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

Theoremmpbir2an 701 Detach a conjunction of truths in a biconditional. (Contributed by NM, 10-May-2005.)
𝜓    &   𝜒    &   (𝜑 ↔ (𝜓𝜒))       𝜑

Theoremmpbi2and 702 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 703 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 704 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 705 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 706 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 707 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 708 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 709 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 710 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 711 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 712 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 713 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 714 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 715 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 716 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 717 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 718 Deduction adding two conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.)
(𝜑𝜓)       ((𝜒 ∧ (𝜑𝜃)) → 𝜓)

(𝜑𝜓)       ((𝜒 ∧ (𝜃𝜑)) → 𝜓)

Theoremad3antrrr 720 Deduction adding three conjuncts to antecedent. (Contributed by NM, 28-Jul-2012.)
(𝜑𝜓)       ((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜓)

Theoremad3antlr 721 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 722 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 723 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 724 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 725 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 726 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 727 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 728 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 729 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 730 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 731 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 732 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 733 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 734 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 735 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 736 Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓) → 𝜒)       (((𝜃𝜑) ∧ (𝜏𝜓)) → 𝜒)

Theoremad2ant2r 737 Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓) → 𝜒)       (((𝜑𝜃) ∧ (𝜓𝜏)) → 𝜒)

Theoremad2ant2lr 738 Deduction adding two conjuncts to antecedent. (Contributed by NM, 23-Nov-2007.)
((𝜑𝜓) → 𝜒)       (((𝜃𝜑) ∧ (𝜓𝜏)) → 𝜒)

Theoremad2ant2rl 739 Deduction adding two conjuncts to antecedent. (Contributed by NM, 24-Nov-2007.)
((𝜑𝜓) → 𝜒)       (((𝜑𝜃) ∧ (𝜏𝜓)) → 𝜒)

Theoremadantl3r 740 Deduction adding 1 conjunct to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       (((((𝜑𝜂) ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theoremad4ant13 741 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 742 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 743 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 744 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 745 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 746 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓) → 𝜒)       (((((𝜑𝜓) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜒)

Theoremad5ant13 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) ∧ 𝜂) → 𝜒)

Theoremad5ant14 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) ∧ 𝜂) → 𝜒)

Theoremad5ant15 749 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 750 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 751 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 752 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 753 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 754 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)

Theorempm3.33 755 Theorem *3.33 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜓𝜒)) → (𝜑𝜒))

Theorempm3.34 756 Theorem *3.34 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
(((𝜓𝜒) ∧ (𝜑𝜓)) → (𝜑𝜒))

Theoremsimpll 757 Simplification of a conjunction. (Contributed by NM, 18-Mar-2007.)
(((𝜑𝜓) ∧ 𝜒) → 𝜑)

Theoremsimplld 758 Deduction form of simpll 757, eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → ((𝜓𝜒) ∧ 𝜃))       (𝜑𝜓)

Theoremsimplr 759 Simplification of a conjunction. (Contributed by NM, 20-Mar-2007.)
(((𝜑𝜓) ∧ 𝜒) → 𝜓)

Theoremsimplrd 760 Deduction eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → ((𝜓𝜒) ∧ 𝜃))       (𝜑𝜒)

Theoremsimprl 761 Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
((𝜑 ∧ (𝜓𝜒)) → 𝜓)

Theoremsimprld 762 Deduction eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝜓 ∧ (𝜒𝜃)))       (𝜑𝜒)

Theoremsimprr 763 Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
((𝜑 ∧ (𝜓𝜒)) → 𝜒)

Theoremsimprrd 764 Deduction form of simprr 763, eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝜓 ∧ (𝜒𝜃)))       (𝜑𝜃)

Theoremsimplll 765 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.) (Proof shortened by Wolf Lammen, 6-Apr-2022.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜑)

Theoremsimpllr 766 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.) (Proof shortened by Wolf Lammen, 6-Apr-2022.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜓)

Theoremsimplrl 767 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
(((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜓)

Theoremsimplrr 768 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
(((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜒)

Theoremsimprll 769 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)) → 𝜓)

Theoremsimprlr 770 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)) → 𝜒)

Theoremsimprrl 771 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜒)

Theoremsimprrr 772 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜃)

Theoremsimp-4l 773 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 774 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 775 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 776 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 777 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 778 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 779 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 780 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 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) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) → 𝜑)

Theoremsimp-8r 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) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) → 𝜓)

Theoremsimp-9l 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) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) → 𝜑)

Theoremsimp-9r 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) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) → 𝜓)

Theoremsimp-10l 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) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜑)

Theoremsimp-10r 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) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜓)

Theoremsimp-11l 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) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) → 𝜑)

Theoremsimp-11r 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) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) → 𝜓)

Theorempm2.01da 789 Deduction based on reductio ad absurdum. See pm2.01 181. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑𝜓) → ¬ 𝜓)       (𝜑 → ¬ 𝜓)

Theorempm2.18da 790 Deduction based on reductio ad absurdum. See pm2.18 125. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑 ∧ ¬ 𝜓) → 𝜓)       (𝜑𝜓)

Theoremimpbida 791 Deduce an equivalence from two implications. Variant of impbid 204. (Contributed by NM, 17-Feb-2007.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜒) → 𝜓)       (𝜑 → (𝜓𝜒))

Theorempm5.21nd 792 Eliminate an antecedent implied by each side of a biconditional. Variant of pm5.21ndd 371. (Contributed by NM, 20-Nov-2005.) (Proof shortened by Wolf Lammen, 4-Nov-2013.)
((𝜑𝜓) → 𝜃)    &   ((𝜑𝜒) → 𝜃)    &   (𝜃 → (𝜓𝜒))       (𝜑 → (𝜓𝜒))

Theorempm3.35 793 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 794 Distribution of implication over biconditional (deduction form). Variant of pm5.74d 265. (Contributed by NM, 4-May-2007.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))

Theorembitr 795 Theorem *4.22 of [WhiteheadRussell] p. 117. bitri 267 in closed form. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜓𝜒)) → (𝜑𝜒))

Theorembiantr 796 A transitive law of equivalence. Compare Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 18-Aug-1993.)
(((𝜑𝜓) ∧ (𝜒𝜓)) → (𝜑𝜒))

Theorempm4.14 797 Theorem *4.14 of [WhiteheadRussell] p. 117. Related to con34b 308. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Oct-2012.)
(((𝜑𝜓) → 𝜒) ↔ ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓))

Theorempm3.37 798 Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Oct-2012.)
(((𝜑𝜓) → 𝜒) → ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓))

Theoremprth 799 Conjoin antecedents and consequents of two premises. This is the closed theorem form of anim12d 602. Theorem *3.47 of [WhiteheadRussell] p. 113. It was proved by Leibniz, and it evidently pleased him enough to call it praeclarum theorema (splendid theorem). (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜒) → (𝜓𝜃)))

Theorempm3.4 800 Conjunction implies implication. Theorem *3.4 of [WhiteheadRussell] p. 113. (Contributed by NM, 31-Jul-1995.)
((𝜑𝜓) → (𝜑𝜓))

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