction implies disjunction with one common formula (4/4). (Contributed by BJ, 4-Oct-2019.)
((𝜑𝜓) → (𝜓𝜒))

Theoremianor 977 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.)
(¬ (𝜑𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))

Theoremanor 978 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.)
((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))

Theoremioran 979 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.)
(¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))

Theorempm4.52 980 Theorem *4.52 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Nov-2012.)
((𝜑 ∧ ¬ 𝜓) ↔ ¬ (¬ 𝜑𝜓))

Theorempm4.53 981 Theorem *4.53 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑 ∧ ¬ 𝜓) ↔ (¬ 𝜑𝜓))

Theorempm4.54 982 Theorem *4.54 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Nov-2012.)
((¬ 𝜑𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓))

Theorempm4.55 983 Theorem *4.55 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
(¬ (¬ 𝜑𝜓) ↔ (𝜑 ∨ ¬ 𝜓))

Theorempm4.56 984 Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓))

Theoremoran 985 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.)
((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓))

Theorempm4.57 986 Theorem *4.57 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
(¬ (¬ 𝜑 ∧ ¬ 𝜓) ↔ (𝜑𝜓))

Theorempm3.1 987 Theorem *3.1 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ¬ (¬ 𝜑 ∨ ¬ 𝜓))

Theorempm3.11 988 Theorem *3.11 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.)
(¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑𝜓))

Theorempm3.12 989 Theorem *3.12 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓))

Theorempm3.13 990 Theorem *3.13 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))

Theorempm3.14 991 Theorem *3.14 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑 ∨ ¬ 𝜓) → ¬ (𝜑𝜓))

Theorempm4.44 992 Theorem *4.44 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
(𝜑 ↔ (𝜑 ∨ (𝜑𝜓)))

Theorempm4.45 993 Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
(𝜑 ↔ (𝜑 ∧ (𝜑𝜓)))

Theoremorabs 994 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 ↔ ((𝜑𝜓) ∧ 𝜑))

Theoremoranabs 995 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) ↔ (𝜑𝜓))

Theorempm5.61 996 Theorem *5.61 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 30-Jun-2013.)
(((𝜑𝜓) ∧ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓))

Theorempm5.6 997 Conjunction in antecedent versus disjunction in consequent. Theorem *5.6 of [WhiteheadRussell] p. 125. (Contributed by NM, 8-Jun-1994.)
(((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒)))

Theoremorcanai 998 Change disjunction in consequent to conjunction in antecedent. (Contributed by NM, 8-Jun-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑 ∧ ¬ 𝜓) → 𝜒)

Theorempm4.79 999 Theorem *4.79 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 27-Jun-2013.)
(((𝜓𝜑) ∨ (𝜒𝜑)) ↔ ((𝜓𝜒) → 𝜑))

Theorempm5.53 1000 Theorem *5.53 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((((𝜑𝜓) ∨ 𝜒) → 𝜃) ↔ (((𝜑𝜃) ∧ (𝜓𝜃)) ∧ (𝜒𝜃)))

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