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Theorem List for Metamath Proof Explorer - 501-600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempm1.2 501 Axiom *1.2 of [WhiteheadRussell] p. 96, which they call "Taut". (Contributed by NM, 3-Jan-2005.)

Theoremoridm 502 Idempotent law for disjunction. Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 16-Apr-2011.) (Proof shortened by Wolf Lammen, 10-Mar-2013.)

Theorempm4.25 503 Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)

Theoremorim12i 504 Disjoin antecedents and consequents of two premises. (Contributed by NM, 6-Jun-1994.) (Proof shortened by Wolf Lammen, 25-Jul-2012.)

Theoremorim1i 505 Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)

Theoremorim2i 506 Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)

Theoremorbi2i 507 Inference adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Dec-2012.)

Theoremorbi1i 508 Inference adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)

Theoremorbi12i 509 Infer the disjunction of two equivalences. (Contributed by NM, 5-Aug-1993.)

Theorempm1.5 510 Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)

Theoremor12 511 Swap two disjuncts. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)

Theoremorass 512 Associative law for disjunction. Theorem *4.33 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theorempm2.31 513 Theorem *2.31 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)

Theorempm2.32 514 Theorem *2.32 of [WhiteheadRussell] p. 105. (Contributed by NM, 3-Jan-2005.)

Theoremor32 515 A rearrangement of disjuncts. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremor4 516 Rearrangement of 4 disjuncts. (Contributed by NM, 12-Aug-1994.)

Theoremor42 517 Rearrangement of 4 disjuncts. (Contributed by NM, 10-Jan-2005.)

Theoremorordi 518 Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)

Theoremorordir 519 Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)

Theoremjca 520 Deduce conjunction of the consequents of two implications ("join consequents with 'and'"). Equivalent to the natural deduction rule I ( introduction), see natded 4. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)

Theoremjcad 521 Deduction conjoining the consequents of two implications. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 23-Jul-2013.)

Theoremjca31 522 Join three consequents. (Contributed by Jeff Hankins, 1-Aug-2009.)

Theoremjca32 523 Join three consequents. (Contributed by FL, 1-Aug-2009.)

Theoremjcai 524 Deduction replacing implication with conjunction. (Contributed by NM, 5-Aug-1993.)

Theoremjctil 525 Inference conjoining a theorem to left of consequent in an implication. (Contributed by NM, 31-Dec-1993.)

Theoremjctir 526 Inference conjoining a theorem to right of consequent in an implication. (Contributed by NM, 31-Dec-1993.)

Theoremjctl 527 Inference conjoining a theorem to the left of a consequent. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 24-Oct-2012.)

Theoremjctr 528 Inference conjoining a theorem to the right of a consequent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 24-Oct-2012.)

Theoremjctild 529 Deduction conjoining a theorem to left of consequent in an implication. (Contributed by NM, 21-Apr-2005.)

Theoremjctird 530 Deduction conjoining a theorem to right of consequent in an implication. (Contributed by NM, 21-Apr-2005.)

Theoremancl 531 Conjoin antecedent to left of consequent. (Contributed by NM, 15-Aug-1994.)

Theoremanclb 532 Conjoin antecedent to left of consequent. Theorem *4.7 of [WhiteheadRussell] p. 120. (Contributed by NM, 25-Jul-1999.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)

Theorempm5.42 533 Theorem *5.42 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)

Theoremancr 534 Conjoin antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)

Theoremancrb 535 Conjoin antecedent to right of consequent. (Contributed by NM, 25-Jul-1999.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)

Theoremancli 536 Deduction conjoining antecedent to left of consequent. (Contributed by NM, 12-Aug-1993.)

Theoremancri 537 Deduction conjoining antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)

Theoremancld 538 Deduction conjoining antecedent to left of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 1-Nov-2012.)

Theoremancrd 539 Deduction conjoining antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 1-Nov-2012.)

Theoremanc2l 540 Conjoin antecedent to left of consequent in nested implication. (Contributed by NM, 10-Aug-1994.) (Proof shortened by Wolf Lammen, 14-Jul-2013.)

Theoremanc2r 541 Conjoin antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.)

Theoremanc2li 542 Deduction conjoining antecedent to left of consequent in nested implication. (Contributed by NM, 10-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)

Theoremanc2ri 543 Deduction conjoining antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)

Theorempm3.41 544 Theorem *3.41 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)

Theorempm3.42 545 Theorem *3.42 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)

Theorempm3.4 546 Conjunction implies implication. Theorem *3.4 of [WhiteheadRussell] p. 113. (Contributed by NM, 31-Jul-1995.)

Theorempm4.45im 547 Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 17-May-1998.)

Theoremanim12d 548 Conjoin antecedents and consequents in a deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 18-Dec-2013.)

Theoremanim1d 549 Add a conjunct to right of antecedent and consequent in a deduction. (Contributed by NM, 3-Apr-1994.)

Theoremanim2d 550 Add a conjunct to left of antecedent and consequent in a deduction. (Contributed by NM, 5-Aug-1993.)

Theoremanim12i 551 Conjoin antecedents and consequents of two premises. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Dec-2013.)

Theoremanim12ci 552 Variant of anim12i 551 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremanim12OLD 553 Conjoin antecedents and consequents of two premises. Obsolete as of 22-May-2016. Note: this is a duplicate of prth 557. (Contributed by Mario Carneiro, 8-May-2016.) (New usage is discouraged.)

Theoremanim1i 554 Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.)

Theoremanim2i 555 Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.)

Theoremanim12ii 556 Conjoin antecedents and consequents in a deduction. (Contributed by NM, 11-Nov-2007.) (Proof shortened by Wolf Lammen, 19-Jul-2013.)

Theoremprth 557 Conjoin antecedents and consequents of two premises. This is the closed theorem form of anim12d 548. 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.)

Theorempm2.3 558 Theorem *2.3 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)

Theorempm2.41 559 Theorem *2.41 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)

Theorempm2.42 560 Theorem *2.42 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)

Theorempm2.4 561 Theorem *2.4 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)

Theorempm2.65da 562 Deduction rule for proof by contradiction. (Contributed by NM, 12-Jun-2014.)

Theorempm4.44 563 Theorem *4.44 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)

Theorempm4.14 564 Theorem *4.14 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Oct-2012.)

Theorempm3.37 565 Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Oct-2012.)

Theoremnan 566 Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)

Theorempm4.15 567 Theorem *4.15 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 18-Nov-2012.)

Theorempm4.78 568 Theorem *4.78 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)

Theorempm4.79 569 Theorem *4.79 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 27-Jun-2013.)

Theorempm4.87 570 Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Eric Schmidt, 26-Oct-2006.)

Theorempm3.33 571 Theorem *3.33 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)

Theorempm3.34 572 Theorem *3.34 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)

Theorempm3.35 573 Conjunctive detachment. Theorem *3.35 of [WhiteheadRussell] p. 112. (Contributed by NM, 14-Dec-2002.)

Theorempm5.31 574 Theorem *5.31 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)

Theoremimp4a 575 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp4b 576 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp4c 577 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp4d 578 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp41 579 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp42 580 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp43 581 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp44 582 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp45 583 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp5a 584 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremimp5d 585 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremimp5g 586 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremimp55 587 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremimp511 588 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremexpimpd 589 Exportation followed by a deduction version of importation. (Contributed by NM, 6-Sep-2008.)

Theoremexp31 590 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp32 591 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp4a 592 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp4b 593 An exportation inference. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)

Theoremexp4c 594 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp4d 595 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp41 596 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp42 597 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp43 598 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp44 599 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp45 600 An exportation inference. (Contributed by NM, 26-Apr-1994.)

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