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Theorem List for Metamath Proof Explorer - 701-800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmpanr12 701 An inference based on modus ponens. (Contributed by NM, 24-Jul-2009.)
𝜓    &   𝜒    &   ((𝜑 ∧ (𝜓𝜒)) → 𝜃)       (𝜑𝜃)
 
Theoremmpanlr1 702 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)
 
Theoremmpbirand 703 Detach truth from conjunction in biconditional. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝜒)    &   (𝜑 → (𝜓 ↔ (𝜒𝜃)))       (𝜑 → (𝜓𝜃))
 
Theoremmpbiran2d 704 Detach truth from conjunction in biconditional. Deduction form. (Contributed by Peter Mazsa, 24-Sep-2022.)
(𝜑𝜃)    &   (𝜑 → (𝜓 ↔ (𝜒𝜃)))       (𝜑 → (𝜓𝜒))
 
Theoremmpbiran 705 Detach truth from conjunction in biconditional. (Contributed by NM, 27-Feb-1996.)
𝜓    &   (𝜑 ↔ (𝜓𝜒))       (𝜑𝜒)
 
Theoremmpbiran2 706 Detach truth from conjunction in biconditional. (Contributed by NM, 22-Feb-1996.)
𝜒    &   (𝜑 ↔ (𝜓𝜒))       (𝜑𝜓)
 
Theoremmpbir2an 707 Detach a conjunction of truths in a biconditional. (Contributed by NM, 10-May-2005.)
𝜓    &   𝜒    &   (𝜑 ↔ (𝜓𝜒))       𝜑
 
Theoremmpbi2and 708 Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑 → ((𝜓𝜒) ↔ 𝜃))       (𝜑𝜃)
 
Theoremmpbir2and 709 Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
(𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑 → (𝜓 ↔ (𝜒𝜃)))       (𝜑𝜓)
 
Theoremadantll 710 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 711 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 712 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 713 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 714 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 715 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 716 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 717 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 718 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 719 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 720 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 721 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 722 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 723 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 724 Deduction adding two conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.)
(𝜑𝜓)       ((𝜒 ∧ (𝜑𝜃)) → 𝜓)
 
Theoremad2antll 725 Deduction adding conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.)
(𝜑𝜓)       ((𝜒 ∧ (𝜃𝜑)) → 𝜓)
 
Theoremad3antrrr 726 Deduction adding three conjuncts to antecedent. (Contributed by NM, 28-Jul-2012.)
(𝜑𝜓)       ((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜓)
 
Theoremad3antlr 727 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)
 
Theoremad4antr 728 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)
 
Theoremad4antlr 729 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)
 
Theoremad5antr 730 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)
 
Theoremad5antlr 731 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)
 
Theoremad6antr 732 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)
 
Theoremad6antlr 733 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)
 
Theoremad7antr 734 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)
 
Theoremad7antlr 735 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)
 
Theoremad8antr 736 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)
 
Theoremad8antlr 737 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)
 
Theoremad9antr 738 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)
 
Theoremad9antlr 739 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)
 
Theoremad10antr 740 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)
 
Theoremad10antlr 741 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)
 
Theoremad2ant2l 742 Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓) → 𝜒)       (((𝜃𝜑) ∧ (𝜏𝜓)) → 𝜒)
 
Theoremad2ant2r 743 Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓) → 𝜒)       (((𝜑𝜃) ∧ (𝜓𝜏)) → 𝜒)
 
Theoremad2ant2lr 744 Deduction adding two conjuncts to antecedent. (Contributed by NM, 23-Nov-2007.)
((𝜑𝜓) → 𝜒)       (((𝜃𝜑) ∧ (𝜓𝜏)) → 𝜒)
 
Theoremad2ant2rl 745 Deduction adding two conjuncts to antecedent. (Contributed by NM, 24-Nov-2007.)
((𝜑𝜓) → 𝜒)       (((𝜑𝜃) ∧ (𝜏𝜓)) → 𝜒)
 
Theoremadantl3r 746 Deduction adding 1 conjunct to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       (((((𝜑𝜂) ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
 
Theoremad4ant13 747 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓) → 𝜒)       ((((𝜑𝜃) ∧ 𝜓) ∧ 𝜏) → 𝜒)
 
Theoremad4ant14 748 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓) → 𝜒)       ((((𝜑𝜃) ∧ 𝜏) ∧ 𝜓) → 𝜒)
 
Theoremad4ant23 749 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓) → 𝜒)       ((((𝜃𝜑) ∧ 𝜓) ∧ 𝜏) → 𝜒)
 
Theoremad4ant24 750 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓) → 𝜒)       ((((𝜃𝜑) ∧ 𝜏) ∧ 𝜓) → 𝜒)
 
Theoremadantl4r 751 Deduction adding 1 conjunct to antecedent. (Contributed by Thierry Arnoux, 11-Feb-2018.)
(((((𝜑𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜅)       ((((((𝜑𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜅)
 
Theoremad5ant12 752 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓) → 𝜒)       (((((𝜑𝜓) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜒)
 
Theoremad5ant13 753 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓) → 𝜒)       (((((𝜑𝜃) ∧ 𝜓) ∧ 𝜏) ∧ 𝜂) → 𝜒)
 
Theoremad5ant14 754 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓) → 𝜒)       (((((𝜑𝜃) ∧ 𝜏) ∧ 𝜓) ∧ 𝜂) → 𝜒)
 
Theoremad5ant15 755 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓) → 𝜒)       (((((𝜑𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜓) → 𝜒)
 
Theoremad5ant23 756 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓) → 𝜒)       (((((𝜃𝜑) ∧ 𝜓) ∧ 𝜏) ∧ 𝜂) → 𝜒)
 
Theoremad5ant24 757 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓) → 𝜒)       (((((𝜃𝜑) ∧ 𝜏) ∧ 𝜓) ∧ 𝜂) → 𝜒)
 
Theoremad5ant25 758 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓) → 𝜒)       (((((𝜃𝜑) ∧ 𝜏) ∧ 𝜂) ∧ 𝜓) → 𝜒)
 
Theoremadantl5r 759 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 760 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)
 
Theorempm3.33 761 Theorem *3.33 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜓𝜒)) → (𝜑𝜒))
 
Theorempm3.34 762 Theorem *3.34 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
(((𝜓𝜒) ∧ (𝜑𝜓)) → (𝜑𝜒))
 
Theoremsimpll 763 Simplification of a conjunction. (Contributed by NM, 18-Mar-2007.)
(((𝜑𝜓) ∧ 𝜒) → 𝜑)
 
Theoremsimplld 764 Deduction form of simpll 763, eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → ((𝜓𝜒) ∧ 𝜃))       (𝜑𝜓)
 
Theoremsimplr 765 Simplification of a conjunction. (Contributed by NM, 20-Mar-2007.)
(((𝜑𝜓) ∧ 𝜒) → 𝜓)
 
Theoremsimplrd 766 Deduction eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → ((𝜓𝜒) ∧ 𝜃))       (𝜑𝜒)
 
Theoremsimprl 767 Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
((𝜑 ∧ (𝜓𝜒)) → 𝜓)
 
Theoremsimprld 768 Deduction eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝜓 ∧ (𝜒𝜃)))       (𝜑𝜒)
 
Theoremsimprr 769 Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
((𝜑 ∧ (𝜓𝜒)) → 𝜒)
 
Theoremsimprrd 770 Deduction form of simprr 769, eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝜓 ∧ (𝜒𝜃)))       (𝜑𝜃)
 
Theoremsimplll 771 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.) (Proof shortened by Wolf Lammen, 6-Apr-2022.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜑)
 
Theoremsimpllr 772 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.) (Proof shortened by Wolf Lammen, 6-Apr-2022.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜓)
 
Theoremsimplrl 773 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
(((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜓)
 
Theoremsimplrr 774 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
(((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜒)
 
Theoremsimprll 775 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)) → 𝜓)
 
Theoremsimprlr 776 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)) → 𝜒)
 
Theoremsimprrl 777 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜒)
 
Theoremsimprrr 778 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜃)
 
Theoremsimp-4l 779 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.)
(((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜑)
 
Theoremsimp-4r 780 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.)
(((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜓)
 
Theoremsimp-5l 781 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.)
((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜑)
 
Theoremsimp-5r 782 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.)
((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜓)
 
Theoremsimp-6l 783 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.)
(((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜑)
 
Theoremsimp-6r 784 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.)
(((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜓)
 
Theoremsimp-7l 785 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.)
((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) → 𝜑)
 
Theoremsimp-7r 786 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.)
((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) → 𝜓)
 
Theoremsimp-8l 787 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.)
(((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) → 𝜑)
 
Theoremsimp-8r 788 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.)
(((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) → 𝜓)
 
Theoremsimp-9l 789 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.)
((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) → 𝜑)
 
Theoremsimp-9r 790 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.)
((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) → 𝜓)
 
Theoremsimp-10l 791 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.)
(((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜑)
 
Theoremsimp-10r 792 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.)
(((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜓)
 
Theoremsimp-11l 793 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.)
((((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) → 𝜑)
 
Theoremsimp-11r 794 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.)
((((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) → 𝜓)
 
Theorempm2.01da 795 Deduction based on reductio ad absurdum. See pm2.01 190. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑𝜓) → ¬ 𝜓)       (𝜑 → ¬ 𝜓)
 
Theorempm2.18da 796 Deduction based on reductio ad absurdum. See pm2.18 128. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑 ∧ ¬ 𝜓) → 𝜓)       (𝜑𝜓)
 
Theoremimpbida 797 Deduce an equivalence from two implications. Variant of impbid 213. (Contributed by NM, 17-Feb-2007.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜒) → 𝜓)       (𝜑 → (𝜓𝜒))
 
Theorempm5.21nd 798 Eliminate an antecedent implied by each side of a biconditional. Variant of pm5.21ndd 381. (Contributed by NM, 20-Nov-2005.) (Proof shortened by Wolf Lammen, 4-Nov-2013.)
((𝜑𝜓) → 𝜃)    &   ((𝜑𝜒) → 𝜃)    &   (𝜃 → (𝜓𝜒))       (𝜑 → (𝜓𝜒))
 
Theorempm3.35 799 Conjunctive detachment. Theorem *3.35 of [WhiteheadRussell] p. 112. Variant of pm2.27 42. (Contributed by NM, 14-Dec-2002.)
((𝜑 ∧ (𝜑𝜓)) → 𝜓)
 
Theorempm5.74da 800 Distribution of implication over biconditional (deduction form). Variant of pm5.74d 274. (Contributed by NM, 4-May-2007.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))
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