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Theorem List for Metamath Proof Explorer - 701-800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsylancbr 701 A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)
(𝜓𝜑)    &   (𝜒𝜑)    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
Theoremsylancom 702 Syllogism inference with commutation of antecedents. (Contributed by NM, 2-Jul-2008.)
((𝜑𝜓) → 𝜒)    &   ((𝜒𝜓) → 𝜃)       ((𝜑𝜓) → 𝜃)
 
Theoremmpdan 703 An inference based on modus ponens. (Contributed by NM, 23-May-1999.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)
(𝜑𝜓)    &   ((𝜑𝜓) → 𝜒)       (𝜑𝜒)
 
Theoremmpancom 704 An inference based on modus ponens with commutation of antecedents. (Contributed by NM, 28-Oct-2003.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
(𝜓𝜑)    &   ((𝜑𝜓) → 𝜒)       (𝜓𝜒)
 
Theoremmpidan 705 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 706 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 707 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 708 An inference based on modus ponens. (Contributed by NM, 13-Apr-1995.)
𝜑    &   𝜓    &   ((𝜑𝜓) → 𝜒)       𝜒
 
Theoremmp4an 709 An inference based on modus ponens. (Contributed by Jeff Madsen, 15-Jun-2010.)
𝜑    &   𝜓    &   𝜒    &   𝜃    &   (((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)       𝜏
 
Theoremmpan2d 710 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
(𝜑𝜒)    &   (𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑 → (𝜓𝜃))
 
Theoremmpand 711 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 712 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 713 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 714 An inference based on modus ponens. (Contributed by NM, 12-Dec-2004.)
𝜓    &   𝜒    &   (𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑𝜃)
 
Theoremmp2and 715 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑𝜃)
 
Theoremmpanl1 716 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 717 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 718 An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)
𝜑    &   𝜓    &   (((𝜑𝜓) ∧ 𝜒) → 𝜃)       (𝜒𝜃)
 
Theoremmpanr1 719 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 720 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 721 An inference based on modus ponens. (Contributed by NM, 24-Jul-2009.)
𝜓    &   𝜒    &   ((𝜑 ∧ (𝜓𝜒)) → 𝜃)       (𝜑𝜃)
 
Theoremmpanlr1 722 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 723 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 4-May-2007.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))
 
Theorempm4.45 724 Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
(𝜑 ↔ (𝜑 ∧ (𝜑𝜓)))
 
Theoremimdistan 725 Distribution of implication with conjunction. (Contributed by NM, 31-May-1999.) (Proof shortened by Wolf Lammen, 6-Dec-2012.)
((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒)))
 
Theoremimdistani 726 Distribution of implication with conjunction. (Contributed by NM, 1-Aug-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) → (𝜑𝜒))
 
Theoremimdistanri 727 Distribution of implication with conjunction. (Contributed by NM, 8-Jan-2002.)
(𝜑 → (𝜓𝜒))       ((𝜓𝜑) → (𝜒𝜑))
 
Theoremimdistand 728 Distribution of implication with conjunction (deduction rule). (Contributed by NM, 27-Aug-2004.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → ((𝜓𝜒) → (𝜓𝜃)))
 
Theoremimdistanda 729 Distribution of implication with conjunction (deduction version with conjoined antecedent). (Contributed by Jeff Madsen, 19-Jun-2011.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → ((𝜓𝜒) → (𝜓𝜃)))
 
Theoremanbi2i 730 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 731 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 732 Variant of anbi2i 730 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
(𝜑𝜓)       ((𝜑𝜒) ↔ (𝜒𝜓))
 
Theoremanbi12i 733 Conjoin both sides of two equivalences. (Contributed by NM, 12-Mar-1993.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))
 
Theoremanbi12ci 734 Variant of anbi12i 733 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜃𝜓))
 
Theoremsyldanl 735 A syllogism deduction with conjoined antecedents. (Contributed by Jeff Madsen, 20-Jun-2011.)
((𝜑𝜓) → 𝜒)    &   (((𝜑𝜒) ∧ 𝜃) → 𝜏)       (((𝜑𝜓) ∧ 𝜃) → 𝜏)
 
Theoremsylan9bb 736 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))       ((𝜑𝜃) → (𝜓𝜏))
 
Theoremsylan9bbr 737 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))       ((𝜃𝜑) → (𝜓𝜏))
 
Theoremorbi2d 738 Deduction adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 21-Jun-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) ↔ (𝜃𝜒)))
 
Theoremorbi1d 739 Deduction adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 21-Jun-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))
 
Theoremanbi2d 740 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 741 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 742 Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))
 
Theoremanbi1 743 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 744 Introduce a left conjunct to both sides of a logical equivalence. (Contributed by NM, 16-Nov-2013.)
((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))
 
Theorembitr 745 Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜓𝜒)) → (𝜑𝜒))
 
Theoremorbi12d 746 Deduction joining two equivalences to form equivalence of disjunctions. (Contributed by NM, 21-Jun-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
 
Theoremanbi12d 747 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 26-May-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
 
Theorempm5.3 748 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)))
 
Theorempm5.61 749 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 750 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 751 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 752 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 753 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 754 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 755 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 756 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 757 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 758 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 759 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 760 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 761 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 762 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 763 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 764 Deduction adding two conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.)
(𝜑𝜓)       ((𝜒 ∧ (𝜑𝜃)) → 𝜓)
 
Theoremad2antll 765 Deduction adding conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.)
(𝜑𝜓)       ((𝜒 ∧ (𝜃𝜑)) → 𝜓)
 
Theoremad3antrrr 766 Deduction adding three conjuncts to antecedent. (Contributed by NM, 28-Jul-2012.)
(𝜑𝜓)       ((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜓)
 
Theoremad3antlr 767 Deduction adding three conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.)
(𝜑𝜓)       ((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) → 𝜓)
 
Theoremad3antlrOLD 768 Obsolete version of ad3antlr 767 as of 5-Apr-2022. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       ((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) → 𝜓)
 
Theoremad4antr 769 Deduction adding 4 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.)
(𝜑𝜓)       (((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜓)
 
Theoremad4antrOLD 770 Obsolete version of ad4antr 769 as of 5-Apr-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       (((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜓)
 
Theoremad4antlr 771 Deduction adding 4 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.)
(𝜑𝜓)       (((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜓)
 
Theoremad4antlrOLD 772 Obsolete version of ad4antlr 771 as of 5-Apr-2022. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       (((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜓)
 
Theoremad5antr 773 Deduction adding 5 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.)
(𝜑𝜓)       ((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜓)
 
Theoremad5antrOLD 774 Obsolete version of ad5antr 773 as of 5-Apr-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       ((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜓)
 
Theoremad5antlr 775 Deduction adding 5 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.)
(𝜑𝜓)       ((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜓)
 
Theoremad5antlrOLD 776 Obsolete version of ad5antlr 775 as of 5-Apr-2022. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       ((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜓)
 
Theoremad6antr 777 Deduction adding 6 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.)
(𝜑𝜓)       (((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) → 𝜓)
 
Theoremad6antrOLD 778 Obsolete version of ad6antr 777 as of 5-Apr-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       (((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) → 𝜓)
 
Theoremad6antlr 779 Deduction adding 6 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.)
(𝜑𝜓)       (((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) → 𝜓)
 
Theoremad6antlrOLD 780 Obsolete version of ad6antlr 779 as of 5-Apr-2022. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       (((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) → 𝜓)
 
Theoremad7antr 781 Deduction adding 7 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.)
(𝜑𝜓)       ((((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) → 𝜓)
 
Theoremad7antrOLD 782 Obsolete version of ad7antr 781 as of 5-Apr-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       ((((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) → 𝜓)
 
Theoremad7antlr 783 Deduction adding 7 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.)
(𝜑𝜓)       ((((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) → 𝜓)
 
Theoremad7antlrOLD 784 Obsolete version of ad7antlr 783 as of 5-Apr-2022. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       ((((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) → 𝜓)
 
Theoremad8antr 785 Deduction adding 8 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.)
(𝜑𝜓)       (((((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) → 𝜓)
 
Theoremad8antrOLD 786 Obsolete version of ad8antr 785 as of 5-Apr-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       (((((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) → 𝜓)
 
Theoremad8antlr 787 Deduction adding 8 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.)
(𝜑𝜓)       (((((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) → 𝜓)
 
Theoremad8antlrOLD 788 Obsolete version of ad8antlr 787 as of 5-Apr-2022. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       (((((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) → 𝜓)
 
Theoremad9antr 789 Deduction adding 9 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.)
(𝜑𝜓)       ((((((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜓)
 
Theoremad9antrOLD 790 Obsolete version of ad9antr 789 as of 5-Apr-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       ((((((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜓)
 
Theoremad9antlr 791 Deduction adding 9 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.)
(𝜑𝜓)       ((((((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜓)
 
Theoremad9antlrOLD 792 Obsolete version of ad9antlr 791 as of 5-Apr-2022. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       ((((((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜓)
 
Theoremad10antr 793 Deduction adding 10 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.)
(𝜑𝜓)       (((((((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) → 𝜓)
 
Theoremad10antrOLD 794 Obsolete version of ad10antr 793 as of 5-Apr-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       (((((((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) → 𝜓)
 
Theoremad10antlr 795 Deduction adding 10 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.)
(𝜑𝜓)       (((((((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) → 𝜓)
 
Theoremad10antlrOLD 796 Obsolete version of ad10antlr 795 as of 5-Apr-2022. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       (((((((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) → 𝜓)
 
Theoremad2ant2l 797 Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓) → 𝜒)       (((𝜃𝜑) ∧ (𝜏𝜓)) → 𝜒)
 
Theoremad2ant2r 798 Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓) → 𝜒)       (((𝜑𝜃) ∧ (𝜓𝜏)) → 𝜒)
 
Theoremad2ant2lr 799 Deduction adding two conjuncts to antecedent. (Contributed by NM, 23-Nov-2007.)
((𝜑𝜓) → 𝜒)       (((𝜃𝜑) ∧ (𝜓𝜏)) → 𝜒)
 
Theoremad2ant2rl 800 Deduction adding two conjuncts to antecedent. (Contributed by NM, 24-Nov-2007.)
((𝜑𝜓) → 𝜒)       (((𝜑𝜃) ∧ (𝜏𝜓)) → 𝜒)
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