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Theorem List for Metamath Proof Explorer - 401-500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremexpdcom 401 Commuted form of expd 402. (Contributed by Alan Sare, 18-Mar-2012.) Shorten expd 402. (Revised, 28-Jul-2022.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (𝜓 → (𝜒 → (𝜑𝜃)))
 
Theoremexpd 402 Exportation deduction. (Contributed by NM, 20-Aug-1993.) (Proof shortened by Wolf Lammen, 28-Jul-2022.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑 → (𝜓 → (𝜒𝜃)))
 
TheoremexpdOLD 403 Obsolete version of expd 402 as of 28-Jul-2022. (Contributed by NM, 20-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑 → (𝜓 → (𝜒𝜃)))
 
Theoremexpcomd 404 Deduction form of expcom 400. (Contributed by Alan Sare, 22-Jul-2012.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑 → (𝜒 → (𝜓𝜃)))
 
TheoremexpdcomOLD 405 Obsolete version of expdcom 401 as of 28-Jul-2022. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (𝜓 → (𝜒 → (𝜑𝜃)))
 
Theoremimp31 406 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (((𝜑𝜓) ∧ 𝜒) → 𝜃)
 
Theoremimp32 407 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒𝜃)))       ((𝜑 ∧ (𝜓𝜒)) → 𝜃)
 
Theoremexp31 408 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))
 
Theoremexp32 409 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))
 
Theoremimp4b 410 An importation inference. (Contributed by NM, 26-Apr-1994.) Shorten imp4a 411. (Revised by Wolf Lammen, 19-Jul-2021.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((𝜑𝜓) → ((𝜒𝜃) → 𝜏))
 
Theoremimp4a 411 An importation inference. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Jul-2021.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → (𝜓 → ((𝜒𝜃) → 𝜏)))
 
Theoremimp4c 412 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → (((𝜓𝜒) ∧ 𝜃) → 𝜏))
 
Theoremimp4d 413 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → ((𝜓 ∧ (𝜒𝜃)) → 𝜏))
 
Theoremimp41 414 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
 
Theoremimp42 415 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜏)
 
Theoremimp43 416 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)
 
Theoremimp44 417 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)) → 𝜏)
 
Theoremimp45 418 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜏)
 
Theoremexp4b 419 An exportation inference. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 23-Nov-2012.) Shorten exp4a 420. (Revised by Wolf Lammen, 20-Jul-2021.)
((𝜑𝜓) → ((𝜒𝜃) → 𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp4a 420 An exportation inference. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 20-Jul-2021.)
(𝜑 → (𝜓 → ((𝜒𝜃) → 𝜏)))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp4c 421 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (((𝜓𝜒) ∧ 𝜃) → 𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp4d 422 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → ((𝜓 ∧ (𝜒𝜃)) → 𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp41 423 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp42 424 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp43 425 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp44 426 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp45 427 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremimp5d 428 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (((𝜑𝜓) ∧ 𝜒) → ((𝜃𝜏) → 𝜂))
 
Theoremimp5a 429 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) (Proof shortened by Wolf Lammen, 2-Aug-2022.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (𝜑 → (𝜓 → (𝜒 → ((𝜃𝜏) → 𝜂))))
 
Theoremimp5g 430 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       ((𝜑𝜓) → (((𝜒𝜃) ∧ 𝜏) → 𝜂))
 
Theoremimp55 431 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) ∧ 𝜏) → 𝜂)
 
Theoremimp511 432 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       ((𝜑 ∧ ((𝜓 ∧ (𝜒𝜃)) ∧ 𝜏)) → 𝜂)
 
Theoremexp5c 433 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → ((𝜓𝜒) → ((𝜃𝜏) → 𝜂)))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp5j 434 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → ((((𝜓𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp5l 435 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (((𝜓𝜒) ∧ (𝜃𝜏)) → 𝜂))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp53 436 An exportation inference. (Contributed by Jeff Hankins, 30-Aug-2009.)
((((𝜑𝜓) ∧ (𝜒𝜃)) ∧ 𝜏) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theorempm3.3 437 Theorem *3.3 (Exp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
(((𝜑𝜓) → 𝜒) → (𝜑 → (𝜓𝜒)))
 
Theorempm3.31 438 Theorem *3.31 (Imp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → 𝜒))
 
Theoremimpexp 439 Import-export theorem. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
(((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒)))
 
Theoremimp5aOLD 440 Obsolete version of imp5a 429 as of 2-Aug-2022. (Contributed by Jeff Hankins, 7-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (𝜑 → (𝜓 → (𝜒 → ((𝜃𝜏) → 𝜂))))
 
Theoremimpancom 441 Mixed importation/commutation inference. (Contributed by NM, 22-Jun-2013.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑𝜒) → (𝜓𝜃))
 
Theoremexpdimp 442 A deduction version of exportation, followed by importation. (Contributed by NM, 6-Sep-2008.)
(𝜑 → ((𝜓𝜒) → 𝜃))       ((𝜑𝜓) → (𝜒𝜃))
 
Theoremexpimpd 443 Exportation followed by a deduction version of importation. (Contributed by NM, 6-Sep-2008.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → ((𝜓𝜒) → 𝜃))
 
Theoremimpr 444 Import a wff into a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑 ∧ (𝜓𝜒)) → 𝜃)
 
Theoremimpl 445 Export a wff from a left conjunct. (Contributed by Mario Carneiro, 9-Jul-2014.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (((𝜑𝜓) ∧ 𝜒) → 𝜃)
 
Theoremexpr 446 Export a wff from a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑𝜓) → (𝜒𝜃))
 
Theoremexpl 447 Export a wff from a left conjunct. (Contributed by Jeff Hankins, 28-Aug-2009.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (𝜑 → ((𝜓𝜒) → 𝜃))
 
Theoremancoms 448 Inference commuting conjunction in antecedent. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓) → 𝜒)       ((𝜓𝜑) → 𝜒)
 
Theorempm3.22 449 Theorem *3.22 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Nov-2012.)
((𝜑𝜓) → (𝜓𝜑))
 
Theoremancom 450 Commutative law for conjunction. Theorem *4.3 of [WhiteheadRussell] p. 118. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Wolf Lammen, 4-Nov-2012.)
((𝜑𝜓) ↔ (𝜓𝜑))
 
Theoremancomd 451 Commutation of conjuncts in consequent. (Contributed by Jeff Hankins, 14-Aug-2009.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜒𝜓))
 
Theoremancomst 452 Closed form of ancoms 448. (Contributed by Alan Sare, 31-Dec-2011.)
(((𝜑𝜓) → 𝜒) ↔ ((𝜓𝜑) → 𝜒))
 
Theoremancomsd 453 Deduction commuting conjunction in antecedent. (Contributed by NM, 12-Dec-2004.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑 → ((𝜒𝜓) → 𝜃))
 
Theoremanasss 454 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜒)) → 𝜃)
 
Theoremanassrs 455 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       (((𝜑𝜓) ∧ 𝜒) → 𝜃)
 
Theoremanass 456 Associative law for conjunction. Theorem *4.32 of [WhiteheadRussell] p. 118. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
(((𝜑𝜓) ∧ 𝜒) ↔ (𝜑 ∧ (𝜓𝜒)))
 
Theorempm3.2 457 Join antecedents with conjunction ("conjunction introduction"). Theorem *3.2 of [WhiteheadRussell] p. 111. Its associated inference is pm3.2i 458 and its associated deduction is jca 503 (and the double deduction is jcad 504). See pm3.2im 158 for a version using only implication and negation. (Contributed by NM, 5-Jan-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.)
(𝜑 → (𝜓 → (𝜑𝜓)))
 
Theorempm3.2i 458 Infer conjunction of premises. Inference associated with pm3.2 457. Its associated deduction is jca 503 (and the double deduction is jcad 504). (Contributed by NM, 21-Jun-1993.)
𝜑    &   𝜓       (𝜑𝜓)
 
Theorempm3.21 459 Join antecedents with conjunction. Theorem *3.21 of [WhiteheadRussell] p. 111. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓 → (𝜓𝜑)))
 
Theorempm3.43i 460 Nested conjunction of antecedents. (Contributed by NM, 4-Jan-1993.)
((𝜑𝜓) → ((𝜑𝜒) → (𝜑 → (𝜓𝜒))))
 
Theorempm3.43 461 Theorem *3.43 (Comp) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜑𝜒)) → (𝜑 → (𝜓𝜒)))
 
Theoremdfbi2 462 A theorem similar to the standard definition of the biconditional. Definition of [Margaris] p. 49. (Contributed by NM, 24-Jan-1993.)
((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
 
Theoremdfbi 463 Definition df-bi 198 rewritten in an abbreviated form to help intuitive understanding of that definition. Note that it is a conjunction of two implications; one which asserts properties that follow from the biconditional and one which asserts properties that imply the biconditional. (Contributed by NM, 15-Aug-2008.)
(((𝜑𝜓) → ((𝜑𝜓) ∧ (𝜓𝜑))) ∧ (((𝜑𝜓) ∧ (𝜓𝜑)) → (𝜑𝜓)))
 
Theorembiimpa 464 Importation inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) → 𝜒)
 
Theorembiimpar 465 Importation inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜒) → 𝜓)
 
Theorembiimpac 466 Importation inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))       ((𝜓𝜑) → 𝜒)
 
Theorembiimparc 467 Importation inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))       ((𝜒𝜑) → 𝜓)
 
Theoremadantr 468 Inference adding a conjunct to the right of an antecedent. (Contributed by NM, 30-Aug-1993.)
(𝜑𝜓)       ((𝜑𝜒) → 𝜓)
 
Theoremadantl 469 Inference adding a conjunct to the left of an antecedent. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)
(𝜑𝜓)       ((𝜒𝜑) → 𝜓)
 
Theoremsimpl 470 Elimination of a conjunct. Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 14-Jun-2022.)
((𝜑𝜓) → 𝜑)
 
TheoremsimplOLD 471 Obsolete proof of simpl 470 as of 14-Jun-2022. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 13-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → 𝜑)
 
Theoremsimpli 472 Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.)
(𝜑𝜓)       𝜑
 
Theoremsimpr 473 Elimination of a conjunct. Theorem *3.27 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 14-Jun-2022.)
((𝜑𝜓) → 𝜓)
 
TheoremsimprOLD 474 Obsolete proof of simpr 473 as of 14-Jun-2022. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 13-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → 𝜓)
 
Theoremsimpri 475 Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.)
(𝜑𝜓)       𝜓
 
Theoremintnan 476 Introduction of conjunct inside of a contradiction. (Contributed by NM, 16-Sep-1993.)
¬ 𝜑        ¬ (𝜓𝜑)
 
Theoremintnanr 477 Introduction of conjunct inside of a contradiction. (Contributed by NM, 3-Apr-1995.)
¬ 𝜑        ¬ (𝜑𝜓)
 
Theoremintnand 478 Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.)
(𝜑 → ¬ 𝜓)       (𝜑 → ¬ (𝜒𝜓))
 
Theoremintnanrd 479 Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.)
(𝜑 → ¬ 𝜓)       (𝜑 → ¬ (𝜓𝜒))
 
Theoremadantld 480 Deduction adding a conjunct to the left of an antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2012.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) → 𝜒))
 
Theoremadantrd 481 Deduction adding a conjunct to the right of an antecedent. (Contributed by NM, 4-May-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) → 𝜒))
 
Theorempm3.41 482 Theorem *3.41 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜒) → ((𝜑𝜓) → 𝜒))
 
Theorempm3.42 483 Theorem *3.42 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
((𝜓𝜒) → ((𝜑𝜓) → 𝜒))
 
Theoremsimpld 484 Deduction eliminating a conjunct. A translation of natural deduction rule EL ( elimination left), see natded 27589. (Contributed by NM, 26-May-1993.)
(𝜑 → (𝜓𝜒))       (𝜑𝜓)
 
Theoremsimprd 485 Deduction eliminating a conjunct. (Contributed by NM, 14-May-1993.) A translation of natural deduction rule ER ( elimination right), see natded 27589. (Proof shortened by Wolf Lammen, 3-Oct-2013.)
(𝜑 → (𝜓𝜒))       (𝜑𝜒)
 
Theoremsimprbi 486 Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
(𝜑 ↔ (𝜓𝜒))       (𝜑𝜒)
 
Theoremsimplbi 487 Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
(𝜑 ↔ (𝜓𝜒))       (𝜑𝜓)
 
Theoremsimprbda 488 Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜓) → 𝜒)
 
Theoremsimplbda 489 Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜓) → 𝜃)
 
Theoremsimplbi2 490 Deduction eliminating a conjunct. (Contributed by Alan Sare, 31-Dec-2011.)
(𝜑 ↔ (𝜓𝜒))       (𝜓 → (𝜒𝜑))
 
Theoremsimplbi2comt 491 Closed form of simplbi2com 492. (Contributed by Alan Sare, 22-Jul-2012.)
((𝜑 ↔ (𝜓𝜒)) → (𝜒 → (𝜓𝜑)))
 
Theoremsimplbi2com 492 A deduction eliminating a conjunct, similar to simplbi2 490. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.)
(𝜑 ↔ (𝜓𝜒))       (𝜒 → (𝜓𝜑))
 
Theoremsimpl2im 493 Implication from an eliminated conjunct implied by the antecedent. (Contributed by BJ/AV, 5-Apr-2021.) (Proof shortened by Wolf Lammen, 26-Mar-2022.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜃)       (𝜑𝜃)
 
Theoremsimpl2imOLD 494 Obsolete proof of simpl2im 493 as of 26-Mar-2022. (Contributed by BJ/AV, 5-Apr-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜃)       (𝜑𝜃)
 
Theoremsimplbiim 495 Implication from an eliminated conjunct equivalent to the antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Wolf Lammen, 26-Mar-2022.)
(𝜑 ↔ (𝜓𝜒))    &   (𝜒𝜃)       (𝜑𝜃)
 
TheoremsimplbiimOLD 496 Obsolete proof of simplbiim 495 as of 26-Mar-2022. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 ↔ (𝜓𝜒))    &   (𝜒𝜃)       (𝜑𝜃)
 
Theoremimpel 497 An inference for implication elimination. (Contributed by Giovanni Mascellani, 23-May-2019.) (Proof shortened by Wolf Lammen, 2-Sep-2020.)
(𝜑 → (𝜓𝜒))    &   (𝜃𝜓)       ((𝜑𝜃) → 𝜒)
 
Theoremmpan9 498 Modus ponens conjoining dissimilar antecedents. (Contributed by NM, 1-Feb-2008.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(𝜑𝜓)    &   (𝜒 → (𝜓𝜃))       ((𝜑𝜒) → 𝜃)
 
Theoremsylan9 499 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 14-May-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))       ((𝜑𝜃) → (𝜓𝜏))
 
Theoremsylan9r 500 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 14-May-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))       ((𝜃𝜑) → (𝜓𝜏))
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