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Theorem List for Metamath Proof Explorer - 701-800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmpidan 701 A deduction which "stacks" a hypothesis. (Contributed by Stanislas Polu, 9-Mar-2020.) (Proof shortened by Wolf Lammen, 28-Mar-2021.)
(𝜑𝜒)    &   (((𝜑𝜓) ∧ 𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)

Theoremmpan 702 An inference based on modus ponens. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
𝜑    &   ((𝜑𝜓) → 𝜒)       (𝜓𝜒)

Theoremmpan2 703 An inference based on modus ponens. (Contributed by NM, 16-Sep-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)
𝜓    &   ((𝜑𝜓) → 𝜒)       (𝜑𝜒)

Theoremmp2an 704 An inference based on modus ponens. (Contributed by NM, 13-Apr-1995.)
𝜑    &   𝜓    &   ((𝜑𝜓) → 𝜒)       𝜒

Theoremmp4an 705 An inference based on modus ponens. (Contributed by Jeff Madsen, 15-Jun-2010.)
𝜑    &   𝜓    &   𝜒    &   𝜃    &   (((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)       𝜏

Theoremmpan2d 706 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
(𝜑𝜒)    &   (𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑 → (𝜓𝜃))

Theoremmpand 707 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
(𝜑𝜓)    &   (𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑 → (𝜒𝜃))

Theoremmpani 708 An inference based on modus ponens. (Contributed by NM, 10-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)
𝜓    &   (𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑 → (𝜒𝜃))

Theoremmpan2i 709 An inference based on modus ponens. (Contributed by NM, 10-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)
𝜒    &   (𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑 → (𝜓𝜃))

Theoremmp2ani 710 An inference based on modus ponens. (Contributed by NM, 12-Dec-2004.)
𝜓    &   𝜒    &   (𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑𝜃)

Theoremmp2and 711 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑𝜃)

Theoremmpanl1 712 An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
𝜑    &   (((𝜑𝜓) ∧ 𝜒) → 𝜃)       ((𝜓𝜒) → 𝜃)

Theoremmpanl2 713 An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.)
𝜓    &   (((𝜑𝜓) ∧ 𝜒) → 𝜃)       ((𝜑𝜒) → 𝜃)

Theoremmpanl12 714 An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)
𝜑    &   𝜓    &   (((𝜑𝜓) ∧ 𝜒) → 𝜃)       (𝜒𝜃)

Theoremmpanr1 715 An inference based on modus ponens. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.)
𝜓    &   ((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑𝜒) → 𝜃)

Theoremmpanr2 716 An inference based on modus ponens. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
𝜒    &   ((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑𝜓) → 𝜃)

Theoremmpanr12 717 An inference based on modus ponens. (Contributed by NM, 24-Jul-2009.)
𝜓    &   𝜒    &   ((𝜑 ∧ (𝜓𝜒)) → 𝜃)       (𝜑𝜃)

Theoremmpanlr1 718 An inference based on modus ponens. (Contributed by NM, 30-Dec-2004.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
𝜓    &   (((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜏)       (((𝜑𝜒) ∧ 𝜃) → 𝜏)

Theorempm5.74da 719 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 4-May-2007.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))

Theorempm4.45 720 Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
(𝜑 ↔ (𝜑 ∧ (𝜑𝜓)))

Theoremimdistan 721 Distribution of implication with conjunction. (Contributed by NM, 31-May-1999.) (Proof shortened by Wolf Lammen, 6-Dec-2012.)
((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒)))

Theoremimdistani 722 Distribution of implication with conjunction. (Contributed by NM, 1-Aug-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) → (𝜑𝜒))

Theoremimdistanri 723 Distribution of implication with conjunction. (Contributed by NM, 8-Jan-2002.)
(𝜑 → (𝜓𝜒))       ((𝜓𝜑) → (𝜒𝜑))

Theoremimdistand 724 Distribution of implication with conjunction (deduction rule). (Contributed by NM, 27-Aug-2004.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → ((𝜓𝜒) → (𝜓𝜃)))

Theoremimdistanda 725 Distribution of implication with conjunction (deduction version with conjoined antecedent). (Contributed by Jeff Madsen, 19-Jun-2011.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → ((𝜓𝜒) → (𝜓𝜃)))

Theoremanbi2i 726 Introduce a left conjunct to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
(𝜑𝜓)       ((𝜒𝜑) ↔ (𝜒𝜓))

Theoremanbi1i 727 Introduce a right conjunct to both sides of a logical equivalence. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
(𝜑𝜓)       ((𝜑𝜒) ↔ (𝜓𝜒))

Theoremanbi2ci 728 Variant of anbi2i 726 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
(𝜑𝜓)       ((𝜑𝜒) ↔ (𝜒𝜓))

Theoremanbi12i 729 Conjoin both sides of two equivalences. (Contributed by NM, 12-Mar-1993.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))

Theoremanbi12ci 730 Variant of anbi12i 729 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜃𝜓))

Theoremsyldanl 731 A syllogism deduction with conjoined antecedents. (Contributed by Jeff Madsen, 20-Jun-2011.)
((𝜑𝜓) → 𝜒)    &   (((𝜑𝜒) ∧ 𝜃) → 𝜏)       (((𝜑𝜓) ∧ 𝜃) → 𝜏)

Theoremsylan9bb 732 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))       ((𝜑𝜃) → (𝜓𝜏))

Theoremsylan9bbr 733 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))       ((𝜃𝜑) → (𝜓𝜏))

Theoremorbi2d 734 Deduction adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 21-Jun-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) ↔ (𝜃𝜒)))

Theoremorbi1d 735 Deduction adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 21-Jun-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))

Theoremanbi2d 736 Deduction adding a left conjunct to both sides of a logical equivalence. (Contributed by NM, 11-May-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) ↔ (𝜃𝜒)))

Theoremanbi1d 737 Deduction adding a right conjunct to both sides of a logical equivalence. (Contributed by NM, 11-May-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))

Theoremorbi1 738 Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))

Theoremanbi1 739 Introduce a right conjunct to both sides of a logical equivalence. Theorem *4.36 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))

Theoremanbi2 740 Introduce a left conjunct to both sides of a logical equivalence. (Contributed by NM, 16-Nov-2013.)
((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))

Theorembitr 741 Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜓𝜒)) → (𝜑𝜒))

Theoremorbi12d 742 Deduction joining two equivalences to form equivalence of disjunctions. (Contributed by NM, 21-Jun-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))

Theoremanbi12d 743 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 26-May-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))

Theorempm5.3 744 Theorem *5.3 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(((𝜑𝜓) → 𝜒) ↔ ((𝜑𝜓) → (𝜑𝜒)))

Theorempm5.61 745 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))

Theoremadantll 746 Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
((𝜑𝜓) → 𝜒)       (((𝜃𝜑) ∧ 𝜓) → 𝜒)

Theoremadantlr 747 Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
((𝜑𝜓) → 𝜒)       (((𝜑𝜃) ∧ 𝜓) → 𝜒)

Theoremadantrl 748 Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
((𝜑𝜓) → 𝜒)       ((𝜑 ∧ (𝜃𝜓)) → 𝜒)

Theoremadantrr 749 Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
((𝜑𝜓) → 𝜒)       ((𝜑 ∧ (𝜓𝜃)) → 𝜒)

Theoremadantlll 750 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 2-Dec-2012.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       ((((𝜏𝜑) ∧ 𝜓) ∧ 𝜒) → 𝜃)

Theoremadantllr 751 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       ((((𝜑𝜏) ∧ 𝜓) ∧ 𝜒) → 𝜃)

Theoremadantlrl 752 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜑 ∧ (𝜏𝜓)) ∧ 𝜒) → 𝜃)

Theoremadantlrr 753 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜑 ∧ (𝜓𝜏)) ∧ 𝜒) → 𝜃)

Theoremadantrll 754 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑 ∧ ((𝜏𝜓) ∧ 𝜒)) → 𝜃)

Theoremadantrlr 755 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑 ∧ ((𝜓𝜏) ∧ 𝜒)) → 𝜃)

Theoremadantrrl 756 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑 ∧ (𝜓 ∧ (𝜏𝜒))) → 𝜃)

Theoremadantrrr 757 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑 ∧ (𝜓 ∧ (𝜒𝜏))) → 𝜃)

Theoremad2antrr 758 Deduction adding two conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.) (Proof shortened by Wolf Lammen, 20-Nov-2012.)
(𝜑𝜓)       (((𝜑𝜒) ∧ 𝜃) → 𝜓)

Theoremad2antlr 759 Deduction adding two conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.) (Proof shortened by Wolf Lammen, 20-Nov-2012.)
(𝜑𝜓)       (((𝜒𝜑) ∧ 𝜃) → 𝜓)

Theoremad2antrl 760 Deduction adding two conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.)
(𝜑𝜓)       ((𝜒 ∧ (𝜑𝜃)) → 𝜓)

(𝜑𝜓)       ((𝜒 ∧ (𝜃𝜑)) → 𝜓)

Theoremad3antrrr 762 Deduction adding three conjuncts to antecedent. (Contributed by NM, 28-Jul-2012.)
(𝜑𝜓)       ((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜓)

Theoremad3antlr 763 Deduction adding three conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝜓)       ((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) → 𝜓)

Theoremad4antr 764 Deduction adding 4 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝜓)       (((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜓)

Theoremad4antlr 765 Deduction adding 4 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝜓)       (((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜓)

Theoremad5antr 766 Deduction adding 5 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝜓)       ((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜓)

Theoremad5antlr 767 Deduction adding 5 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝜓)       ((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜓)

Theoremad6antr 768 Deduction adding 6 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝜓)       (((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) → 𝜓)

Theoremad6antlr 769 Deduction adding 6 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝜓)       (((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) → 𝜓)

Theoremad7antr 770 Deduction adding 7 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝜓)       ((((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) → 𝜓)

Theoremad7antlr 771 Deduction adding 7 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝜓)       ((((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) → 𝜓)

Theoremad8antr 772 Deduction adding 8 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝜓)       (((((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) → 𝜓)

Theoremad8antlr 773 Deduction adding 8 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝜓)       (((((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) → 𝜓)

Theoremad9antr 774 Deduction adding 9 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝜓)       ((((((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜓)

Theoremad9antlr 775 Deduction adding 9 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝜓)       ((((((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜓)

Theoremad10antr 776 Deduction adding 10 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝜓)       (((((((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) → 𝜓)

Theoremad10antlr 777 Deduction adding 10 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝜓)       (((((((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) → 𝜓)

Theoremad2ant2l 778 Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓) → 𝜒)       (((𝜃𝜑) ∧ (𝜏𝜓)) → 𝜒)

Theoremad2ant2r 779 Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓) → 𝜒)       (((𝜑𝜃) ∧ (𝜓𝜏)) → 𝜒)

Theoremad2ant2lr 780 Deduction adding two conjuncts to antecedent. (Contributed by NM, 23-Nov-2007.)
((𝜑𝜓) → 𝜒)       (((𝜃𝜑) ∧ (𝜓𝜏)) → 𝜒)

Theoremad2ant2rl 781 Deduction adding two conjuncts to antecedent. (Contributed by NM, 24-Nov-2007.)
((𝜑𝜓) → 𝜒)       (((𝜑𝜃) ∧ (𝜏𝜓)) → 𝜒)

Theoremadantl3r 782 Deduction adding 1 conjunct to antecedent. (Contributed by Thierry Arnoux, 11-Feb-2018.)
((((𝜑𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜅)       (((((𝜑𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜅)

Theoremadantl4r 783 Deduction adding 1 conjunct to antecedent. (Contributed by Thierry Arnoux, 11-Feb-2018.)
(((((𝜑𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜅)       ((((((𝜑𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜅)

Theoremadantl5r 784 Deduction adding 1 conjunct to antecedent. (Contributed by Thierry Arnoux, 11-Feb-2018.)
((((((𝜑𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜅)       (((((((𝜑𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜅)

Theoremadantl6r 785 Deduction adding 1 conjunct to antecedent. (Contributed by Thierry Arnoux, 11-Feb-2018.)
(((((((𝜑𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜅)       ((((((((𝜑𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜅)

Theoremsimpll 786 Simplification of a conjunction. (Contributed by NM, 18-Mar-2007.)
(((𝜑𝜓) ∧ 𝜒) → 𝜑)

Theoremsimplld 787 Deduction form of simpll 786, eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → ((𝜓𝜒) ∧ 𝜃))       (𝜑𝜓)

Theoremsimplr 788 Simplification of a conjunction. (Contributed by NM, 20-Mar-2007.)
(((𝜑𝜓) ∧ 𝜒) → 𝜓)

Theoremsimplrd 789 Deduction eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → ((𝜓𝜒) ∧ 𝜃))       (𝜑𝜒)

Theoremsimprl 790 Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
((𝜑 ∧ (𝜓𝜒)) → 𝜓)

Theoremsimprld 791 Deduction eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝜓 ∧ (𝜒𝜃)))       (𝜑𝜒)

Theoremsimprr 792 Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
((𝜑 ∧ (𝜓𝜒)) → 𝜒)

Theoremsimprrd 793 Deduction form of simprr 792, eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝜓 ∧ (𝜒𝜃)))       (𝜑𝜃)

Theoremsimplll 794 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜑)

Theoremsimpllr 795 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜓)

Theoremsimplrl 796 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
(((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜓)

Theoremsimplrr 797 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
(((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜒)

Theoremsimprll 798 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)) → 𝜓)

Theoremsimprlr 799 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)) → 𝜒)

Theoremsimprrl 800 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜒)

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