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

Theoremorcom 401 Commutative law for disjunction. Theorem *4.31 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 15-Nov-2012.)
((𝜑𝜓) ↔ (𝜓𝜑))

Theoremorcomd 402 Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜒𝜓))

Theoremorcoms 403 Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.)
((𝜑𝜓) → 𝜒)       ((𝜓𝜑) → 𝜒)

Theoremorci 404 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
𝜑       (𝜑𝜓)

Theoremolci 405 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
𝜑       (𝜓𝜑)

Theoremorcd 406 Deduction introducing a disjunct. A translation of natural deduction rule IR ( insertion right), see natded 26652. (Contributed by NM, 20-Sep-2007.)
(𝜑𝜓)       (𝜑 → (𝜓𝜒))

Theoremolcd 407 Deduction introducing a disjunct. A translation of natural deduction rule IL ( insertion left), see natded 26652. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
(𝜑𝜓)       (𝜑 → (𝜒𝜓))

Theoremorcs 408 Deduction eliminating disjunct. Notational convention: We sometimes suffix with "s" the label of an inference that manipulates an antecedent, leaving the consequent unchanged. The "s" means that the inference eliminates the need for a syllogism (syl 17) -type inference in a proof. (Contributed by NM, 21-Jun-1994.)
((𝜑𝜓) → 𝜒)       (𝜑𝜒)

Theoremolcs 409 Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
((𝜑𝜓) → 𝜒)       (𝜓𝜒)

Theorempm2.07 410 Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)
(𝜑 → (𝜑𝜑))

Theorempm2.45 411 Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → ¬ 𝜑)

Theorempm2.46 412 Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → ¬ 𝜓)

Theorempm2.47 413 Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (¬ 𝜑𝜓))

Theorempm2.48 414 Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (𝜑 ∨ ¬ 𝜓))

Theorempm2.49 415 Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))

Theorempm2.67-2 416 Slight generalization of Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜒) → 𝜓) → (𝜑𝜓))

Theorempm2.67 417 Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) → 𝜓) → (𝜑𝜓))

Theorempm2.25 418 Theorem *2.25 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
(𝜑 ∨ ((𝜑𝜓) → 𝜓))

Theorembiorf 419 A wff is equivalent to its disjunction with falsehood. Theorem *4.74 of [WhiteheadRussell] p. 121. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2012.)
𝜑 → (𝜓 ↔ (𝜑𝜓)))

Theorembiortn 420 A wff is equivalent to its negated disjunction with falsehood. (Contributed by NM, 9-Jul-2012.)
(𝜑 → (𝜓 ↔ (¬ 𝜑𝜓)))

Theorembiorfi 421 A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 16-Jul-2021.)
¬ 𝜑       (𝜓 ↔ (𝜓𝜑))

TheorembiorfiOLD 422 Obsolete proof of biorfi 421 as of 16-Jul-2021. (Contributed by NM, 23-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ 𝜑       (𝜓 ↔ (𝜓𝜑))

Theorempm2.621 423 Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜓) → 𝜓))

Theorempm2.62 424 Theorem *2.62 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Dec-2013.)
((𝜑𝜓) → ((𝜑𝜓) → 𝜓))

Theorempm2.68 425 Theorem *2.68 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) → 𝜓) → (𝜑𝜓))

Theoremdfor2 426 Logical 'or' expressed in terms of implication only. Theorem *5.25 of [WhiteheadRussell] p. 124. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Wolf Lammen, 20-Oct-2012.)
((𝜑𝜓) ↔ ((𝜑𝜓) → 𝜓))

Theoremimor 427 Implication in terms of disjunction. Theorem *4.6 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-1993.)
((𝜑𝜓) ↔ (¬ 𝜑𝜓))

Theoremimori 428 Infer disjunction from implication. (Contributed by NM, 12-Mar-2012.)
(𝜑𝜓)       𝜑𝜓)

Theoremimorri 429 Infer implication from disjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
𝜑𝜓)       (𝜑𝜓)

Theoremexmid 430 Law of excluded middle, also called the principle of tertium non datur. Theorem *2.11 of [WhiteheadRussell] p. 101. It says that something is either true or not true; there are no in-between values of truth. This is an essential distinction of our classical logic and is not a theorem of intuitionistic logic. In intuitionistic logic, if this statement is true for some 𝜑, then 𝜑 is decideable. (Contributed by NM, 29-Dec-1992.)
(𝜑 ∨ ¬ 𝜑)

Theoremexmidd 431 Law of excluded middle in a context. (Contributed by Mario Carneiro, 9-Feb-2017.)
(𝜑 → (𝜓 ∨ ¬ 𝜓))

Theorempm2.1 432 Theorem *2.1 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)
𝜑𝜑)

Theorempm2.13 433 Theorem *2.13 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)
(𝜑 ∨ ¬ ¬ ¬ 𝜑)

Theorempm4.62 434 Theorem *4.62 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
((𝜑 → ¬ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))

Theorempm4.66 435 Theorem *4.66 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑 → ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓))

Theorempm4.63 436 Theorem *4.63 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑 → ¬ 𝜓) ↔ (𝜑𝜓))

Theoremimnan 437 Express implication in terms of conjunction. (Contributed by NM, 9-Apr-1994.)
((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))

Theoremimnani 438 Infer implication from negated conjunction. (Contributed by Mario Carneiro, 28-Sep-2015.)
¬ (𝜑𝜓)       (𝜑 → ¬ 𝜓)

Theoremiman 439 Express implication in terms of conjunction. 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))

Theoremannim 440 Express conjunction in terms of implication. (Contributed by NM, 2-Aug-1994.)
((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓))

Theorempm4.61 441 Theorem *4.61 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) ↔ (𝜑 ∧ ¬ 𝜓))

Theorempm4.65 442 Theorem *4.65 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
(¬ (¬ 𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))

Theorempm4.67 443 Theorem *4.67 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
(¬ (¬ 𝜑 → ¬ 𝜓) ↔ (¬ 𝜑𝜓))

Theoremimp 444 Importation inference. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) → 𝜒)

Theoremimpcom 445 Importation inference with commuted antecedents. (Contributed by NM, 25-May-2005.)
(𝜑 → (𝜓𝜒))       ((𝜓𝜑) → 𝜒)

Theoremimpd 446 Importation deduction. (Contributed by NM, 31-Mar-1994.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → ((𝜓𝜒) → 𝜃))

Theoremimp31 447 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (((𝜑𝜓) ∧ 𝜒) → 𝜃)

Theoremimp32 448 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒𝜃)))       ((𝜑 ∧ (𝜓𝜒)) → 𝜃)

Theoremex 449 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 26652. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
((𝜑𝜓) → 𝜒)       (𝜑 → (𝜓𝜒))

Theoremexpcom 450 Exportation inference with commuted antecedents. (Contributed by NM, 25-May-2005.)
((𝜑𝜓) → 𝜒)       (𝜓 → (𝜑𝜒))

Theoremexpd 451 Exportation deduction. (Contributed by NM, 20-Aug-1993.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑 → (𝜓 → (𝜒𝜃)))

Theoremexpdimp 452 A deduction version of exportation, followed by importation. (Contributed by NM, 6-Sep-2008.)
(𝜑 → ((𝜓𝜒) → 𝜃))       ((𝜑𝜓) → (𝜒𝜃))

Theoremexpcomd 453 Deduction form of expcom 450. (Contributed by Alan Sare, 22-Jul-2012.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑 → (𝜒 → (𝜓𝜃)))

Theoremexpdcom 454 Commuted form of expd 451. (Contributed by Alan Sare, 18-Mar-2012.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (𝜓 → (𝜒 → (𝜑𝜃)))

Theoremimpancom 455 Mixed importation/commutation inference. (Contributed by NM, 22-Jun-2013.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑𝜒) → (𝜓𝜃))

Theoremcon3dimp 456 Variant of con3d 147 with importation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑 → (𝜓𝜒))       ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓)

Theorempm2.01da 457 Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑𝜓) → ¬ 𝜓)       (𝜑 → ¬ 𝜓)

Theorempm2.18da 458 Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑 ∧ ¬ 𝜓) → 𝜓)       (𝜑𝜓)

Theorempm3.3 459 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)))

Theorempm3.31 460 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 461 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)))

Theorempm3.2 462 Join antecedents with conjunction ("conjunction introduction"). Theorem *3.2 of [WhiteheadRussell] p. 111. See pm3.2im 156 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)))

Theorempm3.21 463 Join antecedents with conjunction. Theorem *3.21 of [WhiteheadRussell] p. 111. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓 → (𝜓𝜑)))

Theorempm3.22 464 Theorem *3.22 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Nov-2012.)
((𝜑𝜓) → (𝜓𝜑))

Theoremancom 465 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 466 Commutation of conjuncts in consequent. (Contributed by Jeff Hankins, 14-Aug-2009.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜒𝜓))

Theoremancomst 467 Closed form of ancoms 468. (Contributed by Alan Sare, 31-Dec-2011.)
(((𝜑𝜓) → 𝜒) ↔ ((𝜓𝜑) → 𝜒))

Theoremancoms 468 Inference commuting conjunction in antecedent. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓) → 𝜒)       ((𝜓𝜑) → 𝜒)

Theoremancomsd 469 Deduction commuting conjunction in antecedent. (Contributed by NM, 12-Dec-2004.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑 → ((𝜒𝜓) → 𝜃))

Theorempm3.2i 470 Infer conjunction of premises. (Contributed by NM, 21-Jun-1993.)
𝜑    &   𝜓       (𝜑𝜓)

Theorempm3.43i 471 Nested conjunction of antecedents. (Contributed by NM, 4-Jan-1993.)
((𝜑𝜓) → ((𝜑𝜒) → (𝜑 → (𝜓𝜒))))

Theoremsimpl 472 Elimination of a conjunct. Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 13-Nov-2012.)
((𝜑𝜓) → 𝜑)

Theoremsimpli 473 Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.)
(𝜑𝜓)       𝜑

Theoremsimpld 474 Deduction eliminating a conjunct. A translation of natural deduction rule EL ( elimination left), see natded 26652. (Contributed by NM, 26-May-1993.)
(𝜑 → (𝜓𝜒))       (𝜑𝜓)

Theoremsimplbi 475 Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
(𝜑 ↔ (𝜓𝜒))       (𝜑𝜓)

Theoremsimpr 476 Elimination of a conjunct. Theorem *3.27 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 13-Nov-2012.)
((𝜑𝜓) → 𝜓)

Theoremsimpri 477 Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.)
(𝜑𝜓)       𝜓

Theoremsimprd 478 Deduction eliminating a conjunct. (Contributed by NM, 14-May-1993.) A translation of natural deduction rule ER ( elimination right), see natded 26652. (Proof shortened by Wolf Lammen, 3-Oct-2013.)
(𝜑 → (𝜓𝜒))       (𝜑𝜒)

Theoremsimprbi 479 Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
(𝜑 ↔ (𝜓𝜒))       (𝜑𝜒)

Theoremadantr 480 Inference adding a conjunct to the right of an antecedent. (Contributed by NM, 30-Aug-1993.)
(𝜑𝜓)       ((𝜑𝜒) → 𝜓)

Theoremadantl 481 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)

Theoremadantld 482 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 483 Deduction adding a conjunct to the right of an antecedent. (Contributed by NM, 4-May-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) → 𝜒))

Theoremimpel 484 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 485 Modus ponens conjoining dissimilar antecedents. (Contributed by NM, 1-Feb-2008.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(𝜑𝜓)    &   (𝜒 → (𝜓𝜃))       ((𝜑𝜒) → 𝜃)

Theoremsyldan 486 A syllogism deduction with conjoined antecedents. (Contributed by NM, 24-Feb-2005.) (Proof shortened by Wolf Lammen, 6-Apr-2013.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)

Theoremsylan 487 A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)
(𝜑𝜓)    &   ((𝜓𝜒) → 𝜃)       ((𝜑𝜒) → 𝜃)

Theoremsylanb 488 A syllogism inference. (Contributed by NM, 18-May-1994.)
(𝜑𝜓)    &   ((𝜓𝜒) → 𝜃)       ((𝜑𝜒) → 𝜃)

Theoremsylanbr 489 A syllogism inference. (Contributed by NM, 18-May-1994.)
(𝜓𝜑)    &   ((𝜓𝜒) → 𝜃)       ((𝜑𝜒) → 𝜃)

Theoremsylan2 490 A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)
(𝜑𝜒)    &   ((𝜓𝜒) → 𝜃)       ((𝜓𝜑) → 𝜃)

Theoremsylan2b 491 A syllogism inference. (Contributed by NM, 21-Apr-1994.)
(𝜑𝜒)    &   ((𝜓𝜒) → 𝜃)       ((𝜓𝜑) → 𝜃)

Theoremsylan2br 492 A syllogism inference. (Contributed by NM, 21-Apr-1994.)
(𝜒𝜑)    &   ((𝜓𝜒) → 𝜃)       ((𝜓𝜑) → 𝜃)

Theoremsyl2an 493 A double syllogism inference. For an implication-only version, see syl2im 39. (Contributed by NM, 31-Jan-1997.)
(𝜑𝜓)    &   (𝜏𝜒)    &   ((𝜓𝜒) → 𝜃)       ((𝜑𝜏) → 𝜃)

Theoremsyl2anr 494 A double syllogism inference. For an implication-only version, see syl2imc 40. (Contributed by NM, 17-Sep-2013.)
(𝜑𝜓)    &   (𝜏𝜒)    &   ((𝜓𝜒) → 𝜃)       ((𝜏𝜑) → 𝜃)

Theoremsyl2anb 495 A double syllogism inference. (Contributed by NM, 29-Jul-1999.)
(𝜑𝜓)    &   (𝜏𝜒)    &   ((𝜓𝜒) → 𝜃)       ((𝜑𝜏) → 𝜃)

Theoremsyl2anbr 496 A double syllogism inference. (Contributed by NM, 29-Jul-1999.)
(𝜓𝜑)    &   (𝜒𝜏)    &   ((𝜓𝜒) → 𝜃)       ((𝜑𝜏) → 𝜃)

Theoremsyland 497 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → ((𝜒𝜃) → 𝜏))       (𝜑 → ((𝜓𝜃) → 𝜏))

Theoremsylan2d 498 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → ((𝜃𝜒) → 𝜏))       (𝜑 → ((𝜃𝜓) → 𝜏))

Theoremsyl2and 499 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))    &   (𝜑 → ((𝜒𝜏) → 𝜂))       (𝜑 → ((𝜓𝜃) → 𝜂))

Theorembiimpa 500 Importation inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) → 𝜒)

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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