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Theorem List for Metamath Proof Explorer - 701-800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmpan 701 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 702 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 703 An inference based on modus ponens. (Contributed by NM, 13-Apr-1995.)
𝜑    &   𝜓    &   ((𝜑𝜓) → 𝜒)       𝜒
 
Theoremmp4an 704 An inference based on modus ponens. (Contributed by Jeff Madsen, 15-Jun-2010.)
𝜑    &   𝜓    &   𝜒    &   𝜃    &   (((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)       𝜏
 
Theoremmpan2d 705 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
(𝜑𝜒)    &   (𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑 → (𝜓𝜃))
 
Theoremmpand 706 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 707 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 708 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 709 An inference based on modus ponens. (Contributed by NM, 12-Dec-2004.)
𝜓    &   𝜒    &   (𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑𝜃)
 
Theoremmp2and 710 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑𝜃)
 
Theoremmpanl1 711 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 712 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 713 An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)
𝜑    &   𝜓    &   (((𝜑𝜓) ∧ 𝜒) → 𝜃)       (𝜒𝜃)
 
Theoremmpanr1 714 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 715 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 716 An inference based on modus ponens. (Contributed by NM, 24-Jul-2009.)
𝜓    &   𝜒    &   ((𝜑 ∧ (𝜓𝜒)) → 𝜃)       (𝜑𝜃)
 
Theoremmpanlr1 717 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 718 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 4-May-2007.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))
 
Theorempm4.45 719 Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
(𝜑 ↔ (𝜑 ∧ (𝜑𝜓)))
 
Theoremimdistan 720 Distribution of implication with conjunction. (Contributed by NM, 31-May-1999.) (Proof shortened by Wolf Lammen, 6-Dec-2012.)
((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒)))
 
Theoremimdistani 721 Distribution of implication with conjunction. (Contributed by NM, 1-Aug-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) → (𝜑𝜒))
 
Theoremimdistanri 722 Distribution of implication with conjunction. (Contributed by NM, 8-Jan-2002.)
(𝜑 → (𝜓𝜒))       ((𝜓𝜑) → (𝜒𝜑))
 
Theoremimdistand 723 Distribution of implication with conjunction (deduction rule). (Contributed by NM, 27-Aug-2004.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → ((𝜓𝜒) → (𝜓𝜃)))
 
Theoremimdistanda 724 Distribution of implication with conjunction (deduction version with conjoined antecedent). (Contributed by Jeff Madsen, 19-Jun-2011.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → ((𝜓𝜒) → (𝜓𝜃)))
 
Theoremanbi2i 725 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 726 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 727 Variant of anbi2i 725 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
(𝜑𝜓)       ((𝜑𝜒) ↔ (𝜒𝜓))
 
Theoremanbi12i 728 Conjoin both sides of two equivalences. (Contributed by NM, 12-Mar-1993.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))
 
Theoremanbi12ci 729 Variant of anbi12i 728 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜃𝜓))
 
Theoremsyldanl 730 A syllogism deduction with conjoined antecedents. (Contributed by Jeff Madsen, 20-Jun-2011.)
((𝜑𝜓) → 𝜒)    &   (((𝜑𝜒) ∧ 𝜃) → 𝜏)       (((𝜑𝜓) ∧ 𝜃) → 𝜏)
 
Theoremsylan9bb 731 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))       ((𝜑𝜃) → (𝜓𝜏))
 
Theoremsylan9bbr 732 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))       ((𝜃𝜑) → (𝜓𝜏))
 
Theoremorbi2d 733 Deduction adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 21-Jun-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) ↔ (𝜃𝜒)))
 
Theoremorbi1d 734 Deduction adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 21-Jun-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))
 
Theoremanbi2d 735 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 736 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 737 Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))
 
Theoremanbi1 738 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 739 Introduce a left conjunct to both sides of a logical equivalence. (Contributed by NM, 16-Nov-2013.)
((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))
 
Theorembitr 740 Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜓𝜒)) → (𝜑𝜒))
 
Theoremorbi12d 741 Deduction joining two equivalences to form equivalence of disjunctions. (Contributed by NM, 21-Jun-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
 
Theoremanbi12d 742 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 26-May-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
 
Theorempm5.3 743 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)))
 
Theorempm5.61 744 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 745 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 746 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 747 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 748 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 749 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 750 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 751 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 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 ∧ (𝜓𝜏)) ∧ 𝜒) → 𝜃)
 
Theoremadantrll 753 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 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 ∧ ((𝜓𝜏) ∧ 𝜒)) → 𝜃)
 
Theoremadantrrl 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 ∧ (𝜓 ∧ (𝜏𝜒))) → 𝜃)
 
Theoremadantrrr 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 ∧ (𝜒𝜏))) → 𝜃)
 
Theoremad2antrr 757 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 758 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 759 Deduction adding two conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.)
(𝜑𝜓)       ((𝜒 ∧ (𝜑𝜃)) → 𝜓)
 
Theoremad2antll 760 Deduction adding conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.)
(𝜑𝜓)       ((𝜒 ∧ (𝜃𝜑)) → 𝜓)
 
Theoremad3antrrr 761 Deduction adding three conjuncts to antecedent. (Contributed by NM, 28-Jul-2012.)
(𝜑𝜓)       ((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜓)
 
Theoremad3antlr 762 Deduction adding three conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝜓)       ((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) → 𝜓)
 
Theoremad4antr 763 Deduction adding 4 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝜓)       (((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜓)
 
Theoremad4antlr 764 Deduction adding 4 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝜓)       (((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜓)
 
Theoremad5antr 765 Deduction adding 5 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝜓)       ((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜓)
 
Theoremad5antlr 766 Deduction adding 5 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝜓)       ((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜓)
 
Theoremad6antr 767 Deduction adding 6 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝜓)       (((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) → 𝜓)
 
Theoremad6antlr 768 Deduction adding 6 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝜓)       (((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) → 𝜓)
 
Theoremad7antr 769 Deduction adding 7 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝜓)       ((((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) → 𝜓)
 
Theoremad7antlr 770 Deduction adding 7 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝜓)       ((((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) → 𝜓)
 
Theoremad8antr 771 Deduction adding 8 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝜓)       (((((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) → 𝜓)
 
Theoremad8antlr 772 Deduction adding 8 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝜓)       (((((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) → 𝜓)
 
Theoremad9antr 773 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 774 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 775 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 776 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 777 Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓) → 𝜒)       (((𝜃𝜑) ∧ (𝜏𝜓)) → 𝜒)
 
Theoremad2ant2r 778 Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓) → 𝜒)       (((𝜑𝜃) ∧ (𝜓𝜏)) → 𝜒)
 
Theoremad2ant2lr 779 Deduction adding two conjuncts to antecedent. (Contributed by NM, 23-Nov-2007.)
((𝜑𝜓) → 𝜒)       (((𝜃𝜑) ∧ (𝜓𝜏)) → 𝜒)
 
Theoremad2ant2rl 780 Deduction adding two conjuncts to antecedent. (Contributed by NM, 24-Nov-2007.)
((𝜑𝜓) → 𝜒)       (((𝜑𝜃) ∧ (𝜏𝜓)) → 𝜒)
 
Theoremadantl3r 781 Deduction adding 1 conjunct to antecedent. (Contributed by Thierry Arnoux, 11-Feb-2018.)
((((𝜑𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜅)       (((((𝜑𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜅)
 
Theoremadantl4r 782 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 783 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 784 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 785 Simplification of a conjunction. (Contributed by NM, 18-Mar-2007.)
(((𝜑𝜓) ∧ 𝜒) → 𝜑)
 
Theoremsimplld 786 Deduction form of simpll 785, eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → ((𝜓𝜒) ∧ 𝜃))       (𝜑𝜓)
 
Theoremsimplr 787 Simplification of a conjunction. (Contributed by NM, 20-Mar-2007.)
(((𝜑𝜓) ∧ 𝜒) → 𝜓)
 
Theoremsimplrd 788 Deduction eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → ((𝜓𝜒) ∧ 𝜃))       (𝜑𝜒)
 
Theoremsimprl 789 Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
((𝜑 ∧ (𝜓𝜒)) → 𝜓)
 
Theoremsimprld 790 Deduction eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝜓 ∧ (𝜒𝜃)))       (𝜑𝜒)
 
Theoremsimprr 791 Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
((𝜑 ∧ (𝜓𝜒)) → 𝜒)
 
Theoremsimprrd 792 Deduction form of simprr 791, eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝜓 ∧ (𝜒𝜃)))       (𝜑𝜃)
 
Theoremsimplll 793 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜑)
 
Theoremsimpllr 794 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜓)
 
Theoremsimplrl 795 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
(((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜓)
 
Theoremsimplrr 796 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
(((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜒)
 
Theoremsimprll 797 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)) → 𝜓)
 
Theoremsimprlr 798 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)) → 𝜒)
 
Theoremsimprrl 799 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜒)
 
Theoremsimprrr 800 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜃)
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