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Theorem List for Metamath Proof Explorer - 401-500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempm4.67 401 Theorem *4.67 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
(¬ (¬ 𝜑 → ¬ 𝜓) ↔ (¬ 𝜑𝜓))

Theoremimnan 402 Express an implication in terms of a negated conjunction. (Contributed by NM, 9-Apr-1994.)
((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))

Theoremimnani 403 Infer an implication from a negated conjunction. (Contributed by Mario Carneiro, 28-Sep-2015.)
¬ (𝜑𝜓)       (𝜑 → ¬ 𝜓)

Theoremiman 404 Implication in terms of conjunction and negation. Theorem 3.4(27) of [Stoll] p. 176. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 30-Oct-2012.)
((𝜑𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))

Theorempm3.24 405 Law of noncontradiction. Theorem *3.24 of [WhiteheadRussell] p. 111 (who call it the "law of contradiction"). (Contributed by NM, 16-Sep-1993.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
¬ (𝜑 ∧ ¬ 𝜑)

Theoremannim 406 Express a conjunction in terms of a negated implication. (Contributed by NM, 2-Aug-1994.)
((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓))

Theorempm4.61 407 Theorem *4.61 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) ↔ (𝜑 ∧ ¬ 𝜓))

Theorempm4.65 408 Theorem *4.65 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
(¬ (¬ 𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))

Theoremimp 409 Importation inference. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) → 𝜒)

Theoremimpcom 410 Importation inference with commuted antecedents. (Contributed by NM, 25-May-2005.)
(𝜑 → (𝜓𝜒))       ((𝜓𝜑) → 𝜒)

Theoremcon3dimp 411 Variant of con3d 155 with importation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑 → (𝜓𝜒))       ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓)

Theoremmpnanrd 412 Eliminate the right side of a negated conjunction in an implication. (Contributed by ML, 17-Oct-2020.)
(𝜑𝜓)    &   (𝜑 → ¬ (𝜓𝜒))       (𝜑 → ¬ 𝜒)

Theoremimpd 413 Importation deduction. (Contributed by NM, 31-Mar-1994.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → ((𝜓𝜒) → 𝜃))

Theoremimpcomd 414 Importation deduction with commuted antecedents. (Contributed by Peter Mazsa, 24-Sep-2022.) (Proof shortened by Wolf Lammen, 22-Oct-2022.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → ((𝜒𝜓) → 𝜃))

Theoremex 415 Exportation inference. (This theorem used to be labeled "exp" but was changed to "ex" so as not to conflict with the math token "exp", per the June 2006 Metamath spec change.) A translation of natural deduction rule I ( introduction), see natded 28174. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
((𝜑𝜓) → 𝜒)       (𝜑 → (𝜓𝜒))

Theoremexpcom 416 Exportation inference with commuted antecedents. (Contributed by NM, 25-May-2005.)
((𝜑𝜓) → 𝜒)       (𝜓 → (𝜑𝜒))

Theoremexpdcom 417 Commuted form of expd 418. (Contributed by Alan Sare, 18-Mar-2012.) Shorten expd 418. (Revised by Wolf Lammen, 28-Jul-2022.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (𝜓 → (𝜒 → (𝜑𝜃)))

Theoremexpd 418 Exportation deduction. (Contributed by NM, 20-Aug-1993.) (Proof shortened by Wolf Lammen, 28-Jul-2022.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑 → (𝜓 → (𝜒𝜃)))

Theoremexpcomd 419 Deduction form of expcom 416. (Contributed by Alan Sare, 22-Jul-2012.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑 → (𝜒 → (𝜓𝜃)))

Theoremimp31 420 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (((𝜑𝜓) ∧ 𝜒) → 𝜃)

Theoremimp32 421 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒𝜃)))       ((𝜑 ∧ (𝜓𝜒)) → 𝜃)

Theoremexp31 422 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))

Theoremexp32 423 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))

Theoremimp4b 424 An importation inference. (Contributed by NM, 26-Apr-1994.) Shorten imp4a 425. (Revised by Wolf Lammen, 19-Jul-2021.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((𝜑𝜓) → ((𝜒𝜃) → 𝜏))

Theoremimp4a 425 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 426 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → (((𝜓𝜒) ∧ 𝜃) → 𝜏))

Theoremimp4d 427 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → ((𝜓 ∧ (𝜒𝜃)) → 𝜏))

Theoremimp41 428 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theoremimp42 429 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜏)

Theoremimp43 430 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)

Theoremimp44 431 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)) → 𝜏)

Theoremimp45 432 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜏)

Theoremexp4b 433 An exportation inference. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 23-Nov-2012.) Shorten exp4a 434. (Revised by Wolf Lammen, 20-Jul-2021.)
((𝜑𝜓) → ((𝜒𝜃) → 𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexp4a 434 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 435 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (((𝜓𝜒) ∧ 𝜃) → 𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexp4d 436 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → ((𝜓 ∧ (𝜒𝜃)) → 𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexp41 437 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexp42 438 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexp43 439 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexp44 440 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexp45 441 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremimp5d 442 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (((𝜑𝜓) ∧ 𝜒) → ((𝜃𝜏) → 𝜂))

Theoremimp5a 443 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 444 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       ((𝜑𝜓) → (((𝜒𝜃) ∧ 𝜏) → 𝜂))

Theoremimp55 445 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) ∧ 𝜏) → 𝜂)

Theoremimp511 446 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       ((𝜑 ∧ ((𝜓 ∧ (𝜒𝜃)) ∧ 𝜏)) → 𝜂)

Theoremexp5c 447 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → ((𝜓𝜒) → ((𝜃𝜏) → 𝜂)))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))

Theoremexp5j 448 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → ((((𝜓𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))

Theoremexp5l 449 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (((𝜓𝜒) ∧ (𝜃𝜏)) → 𝜂))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))

Theoremexp53 450 An exportation inference. (Contributed by Jeff Hankins, 30-Aug-2009.)
((((𝜑𝜓) ∧ (𝜒𝜃)) ∧ 𝜏) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))

Theorempm3.3 451 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)))

Theorempm3.31 452 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 453 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)))

Theoremimpancom 454 Mixed importation/commutation inference. (Contributed by NM, 22-Jun-2013.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑𝜒) → (𝜓𝜃))

Theoremexpdimp 455 A deduction version of exportation, followed by importation. (Contributed by NM, 6-Sep-2008.)
(𝜑 → ((𝜓𝜒) → 𝜃))       ((𝜑𝜓) → (𝜒𝜃))

Theoremexpimpd 456 Exportation followed by a deduction version of importation. (Contributed by NM, 6-Sep-2008.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → ((𝜓𝜒) → 𝜃))

Theoremimpr 457 Import a wff into a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑 ∧ (𝜓𝜒)) → 𝜃)

Theoremimpl 458 Export a wff from a left conjunct. (Contributed by Mario Carneiro, 9-Jul-2014.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (((𝜑𝜓) ∧ 𝜒) → 𝜃)

Theoremexpr 459 Export a wff from a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑𝜓) → (𝜒𝜃))

Theoremexpl 460 Export a wff from a left conjunct. (Contributed by Jeff Hankins, 28-Aug-2009.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (𝜑 → ((𝜓𝜒) → 𝜃))

Theoremancoms 461 Inference commuting conjunction in antecedent. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓) → 𝜒)       ((𝜓𝜑) → 𝜒)

Theorempm3.22 462 Theorem *3.22 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Nov-2012.)
((𝜑𝜓) → (𝜓𝜑))

Theoremancom 463 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 464 Commutation of conjuncts in consequent. (Contributed by Jeff Hankins, 14-Aug-2009.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜒𝜓))

Theorembiancomi 465 Commuting conjunction in a biconditional. (Contributed by Peter Mazsa, 17-Jun-2018.)
(𝜑 ↔ (𝜒𝜓))       (𝜑 ↔ (𝜓𝜒))

Theorembiancomd 466 Commuting conjunction in a biconditional, deduction form. (Contributed by Peter Mazsa, 3-Oct-2018.)
(𝜑 → (𝜓 ↔ (𝜃𝜒)))       (𝜑 → (𝜓 ↔ (𝜒𝜃)))

Theoremancomst 467 Closed form of ancoms 461. (Contributed by Alan Sare, 31-Dec-2011.)
(((𝜑𝜓) → 𝜒) ↔ ((𝜓𝜑) → 𝜒))

Theoremancomsd 468 Deduction commuting conjunction in antecedent. (Contributed by NM, 12-Dec-2004.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑 → ((𝜒𝜓) → 𝜃))

Theoremanasss 469 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜒)) → 𝜃)

Theoremanassrs 470 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       (((𝜑𝜓) ∧ 𝜒) → 𝜃)

Theoremanass 471 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)))

Theorempm3.2 472 Join antecedents with conjunction ("conjunction introduction"). Theorem *3.2 of [WhiteheadRussell] p. 111. Its associated inference is pm3.2i 473 and its associated deduction is jca 514 (and the double deduction is jcad 515). See pm3.2im 162 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 473 Infer conjunction of premises. Inference associated with pm3.2 472. Its associated deduction is jca 514 (and the double deduction is jcad 515). (Contributed by NM, 21-Jun-1993.)
𝜑    &   𝜓       (𝜑𝜓)

Theorempm3.21 474 Join antecedents with conjunction. Theorem *3.21 of [WhiteheadRussell] p. 111. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓 → (𝜓𝜑)))

Theorempm3.43i 475 Nested conjunction of antecedents. (Contributed by NM, 4-Jan-1993.)
((𝜑𝜓) → ((𝜑𝜒) → (𝜑 → (𝜓𝜒))))

Theorempm3.43 476 Theorem *3.43 (Comp) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜑𝜒)) → (𝜑 → (𝜓𝜒)))

Theoremdfbi2 477 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 478 Definition df-bi 209 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 479 Importation inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) → 𝜒)

Theorembiimpar 480 Importation inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜒) → 𝜓)

Theorembiimpac 481 Importation inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))       ((𝜓𝜑) → 𝜒)

Theorembiimparc 482 Importation inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))       ((𝜒𝜑) → 𝜓)

Theoremadantr 483 Inference adding a conjunct to the right of an antecedent. (Contributed by NM, 30-Aug-1993.)
(𝜑𝜓)       ((𝜑𝜒) → 𝜓)

Theoremadantl 484 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 485 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)

Theoremsimpli 486 Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.)
(𝜑𝜓)       𝜑

Theoremsimpr 487 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)

Theoremsimpri 488 Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.)
(𝜑𝜓)       𝜓

Theoremintnan 489 Introduction of conjunct inside of a contradiction. (Contributed by NM, 16-Sep-1993.)
¬ 𝜑        ¬ (𝜓𝜑)

Theoremintnanr 490 Introduction of conjunct inside of a contradiction. (Contributed by NM, 3-Apr-1995.)
¬ 𝜑        ¬ (𝜑𝜓)

Theoremintnand 491 Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.)
(𝜑 → ¬ 𝜓)       (𝜑 → ¬ (𝜒𝜓))

Theoremintnanrd 492 Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.)
(𝜑 → ¬ 𝜓)       (𝜑 → ¬ (𝜓𝜒))

Theoremadantld 493 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 494 Deduction adding a conjunct to the right of an antecedent. (Contributed by NM, 4-May-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) → 𝜒))

Theorempm3.41 495 Theorem *3.41 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜒) → ((𝜑𝜓) → 𝜒))

Theorempm3.42 496 Theorem *3.42 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
((𝜓𝜒) → ((𝜑𝜓) → 𝜒))

Theoremsimpld 497 Deduction eliminating a conjunct. A translation of natural deduction rule EL ( elimination left), see natded 28174. (Contributed by NM, 26-May-1993.)
(𝜑 → (𝜓𝜒))       (𝜑𝜓)

Theoremsimprd 498 Deduction eliminating a conjunct. (Contributed by NM, 14-May-1993.) A translation of natural deduction rule ER ( elimination right), see natded 28174. (Proof shortened by Wolf Lammen, 3-Oct-2013.)
(𝜑 → (𝜓𝜒))       (𝜑𝜒)

Theoremsimprbi 499 Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
(𝜑 ↔ (𝜓𝜒))       (𝜑𝜒)

Theoremsimplbi 500 Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
(𝜑 ↔ (𝜓𝜒))       (𝜑𝜓)

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