HomeHome Intuitionistic Logic Explorer
Theorem List (p. 6 of 102)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 501-600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoreman13s 501 Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ch  /\  ( ps  /\  ph )
 )  ->  th )
 
Theoreman32s 502 Swap two conjuncts in antecedent. (Contributed by NM, 13-Mar-1996.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ( ph  /\ 
 ch )  /\  ps )  ->  th )
 
Theoremancom1s 503 Inference commuting a nested conjunction in antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ( ps 
 /\  ph )  /\  ch )  ->  th )
 
Theoreman31s 504 Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ( ch 
 /\  ps )  /\  ph )  ->  th )
 
Theoremanass1rs 505 Commutative-associative law for conjunction in an antecedent. (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ( ph  /\ 
 ch )  /\  ps )  ->  th )
 
Theoremanabs1 506 Absorption into embedded conjunct. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ph )  <->  (
 ph  /\  ps )
 )
 
Theoremanabs5 507 Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)
 |-  ( ( ph  /\  ( ph  /\  ps ) )  <-> 
 ( ph  /\  ps )
 )
 
Theoremanabs7 508 Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 17-Nov-2013.)
 |-  ( ( ps  /\  ( ph  /\  ps )
 ) 
 <->  ( ph  /\  ps ) )
 
Theoremanabsan 509 Absorption of antecedent with conjunction. (Contributed by NM, 24-Mar-1996.) (Revised by NM, 18-Nov-2013.)
 |-  ( ( ( ph  /\  ph )  /\  ps )  ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theoremanabss1 510 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ph )  ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theoremanabss4 511 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.)
 |-  ( ( ( ps 
 /\  ph )  /\  ps )  ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theoremanabss5 512 Absorption of antecedent into conjunction. (Contributed by NM, 10-May-1994.) (Proof shortened by Wolf Lammen, 1-Jan-2013.)
 |-  ( ( ph  /\  ( ph  /\  ps ) ) 
 ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theoremanabsi5 513 Absorption of antecedent into conjunction. (Contributed by NM, 11-Jun-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2013.)
 |-  ( ph  ->  (
 ( ph  /\  ps )  ->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ch )
 
Theoremanabsi6 514 Absorption of antecedent into conjunction. (Contributed by NM, 14-Aug-2000.)
 |-  ( ph  ->  (
 ( ps  /\  ph )  ->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ch )
 
Theoremanabsi7 515 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 18-Nov-2013.)
 |-  ( ps  ->  (
 ( ph  /\  ps )  ->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ch )
 
Theoremanabsi8 516 Absorption of antecedent into conjunction. (Contributed by NM, 26-Sep-1999.)
 |-  ( ps  ->  (
 ( ps  /\  ph )  ->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ch )
 
Theoremanabss7 517 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 19-Nov-2013.)
 |-  ( ( ps  /\  ( ph  /\  ps )
 )  ->  ch )   =>    |-  (
 ( ph  /\  ps )  ->  ch )
 
Theoremanabsan2 518 Absorption of antecedent with conjunction. (Contributed by NM, 10-May-2004.) (Revised by NM, 1-Jan-2013.)
 |-  ( ( ph  /\  ( ps  /\  ps ) ) 
 ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theoremanabss3 519 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 1-Jan-2013.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ps )  ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theoreman4 520 Rearrangement of 4 conjuncts. (Contributed by NM, 10-Jul-1994.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  th ) )  <-> 
 ( ( ph  /\  ch )  /\  ( ps  /\  th ) ) )
 
Theoreman42 521 Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  th ) )  <-> 
 ( ( ph  /\  ch )  /\  ( th  /\  ps ) ) )
 
Theoreman4s 522 Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  th ) ) 
 ->  ta )   =>    |-  ( ( ( ph  /\ 
 ch )  /\  ( ps  /\  th ) ) 
 ->  ta )
 
Theoreman42s 523 Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  th ) ) 
 ->  ta )   =>    |-  ( ( ( ph  /\ 
 ch )  /\  ( th  /\  ps ) ) 
 ->  ta )
 
Theoremanandi 524 Distribution of conjunction over conjunction. (Contributed by NM, 14-Aug-1995.)
 |-  ( ( ph  /\  ( ps  /\  ch ) )  <-> 
 ( ( ph  /\  ps )  /\  ( ph  /\  ch ) ) )
 
Theoremanandir 525 Distribution of conjunction over conjunction. (Contributed by NM, 24-Aug-1995.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch ) 
 <->  ( ( ph  /\  ch )  /\  ( ps  /\  ch ) ) )
 
Theoremanandis 526 Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ph  /\  ch ) ) 
 ->  ta )   =>    |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  ta )
 
Theoremanandirs 527 Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
 |-  ( ( ( ph  /\ 
 ch )  /\  ( ps  /\  ch ) ) 
 ->  ta )   =>    |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  ta )
 
Theoremimpbida 528 Deduce an equivalence from two implications. (Contributed by NM, 17-Feb-2007.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 ch )  ->  ps )   =>    |-  ( ph  ->  ( ps  <->  ch ) )
 
Theorempm3.45 529 Theorem *3.45 (Fact) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  ->  ps )  ->  ( ( ph  /\  ch )  ->  ( ps  /\  ch )
 ) )
 
Theoremim2anan9 530 Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  ( ta  ->  et ) )   =>    |-  ( ( ph  /\  th )  ->  ( ( ps 
 /\  ta )  ->  ( ch  /\  et ) ) )
 
Theoremim2anan9r 531 Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  ( ta  ->  et ) )   =>    |-  ( ( th  /\  ph )  ->  ( ( ps  /\  ta )  ->  ( ch  /\  et )
 ) )
 
Theoremanim12dan 532 Conjoin antecedents and consequents in a deduction. (Contributed by Mario Carneiro, 12-May-2014.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 th )  ->  ta )   =>    |-  (
 ( ph  /\  ( ps 
 /\  th ) )  ->  ( ch  /\  ta )
 )
 
Theorempm5.1 533 Two propositions are equivalent if they are both true. Theorem *5.1 of [WhiteheadRussell] p. 123. (Contributed by NM, 21-May-1994.)
 |-  ( ( ph  /\  ps )  ->  ( ph  <->  ps ) )
 
Theorempm3.43 534 Theorem *3.43 (Comp) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 27-Nov-2013.)
 |-  ( ( ( ph  ->  ps )  /\  ( ph  ->  ch ) )  ->  ( ph  ->  ( ps  /\ 
 ch ) ) )
 
Theoremjcab 535 Distributive law for implication over conjunction. Compare Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 27-Nov-2013.)
 |-  ( ( ph  ->  ( ps  /\  ch )
 ) 
 <->  ( ( ph  ->  ps )  /\  ( ph  ->  ch ) ) )
 
Theorempm4.76 536 Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  ->  ps )  /\  ( ph  ->  ch ) )  <->  ( ph  ->  ( ps  /\  ch )
 ) )
 
Theorempm4.38 537 Theorem *4.38 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  <->  ch )  /\  ( ps  <->  th ) )  ->  ( ( ph  /\  ps ) 
 <->  ( ch  /\  th ) ) )
 
Theorembi2anan9 538 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 31-Jul-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( th  ->  ( ta  <->  et ) )   =>    |-  ( ( ph  /\ 
 th )  ->  (
 ( ps  /\  ta ) 
 <->  ( ch  /\  et ) ) )
 
Theorembi2anan9r 539 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 19-Feb-1996.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( th  ->  ( ta  <->  et ) )   =>    |-  ( ( th  /\  ph )  ->  ( ( ps  /\  ta )  <->  ( ch  /\  et )
 ) )
 
Theorembi2bian9 540 Deduction joining two biconditionals with different antecedents. (Contributed by NM, 12-May-2004.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( th  ->  ( ta  <->  et ) )   =>    |-  ( ( ph  /\ 
 th )  ->  (
 ( ps  <->  ta )  <->  ( ch  <->  et ) ) )
 
Theorempm5.33 541 Theorem *5.33 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  /\  ( ps  ->  ch ) )  <->  ( ph  /\  (
 ( ph  /\  ps )  ->  ch ) ) )
 
Theorempm5.36 542 Theorem *5.36 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  /\  ( ph 
 <->  ps ) )  <->  ( ps  /\  ( ph  <->  ps ) ) )
 
Theorembianabs 543 Absorb a hypothesis into the second member of a biconditional. (Contributed by FL, 15-Feb-2007.)
 |-  ( ph  ->  ( ps 
 <->  ( ph  /\  ch ) ) )   =>    |-  ( ph  ->  ( ps  <->  ch ) )
 
1.2.5  Logical negation (intuitionistic)
 
Axiomax-in1 544 'Not' introduction. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
 |-  ( ( ph  ->  -.  ph )  ->  -.  ph )
 
Axiomax-in2 545 'Not' elimination. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
 |-  ( -.  ph  ->  (
 ph  ->  ps ) )
 
Theorempm2.01 546 Reductio ad absurdum. Theorem *2.01 of [WhiteheadRussell] p. 100. This is valid intuitionistically (in the terminology of Section 1.2 of [Bauer] p. 482 it is a proof of negation not a proof by contradiction); compare with pm2.18dc 750 which only holds for some propositions. (Contributed by Mario Carneiro, 12-May-2015.)
 |-  ( ( ph  ->  -.  ph )  ->  -.  ph )
 
Theorempm2.21 547 From a wff and its negation, anything is true. Theorem *2.21 of [WhiteheadRussell] p. 104. Also called the Duns Scotus law. (Contributed by Mario Carneiro, 12-May-2015.)
 |-  ( -.  ph  ->  (
 ph  ->  ps ) )
 
Theorempm2.01d 548 Deduction based on reductio ad absurdum. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ph  ->  ( ps  ->  -.  ps )
 )   =>    |-  ( ph  ->  -.  ps )
 
Theorempm2.21d 549 A contradiction implies anything. Deduction from pm2.21 547. (Contributed by NM, 10-Feb-1996.)
 |-  ( ph  ->  -.  ps )   =>    |-  ( ph  ->  ( ps  ->  ch ) )
 
Theorempm2.21dd 550 A contradiction implies anything. Deduction from pm2.21 547. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  -.  ps )   =>    |-  ( ph  ->  ch )
 
Theorempm2.24 551 Theorem *2.24 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  ->  ( -.  ph  ->  ps )
 )
 
Theorempm2.24d 552 Deduction version of pm2.24 551. (Contributed by NM, 30-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( -.  ps  ->  ch ) )
 
Theorempm2.24i 553 Inference version of pm2.24 551. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ph   =>    |-  ( -.  ph  ->  ps )
 
Theoremcon2d 554 A contraposition deduction. (Contributed by NM, 19-Aug-1993.) (Revised by NM, 12-Feb-2013.)
 |-  ( ph  ->  ( ps  ->  -.  ch )
 )   =>    |-  ( ph  ->  ( ch  ->  -.  ps )
 )
 
Theoremmt2d 555 Modus tollens deduction. (Contributed by NM, 4-Jul-1994.)
 |-  ( ph  ->  ch )   &    |-  ( ph  ->  ( ps  ->  -. 
 ch ) )   =>    |-  ( ph  ->  -. 
 ps )
 
Theoremnsyl3 556 A negated syllogism inference. (Contributed by NM, 1-Dec-1995.) (Revised by NM, 13-Jun-2013.)
 |-  ( ph  ->  -.  ps )   &    |-  ( ch  ->  ps )   =>    |-  ( ch  ->  -.  ph )
 
Theoremcon2i 557 A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.) (Proof shortened by Wolf Lammen, 13-Jun-2013.)
 |-  ( ph  ->  -.  ps )   =>    |-  ( ps  ->  -.  ph )
 
Theoremnsyl 558 A negated syllogism inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
 |-  ( ph  ->  -.  ps )   &    |-  ( ch  ->  ps )   =>    |-  ( ph  ->  -.  ch )
 
Theoremnotnot 559 Double negation introduction. Theorem *2.12 of [WhiteheadRussell] p. 101. This one holds for all propositions, but its converse only holds for decidable propositions (see notnotrdc 751). (Contributed by NM, 28-Dec-1992.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
 |-  ( ph  ->  -.  -.  ph )
 
Theoremnotnotd 560 Deduction associated with notnot 559 and notnoti 574. (Contributed by Jarvin Udandy, 2-Sep-2016.) Avoid biconditional. (Revised by Wolf Lammen, 27-Mar-2021.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  -.  -.  ps )
 
Theoremcon3d 561 A contraposition deduction. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 31-Jan-2015.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( -.  ch  ->  -.  ps ) )
 
Theoremcon3i 562 A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 20-Jun-2013.)
 |-  ( ph  ->  ps )   =>    |-  ( -.  ps  ->  -.  ph )
 
Theoremcon3rr3 563 Rotate through consequent right. (Contributed by Wolf Lammen, 3-Nov-2013.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( -.  ch  ->  ( ph  ->  -.  ps ) )
 
Theoremcon3and 564 Variant of con3d 561 with importation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ( ph  /\ 
 -.  ch )  ->  -.  ps )
 
Theorempm2.01da 565 Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ( ph  /\  ps )  ->  -.  ps )   =>    |-  ( ph  ->  -.  ps )
 
Theorempm3.2im 566 In classical logic, this is just a restatement of pm3.2 126. In intuitionistic logic, it still holds, but is weaker than pm3.2. (Contributed by Mario Carneiro, 12-May-2015.)
 |-  ( ph  ->  ( ps  ->  -.  ( ph  ->  -.  ps ) ) )
 
Theoremexpi 567 An exportation inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.)
 |-  ( -.  ( ph  ->  -.  ps )  ->  ch )   =>    |-  ( ph  ->  ( ps  ->  ch ) )
 
Theorempm2.65i 568 Inference rule for proof by contradiction. (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  -.  ps )   =>    |-  -.  ph
 
Theoremmt2 569 A rule similar to modus tollens. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Sep-2013.)
 |- 
 ps   &    |-  ( ph  ->  -.  ps )   =>    |- 
 -.  ph
 
Theorembiijust 570 Theorem used to justify definition of intuitionistic biconditional df-bi 110. (Contributed by NM, 24-Nov-2017.)
 |-  ( ( ( (
 ph  ->  ps )  /\  ( ps  ->  ph ) )  ->  ( ( ph  ->  ps )  /\  ( ps 
 ->  ph ) ) ) 
 /\  ( ( (
 ph  ->  ps )  /\  ( ps  ->  ph ) )  ->  ( ( ph  ->  ps )  /\  ( ps 
 ->  ph ) ) ) )
 
Theoremcon3 571 Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Feb-2013.)
 |-  ( ( ph  ->  ps )  ->  ( -.  ps 
 ->  -.  ph ) )
 
Theoremcon2 572 Contraposition. Theorem *2.03 of [WhiteheadRussell] p. 100. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Feb-2013.)
 |-  ( ( ph  ->  -. 
 ps )  ->  ( ps  ->  -.  ph ) )
 
Theoremmt2i 573 Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
 |- 
 ch   &    |-  ( ph  ->  ( ps  ->  -.  ch )
 )   =>    |-  ( ph  ->  -.  ps )
 
Theoremnotnoti 574 Infer double negation. (Contributed by NM, 27-Feb-2008.)
 |-  ph   =>    |- 
 -.  -.  ph
 
Theorempm2.21i 575 A contradiction implies anything. Inference from pm2.21 547. (Contributed by NM, 16-Sep-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |- 
 -.  ph   =>    |-  ( ph  ->  ps )
 
Theorempm2.24ii 576 A contradiction implies anything. Inference from pm2.24 551. (Contributed by NM, 27-Feb-2008.)
 |-  ph   &    |- 
 -.  ph   =>    |- 
 ps
 
Theoremnsyld 577 A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.)
 |-  ( ph  ->  ( ps  ->  -.  ch )
 )   &    |-  ( ph  ->  ( ta  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  -.  ta )
 )
 
Theoremnsyli 578 A negated syllogism inference. (Contributed by NM, 3-May-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  -.  ch )   =>    |-  ( ph  ->  ( th  ->  -. 
 ps ) )
 
Theoremmth8 579 Theorem 8 of [Margaris] p. 60. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
 |-  ( ph  ->  ( -.  ps  ->  -.  ( ph  ->  ps ) ) )
 
Theoremjc 580 Inference joining the consequents of two premises. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   =>    |-  ( ph  ->  -.  ( ps  ->  -.  ch )
 )
 
Theorempm2.51 581 Theorem *2.51 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  ->  ps )  ->  ( ph  ->  -.  ps )
 )
 
Theorempm2.52 582 Theorem *2.52 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( -.  ( ph  ->  ps )  ->  ( -.  ph  ->  -.  ps )
 )
 
Theoremexpt 583 Exportation theorem expressed with primitive connectives. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( -.  ( ph  ->  -.  ps )  ->  ch )  ->  ( ph  ->  ( ps  ->  ch ) ) )
 
Theoremjarl 584 Elimination of a nested antecedent. (Contributed by Wolf Lammen, 10-May-2013.)
 |-  ( ( ( ph  ->  ps )  ->  ch )  ->  ( -.  ph  ->  ch ) )
 
Theorempm2.65 585 Theorem *2.65 of [WhiteheadRussell] p. 107. Proof by contradiction. Proofs, such as this one, which assume a proposition, here  ph, derive a contradiction, and therefore conclude  -. 
ph, are valid intuitionistically (and can be called "proof of negation", for example by Section 1.2 of [Bauer] p. 482). By contrast, proofs which assume  -.  ph, derive a contradiction, and conclude  ph, such as condandc 775, are only valid for decidable propositions. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 8-Mar-2013.)
 |-  ( ( ph  ->  ps )  ->  ( ( ph  ->  -.  ps )  ->  -.  ph ) )
 
Theorempm2.65d 586 Deduction rule for proof by contradiction. (Contributed by NM, 26-Jun-1994.) (Proof shortened by Wolf Lammen, 26-May-2013.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  -. 
 ch ) )   =>    |-  ( ph  ->  -. 
 ps )
 
Theorempm2.65da 587 Deduction rule for proof by contradiction. (Contributed by NM, 12-Jun-2014.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 ps )  ->  -.  ch )   =>    |-  ( ph  ->  -.  ps )
 
Theoremmto 588 The rule of modus tollens. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
 |- 
 -.  ps   &    |-  ( ph  ->  ps )   =>    |- 
 -.  ph
 
Theoremmtod 589 Modus tollens deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  -. 
 ps )
 
Theoremmtoi 590 Modus tollens inference. (Contributed by NM, 5-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
 |- 
 -.  ch   &    |-  ( ph  ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  -.  ps )
 
Theoremmtand 591 A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ph  ->  -.  ps )
 
Theoremnotbid 592 Deduction negating both sides of a logical equivalence. (Contributed by NM, 21-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( -.  ps  <->  -.  ch ) )
 
Theoremcon2b 593 Contraposition. Bidirectional version of con2 572. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( ph  ->  -. 
 ps )  <->  ( ps  ->  -.  ph ) )
 
Theoremnotbii 594 Negate both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ph  <->  ps )   =>    |-  ( -.  ph  <->  -.  ps )
 
Theoremmtbi 595 An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
 |- 
 -.  ph   &    |-  ( ph  <->  ps )   =>    |- 
 -.  ps
 
Theoremmtbir 596 An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 14-Oct-2012.)
 |- 
 -.  ps   &    |-  ( ph  <->  ps )   =>    |- 
 -.  ph
 
Theoremmtbid 597 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.)
 |-  ( ph  ->  -.  ps )   &    |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  -. 
 ch )
 
Theoremmtbird 598 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  -. 
 ps )
 
Theoremmtbii 599 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.)
 |- 
 -.  ps   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  -. 
 ch )
 
Theoremmtbiri 600 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.)
 |- 
 -.  ch   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  -. 
 ps )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10124
  Copyright terms: Public domain < Previous  Next >