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Theorem List for Intuitionistic Logic Explorer - 401-500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsylanl2 401 A syllogism inference. (Contributed by NM, 1-Jan-2005.)
(𝜑𝜒)    &   (((𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜓𝜑) ∧ 𝜃) → 𝜏)
 
Theoremsylanr1 402 A syllogism inference. (Contributed by NM, 9-Apr-2005.)
(𝜑𝜒)    &   ((𝜓 ∧ (𝜒𝜃)) → 𝜏)       ((𝜓 ∧ (𝜑𝜃)) → 𝜏)
 
Theoremsylanr2 403 A syllogism inference. (Contributed by NM, 9-Apr-2005.)
(𝜑𝜃)    &   ((𝜓 ∧ (𝜒𝜃)) → 𝜏)       ((𝜓 ∧ (𝜒𝜑)) → 𝜏)
 
Theoremsylani 404 A syllogism inference. (Contributed by NM, 2-May-1996.)
(𝜑𝜒)    &   (𝜓 → ((𝜒𝜃) → 𝜏))       (𝜓 → ((𝜑𝜃) → 𝜏))
 
Theoremsylan2i 405 A syllogism inference. (Contributed by NM, 1-Aug-1994.)
(𝜑𝜃)    &   (𝜓 → ((𝜒𝜃) → 𝜏))       (𝜓 → ((𝜒𝜑) → 𝜏))
 
Theoremsyl2ani 406 A syllogism inference. (Contributed by NM, 3-Aug-1999.)
(𝜑𝜒)    &   (𝜂𝜃)    &   (𝜓 → ((𝜒𝜃) → 𝜏))       (𝜓 → ((𝜑𝜂) → 𝜏))
 
Theoremsylan9 407 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))       ((𝜑𝜃) → (𝜓𝜏))
 
Theoremsylan9r 408 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))       ((𝜃𝜑) → (𝜓𝜏))
 
Theoremsyl2anc 409 Syllogism inference combined with contraction. (Contributed by NM, 16-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
Theoremsylancl 410 Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝜑𝜓)    &   𝜒    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
Theoremsylancr 411 Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝜓    &   (𝜑𝜒)    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
Theoremsylanblc 412 Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019.)
(𝜑𝜓)    &   𝜒    &   ((𝜓𝜒) ↔ 𝜃)       (𝜑𝜃)
 
Theoremsylanblrc 413 Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019.)
(𝜑𝜓)    &   𝜒    &   (𝜃 ↔ (𝜓𝜒))       (𝜑𝜃)
 
Theoremsylanbrc 414 Syllogism inference. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜃 ↔ (𝜓𝜒))       (𝜑𝜃)
 
Theoremsylancb 415 A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)
(𝜑𝜓)    &   (𝜑𝜒)    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
Theoremsylancbr 416 A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)
(𝜓𝜑)    &   (𝜒𝜑)    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
Theoremsylancom 417 Syllogism inference with commutation of antecents. (Contributed by NM, 2-Jul-2008.)
((𝜑𝜓) → 𝜒)    &   ((𝜒𝜓) → 𝜃)       ((𝜑𝜓) → 𝜃)
 
Theoremmpdan 418 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 419 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 420 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 421 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 422 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 423 An inference based on modus ponens. (Contributed by NM, 13-Apr-1995.)
𝜑    &   𝜓    &   ((𝜑𝜓) → 𝜒)       𝜒
 
Theoremmp4an 424 An inference based on modus ponens. (Contributed by Jeff Madsen, 15-Jun-2011.)
𝜑    &   𝜓    &   𝜒    &   𝜃    &   (((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)       𝜏
 
Theoremmpan2d 425 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
(𝜑𝜒)    &   (𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑 → (𝜓𝜃))
 
Theoremmpand 426 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 427 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 428 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 429 An inference based on modus ponens. (Contributed by NM, 12-Dec-2004.)
𝜓    &   𝜒    &   (𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑𝜃)
 
Theoremmp2and 430 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑𝜃)
 
Theoremmpanl1 431 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 432 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 433 An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)
𝜑    &   𝜓    &   (((𝜑𝜓) ∧ 𝜒) → 𝜃)       (𝜒𝜃)
 
Theoremmpanr1 434 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 435 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 436 An inference based on modus ponens. (Contributed by NM, 24-Jul-2009.)
𝜓    &   𝜒    &   ((𝜑 ∧ (𝜓𝜒)) → 𝜃)       (𝜑𝜃)
 
Theoremmpanlr1 437 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 438 Detach truth from conjunction in biconditional. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝜒)    &   (𝜑 → (𝜓 ↔ (𝜒𝜃)))       (𝜑 → (𝜓𝜃))
 
Theoremmpbiran2d 439 Detach truth from conjunction in biconditional. Deduction form. (Contributed by Peter Mazsa, 24-Sep-2022.)
(𝜑𝜃)    &   (𝜑 → (𝜓 ↔ (𝜒𝜃)))       (𝜑 → (𝜓𝜒))
 
Theorempm5.74da 440 Distribution of implication over biconditional (deduction form). (Contributed by NM, 4-May-2007.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))
 
Theoremimdistan 441 Distribution of implication with conjunction. (Contributed by NM, 31-May-1999.) (Proof shortened by Wolf Lammen, 6-Dec-2012.)
((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒)))
 
Theoremimdistani 442 Distribution of implication with conjunction. (Contributed by NM, 1-Aug-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) → (𝜑𝜒))
 
Theoremimdistanri 443 Distribution of implication with conjunction. (Contributed by NM, 8-Jan-2002.)
(𝜑 → (𝜓𝜒))       ((𝜓𝜑) → (𝜒𝜑))
 
Theoremimdistand 444 Distribution of implication with conjunction (deduction form). (Contributed by NM, 27-Aug-2004.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → ((𝜓𝜒) → (𝜓𝜃)))
 
Theoremimdistanda 445 Distribution of implication with conjunction (deduction version with conjoined antecedent). (Contributed by Jeff Madsen, 19-Jun-2011.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → ((𝜓𝜒) → (𝜓𝜃)))
 
Theorempm5.32d 446 Distribution of implication over biconditional (deduction form). (Contributed by NM, 29-Oct-1996.) (Revised by NM, 31-Jan-2015.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))
 
Theorempm5.32rd 447 Distribution of implication over biconditional (deduction form). (Contributed by NM, 25-Dec-2004.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → ((𝜒𝜓) ↔ (𝜃𝜓)))
 
Theorempm5.32da 448 Distribution of implication over biconditional (deduction form). (Contributed by NM, 9-Dec-2006.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))
 
Theorempm5.32 449 Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.) (Revised by NM, 31-Jan-2015.)
((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))
 
Theorempm5.32i 450 Distribution of implication over biconditional (inference form). (Contributed by NM, 1-Aug-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) ↔ (𝜑𝜒))
 
Theorempm5.32ri 451 Distribution of implication over biconditional (inference form). (Contributed by NM, 12-Mar-1995.)
(𝜑 → (𝜓𝜒))       ((𝜓𝜑) ↔ (𝜒𝜑))
 
Theorembiadan2 452 Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.)
(𝜑𝜓)    &   (𝜓 → (𝜑𝜒))       (𝜑 ↔ (𝜓𝜒))
 
Theoremanbi2i 453 Introduce a left conjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
(𝜑𝜓)       ((𝜒𝜑) ↔ (𝜒𝜓))
 
Theoremanbi1i 454 Introduce a right conjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
(𝜑𝜓)       ((𝜑𝜒) ↔ (𝜓𝜒))
 
Theoremanbi2ci 455 Variant of anbi2i 453 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
(𝜑𝜓)       ((𝜑𝜒) ↔ (𝜒𝜓))
 
Theoremanbi12i 456 Conjoin both sides of two equivalences. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))
 
Theoremanbi12ci 457 Variant of anbi12i 456 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜃𝜓))
 
Theoremsylan9bb 458 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))       ((𝜑𝜃) → (𝜓𝜏))
 
Theoremsylan9bbr 459 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))       ((𝜃𝜑) → (𝜓𝜏))
 
Theoremanbi2d 460 Deduction adding a left conjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) ↔ (𝜃𝜒)))
 
Theoremanbi1d 461 Deduction adding a right conjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))
 
Theoremanbi1 462 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 463 Introduce a left conjunct to both sides of a logical equivalence. (Contributed by NM, 16-Nov-2013.)
((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))
 
Theorembitr 464 Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜓𝜒)) → (𝜑𝜒))
 
Theoremanbi12d 465 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
 
Theoremmpan10 466 Modus ponens mixed with several conjunctions. (Contributed by Jim Kingdon, 7-Jan-2018.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜑) → (𝜓𝜒))
 
Theorempm5.3 467 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)))
 
Theoremadantll 468 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 469 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 470 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 471 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 472 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 473 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 474 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 475 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 476 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 477 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 478 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 479 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 480 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 481 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 482 Deduction adding two conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.)
(𝜑𝜓)       ((𝜒 ∧ (𝜑𝜃)) → 𝜓)
 
Theoremad2antll 483 Deduction adding conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.)
(𝜑𝜓)       ((𝜒 ∧ (𝜃𝜑)) → 𝜓)
 
Theoremad3antrrr 484 Deduction adding three conjuncts to antecedent. (Contributed by NM, 28-Jul-2012.)
(𝜑𝜓)       ((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜓)
 
Theoremad3antlr 485 Deduction adding three conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝜓)       ((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) → 𝜓)
 
Theoremad4antr 486 Deduction adding 4 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝜓)       (((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜓)
 
Theoremad4antlr 487 Deduction adding 4 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝜓)       (((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜓)
 
Theoremad5antr 488 Deduction adding 5 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝜓)       ((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜓)
 
Theoremad5antlr 489 Deduction adding 5 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝜓)       ((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜓)
 
Theoremad6antr 490 Deduction adding 6 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝜓)       (((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) → 𝜓)
 
Theoremad6antlr 491 Deduction adding 6 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝜓)       (((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) → 𝜓)
 
Theoremad7antr 492 Deduction adding 7 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝜓)       ((((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) → 𝜓)
 
Theoremad7antlr 493 Deduction adding 7 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝜓)       ((((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) → 𝜓)
 
Theoremad8antr 494 Deduction adding 8 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝜓)       (((((((((𝜑𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) → 𝜓)
 
Theoremad8antlr 495 Deduction adding 8 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝜓)       (((((((((𝜒𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) → 𝜓)
 
Theoremad9antr 496 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 497 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 498 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 499 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 500 Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓) → 𝜒)       (((𝜃𝜑) ∧ (𝜏𝜓)) → 𝜒)
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