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Theorem List for Metamath Proof Explorer - 801-900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsimprrr 801 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜃)

Theoremsimp-4l 802 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
(((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜑)

Theoremsimp-4r 803 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
(((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜓)

Theoremsimp-5l 804 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜑)

Theoremsimp-5r 805 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜓)

Theoremsimp-6l 806 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
(((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜑)

Theoremsimp-6r 807 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
(((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜓)

Theoremsimp-7l 808 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) → 𝜑)

Theoremsimp-7r 809 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) → 𝜓)

Theoremsimp-8l 810 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
(((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) → 𝜑)

Theoremsimp-8r 811 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
(((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) → 𝜓)

Theoremsimp-9l 812 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) → 𝜑)

Theoremsimp-9r 813 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) → 𝜓)

Theoremsimp-10l 814 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
(((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜑)

Theoremsimp-10r 815 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
(((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜓)

Theoremsimp-11l 816 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
((((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) → 𝜑)

Theoremsimp-11r 817 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
((((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) → 𝜓)

Theoremjaob 818 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)))

Theoremadant423OLD 819 Obsolete as of 2-Oct-2021. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝜑𝜓) → 𝜒)       ((((𝜑𝜃) ∧ 𝜏) ∧ 𝜓) → 𝜒)

Theoremjaoian 820 Inference disjoining the antecedents of two implications. (Contributed by NM, 23-Oct-2005.)
((𝜑𝜓) → 𝜒)    &   ((𝜃𝜓) → 𝜒)       (((𝜑𝜃) ∧ 𝜓) → 𝜒)

Theoremjao1i 821 Add a disjunct in the antecedent of an implication. (Contributed by Rodolfo Medina, 24-Sep-2010.)
(𝜓 → (𝜒𝜑))       ((𝜑𝜓) → (𝜒𝜑))

Theoremjaodan 822 Deduction disjoining the antecedents of two implications. (Contributed by NM, 14-Oct-2005.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜃) → 𝜒)       ((𝜑 ∧ (𝜓𝜃)) → 𝜒)

Theoremmpjaodan 823 Eliminate a disjunction in a deduction. A translation of natural deduction rule E ( elimination), see natded 26652. (Contributed by Mario Carneiro, 29-May-2016.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜃) → 𝜒)    &   (𝜑 → (𝜓𝜃))       (𝜑𝜒)

Theorempm4.77 824 Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
(((𝜓𝜑) ∧ (𝜒𝜑)) ↔ ((𝜓𝜒) → 𝜑))

Theorempm2.63 825 Theorem *2.63 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((¬ 𝜑𝜓) → 𝜓))

Theorempm2.64 826 Theorem *2.64 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑 ∨ ¬ 𝜓) → 𝜑))

Theorempm2.61ian 827 Elimination of an antecedent. (Contributed by NM, 1-Jan-2005.)
((𝜑𝜓) → 𝜒)    &   ((¬ 𝜑𝜓) → 𝜒)       (𝜓𝜒)

Theorempm2.61dan 828 Elimination of an antecedent. (Contributed by NM, 1-Jan-2005.)
((𝜑𝜓) → 𝜒)    &   ((𝜑 ∧ ¬ 𝜓) → 𝜒)       (𝜑𝜒)

Theorempm2.61ddan 829 Elimination of two antecedents. (Contributed by NM, 9-Jul-2013.)
((𝜑𝜓) → 𝜃)    &   ((𝜑𝜒) → 𝜃)    &   ((𝜑 ∧ (¬ 𝜓 ∧ ¬ 𝜒)) → 𝜃)       (𝜑𝜃)

Theorempm2.61dda 830 Elimination of two antecedents. (Contributed by NM, 9-Jul-2013.)
((𝜑 ∧ ¬ 𝜓) → 𝜃)    &   ((𝜑 ∧ ¬ 𝜒) → 𝜃)    &   ((𝜑 ∧ (𝜓𝜒)) → 𝜃)       (𝜑𝜃)

Theoremcondan 831 Proof by contradiction. (Contributed by NM, 9-Feb-2006.) (Proof shortened by Wolf Lammen, 19-Jun-2014.)
((𝜑 ∧ ¬ 𝜓) → 𝜒)    &   ((𝜑 ∧ ¬ 𝜓) → ¬ 𝜒)       (𝜑𝜓)

Theoremabai 832 Introduce one conjunct as an antecedent to the other. "abai" stands for "and, biconditional, and, implication". (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)
((𝜑𝜓) ↔ (𝜑 ∧ (𝜑𝜓)))

Theorempm5.53 833 Theorem *5.53 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((((𝜑𝜓) ∨ 𝜒) → 𝜃) ↔ (((𝜑𝜃) ∧ (𝜓𝜃)) ∧ (𝜒𝜃)))

Theoreman12 834 Swap two conjuncts. Note that the first digit (1) in the label refers to the outer conjunct position, and the next digit (2) to the inner conjunct position. (Contributed by NM, 12-Mar-1995.)
((𝜑 ∧ (𝜓𝜒)) ↔ (𝜓 ∧ (𝜑𝜒)))

Theoreman32 835 A rearrangement of conjuncts. (Contributed by NM, 12-Mar-1995.) (Proof shortened by Wolf Lammen, 25-Dec-2012.)
(((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∧ 𝜓))

Theoreman13 836 A rearrangement of conjuncts. (Contributed by NM, 24-Jun-2012.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
((𝜑 ∧ (𝜓𝜒)) ↔ (𝜒 ∧ (𝜓𝜑)))

Theoreman31 837 A rearrangement of conjuncts. (Contributed by NM, 24-Jun-2012.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
(((𝜑𝜓) ∧ 𝜒) ↔ ((𝜒𝜓) ∧ 𝜑))

Theorembianass 838 An inference to merge two lists of conjuncts. (Contributed by Giovanni Mascellani, 23-May-2019.)
(𝜑 ↔ (𝜓𝜒))       ((𝜂𝜑) ↔ ((𝜂𝜓) ∧ 𝜒))

Theoreman12s 839 Swap two conjuncts in antecedent. The label suffix "s" means that an12 834 is combined with syl 17 (or a variant). (Contributed by NM, 13-Mar-1996.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜓 ∧ (𝜑𝜒)) → 𝜃)

Theoremancom2s 840 Inference commuting a nested conjunction in antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑 ∧ (𝜒𝜓)) → 𝜃)

Theoreman13s 841 Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜒 ∧ (𝜓𝜑)) → 𝜃)

Theoreman32s 842 Swap two conjuncts in antecedent. (Contributed by NM, 13-Mar-1996.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜑𝜒) ∧ 𝜓) → 𝜃)

Theoremancom1s 843 Inference commuting a nested conjunction in antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜓𝜑) ∧ 𝜒) → 𝜃)

Theoreman31s 844 Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜒𝜓) ∧ 𝜑) → 𝜃)

Theoremanass1rs 845 Commutative-associative law for conjunction in an antecedent. (Contributed by Jeff Madsen, 19-Jun-2011.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       (((𝜑𝜒) ∧ 𝜓) → 𝜃)

Theoremanabs1 846 Absorption into embedded conjunct. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
(((𝜑𝜓) ∧ 𝜑) ↔ (𝜑𝜓))

Theoremanabs5 847 Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)
((𝜑 ∧ (𝜑𝜓)) ↔ (𝜑𝜓))

Theoremanabs7 848 Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 17-Nov-2013.)
((𝜓 ∧ (𝜑𝜓)) ↔ (𝜑𝜓))

Theorema2and 849 Deduction distributing a conjunction as embedded antecedent. (Contributed by AV, 25-Oct-2019.) (Proof shortened by Wolf Lammen, 19-Jan-2020.)
(𝜑 → ((𝜓𝜌) → (𝜏𝜃)))    &   (𝜑 → ((𝜓𝜌) → 𝜒))       (𝜑 → (((𝜓𝜒) → 𝜏) → ((𝜓𝜌) → 𝜃)))

Theoremanabsan 850 Absorption of antecedent with conjunction. (Contributed by NM, 24-Mar-1996.)
(((𝜑𝜑) ∧ 𝜓) → 𝜒)       ((𝜑𝜓) → 𝜒)

Theoremanabss1 851 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
(((𝜑𝜓) ∧ 𝜑) → 𝜒)       ((𝜑𝜓) → 𝜒)

Theoremanabss4 852 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.)
(((𝜓𝜑) ∧ 𝜓) → 𝜒)       ((𝜑𝜓) → 𝜒)

Theoremanabss5 853 Absorption of antecedent into conjunction. (Contributed by NM, 10-May-1994.) (Proof shortened by Wolf Lammen, 1-Jan-2013.)
((𝜑 ∧ (𝜑𝜓)) → 𝜒)       ((𝜑𝜓) → 𝜒)

Theoremanabsi5 854 Absorption of antecedent into conjunction. (Contributed by NM, 11-Jun-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2013.)
(𝜑 → ((𝜑𝜓) → 𝜒))       ((𝜑𝜓) → 𝜒)

Theoremanabsi6 855 Absorption of antecedent into conjunction. (Contributed by NM, 14-Aug-2000.)
(𝜑 → ((𝜓𝜑) → 𝜒))       ((𝜑𝜓) → 𝜒)

Theoremanabsi7 856 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 18-Nov-2013.)
(𝜓 → ((𝜑𝜓) → 𝜒))       ((𝜑𝜓) → 𝜒)

Theoremanabsi8 857 Absorption of antecedent into conjunction. (Contributed by NM, 26-Sep-1999.)
(𝜓 → ((𝜓𝜑) → 𝜒))       ((𝜑𝜓) → 𝜒)

Theoremanabss7 858 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 19-Nov-2013.)
((𝜓 ∧ (𝜑𝜓)) → 𝜒)       ((𝜑𝜓) → 𝜒)

Theoremanabsan2 859 Absorption of antecedent with conjunction. (Contributed by NM, 10-May-2004.)
((𝜑 ∧ (𝜓𝜓)) → 𝜒)       ((𝜑𝜓) → 𝜒)

Theoremanabss3 860 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 1-Jan-2013.)
(((𝜑𝜓) ∧ 𝜓) → 𝜒)       ((𝜑𝜓) → 𝜒)

Theoreman4 861 Rearrangement of 4 conjuncts. (Contributed by NM, 10-Jul-1994.)
(((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ((𝜑𝜒) ∧ (𝜓𝜃)))

Theoreman42 862 Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996.)
(((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ((𝜑𝜒) ∧ (𝜃𝜓)))

Theoreman43 863 Rearrangement of 4 conjuncts. (Contributed by Rodolfo Medina, 24-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ((𝜑𝜃) ∧ (𝜓𝜒)))

Theoreman3 864 A rearrangement of conjuncts. (Contributed by Rodolfo Medina, 25-Sep-2010.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → (𝜑𝜃))

Theoreman4s 865 Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)       (((𝜑𝜒) ∧ (𝜓𝜃)) → 𝜏)

Theoreman42s 866 Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)       (((𝜑𝜒) ∧ (𝜃𝜓)) → 𝜏)

Theoremanandi 867 Distribution of conjunction over conjunction. (Contributed by NM, 14-Aug-1995.)
((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))

Theoremanandir 868 Distribution of conjunction over conjunction. (Contributed by NM, 24-Aug-1995.)
(((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∧ (𝜓𝜒)))

Theoremanandis 869 Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
(((𝜑𝜓) ∧ (𝜑𝜒)) → 𝜏)       ((𝜑 ∧ (𝜓𝜒)) → 𝜏)

Theoremanandirs 870 Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
(((𝜑𝜒) ∧ (𝜓𝜒)) → 𝜏)       (((𝜑𝜓) ∧ 𝜒) → 𝜏)

Theoremsyl2an2 871 syl2an 493 with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.)
(𝜑𝜓)    &   ((𝜒𝜑) → 𝜃)    &   ((𝜓𝜃) → 𝜏)       ((𝜒𝜑) → 𝜏)

Theoremsyl2an2r 872 syl2anr 494 with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.)
(𝜑𝜓)    &   ((𝜑𝜒) → 𝜃)    &   ((𝜓𝜃) → 𝜏)       ((𝜑𝜒) → 𝜏)

Theoremimpbida 873 Deduce an equivalence from two implications. (Contributed by NM, 17-Feb-2007.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜒) → 𝜓)       (𝜑 → (𝜓𝜒))

Theorempm3.48 874 Theorem *3.48 of [WhiteheadRussell] p. 114. (Contributed by NM, 28-Jan-1997.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜒) → (𝜓𝜃)))

Theorempm3.45 875 Theorem *3.45 (Fact) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜒) → (𝜓𝜒)))

Theoremim2anan9 876 Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜂))       ((𝜑𝜃) → ((𝜓𝜏) → (𝜒𝜂)))

Theoremim2anan9r 877 Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜂))       ((𝜃𝜑) → ((𝜓𝜏) → (𝜒𝜂)))

Theoremanim12dan 878 Conjoin antecedents and consequents in a deduction. (Contributed by Mario Carneiro, 12-May-2014.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜃) → 𝜏)       ((𝜑 ∧ (𝜓𝜃)) → (𝜒𝜏))

Theoremorim12d 879 Disjoin antecedents and consequents in a deduction. (Contributed by NM, 10-May-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) → (𝜒𝜏)))

Theoremorim1d 880 Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) → (𝜒𝜃)))

Theoremorim2d 881 Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) → (𝜃𝜒)))

Theoremorim2 882 Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97. (Contributed by NM, 3-Jan-2005.)
((𝜓𝜒) → ((𝜑𝜓) → (𝜑𝜒)))

Theorempm2.38 883 Theorem *2.38 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
((𝜓𝜒) → ((𝜓𝜑) → (𝜒𝜑)))

Theorempm2.36 884 Theorem *2.36 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
((𝜓𝜒) → ((𝜑𝜓) → (𝜒𝜑)))

Theorempm2.37 885 Theorem *2.37 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
((𝜓𝜒) → ((𝜓𝜑) → (𝜑𝜒)))

Theorempm2.73 886 Theorem *2.73 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → (((𝜑𝜓) ∨ 𝜒) → (𝜓𝜒)))

Theorempm2.74 887 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)))

Theoremorimdi 888 Disjunction distributes over implication. (Contributed by Wolf Lammen, 5-Jan-2013.)
((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒)))

Theorempm2.76 889 Theorem *2.76 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∨ (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))

Theorempm2.75 890 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)))

Theorempm2.8 891 Theorem *2.8 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Jan-2013.)
((𝜑𝜓) → ((¬ 𝜓𝜒) → (𝜑𝜒)))

Theorempm2.81 892 Theorem *2.81 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
((𝜓 → (𝜒𝜃)) → ((𝜑𝜓) → ((𝜑𝜒) → (𝜑𝜃))))

Theorempm2.82 893 Theorem *2.82 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∨ 𝜒) → (((𝜑 ∨ ¬ 𝜒) ∨ 𝜃) → ((𝜑𝜓) ∨ 𝜃)))

Theorempm2.85 894 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)))

Theorempm3.2ni 895 Infer negated disjunction of negated premises. (Contributed by NM, 4-Apr-1995.)
¬ 𝜑    &    ¬ 𝜓        ¬ (𝜑𝜓)

Theoremorabs 896 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 897 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))

Theorempm5.1 898 Two propositions are equivalent if they are both true. Theorem *5.1 of [WhiteheadRussell] p. 123. (Contributed by NM, 21-May-1994.)
((𝜑𝜓) → (𝜑𝜓))

Theorempm5.21 899 Two propositions are equivalent if they are both false. Theorem *5.21 of [WhiteheadRussell] p. 124. (Contributed by NM, 21-May-1994.)
((¬ 𝜑 ∧ ¬ 𝜓) → (𝜑𝜓))

Theoremnorbi 900 If neither of two propositions is true, then these propositions are equivalent. (Contributed by BJ, 26-Apr-2019.)
(¬ (𝜑𝜓) → (𝜑𝜓))

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