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Theorem List for Intuitionistic Logic Explorer - 401-500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsylan9 401 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  ( ch  ->  ta ) )   =>    |-  ( ( ph  /\  th )  ->  ( ps  ->  ta ) )
 
Theoremsylan9r 402 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  ( ch  ->  ta ) )   =>    |-  ( ( th  /\  ph )  ->  ( ps  ->  ta ) )
 
Theoremsyl2anc 403 Syllogism inference combined with contraction. (Contributed by NM, 16-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  ( ph  ->  th )
 
Theoremsylancl 404 Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ph  ->  ps )   &    |-  ch   &    |-  (
 ( ps  /\  ch )  ->  th )   =>    |-  ( ph  ->  th )
 
Theoremsylancr 405 Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |- 
 ps   &    |-  ( ph  ->  ch )   &    |-  (
 ( ps  /\  ch )  ->  th )   =>    |-  ( ph  ->  th )
 
Theoremsylanblc 406 Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019.)
 |-  ( ph  ->  ps )   &    |-  ch   &    |-  (
 ( ps  /\  ch ) 
 <-> 
 th )   =>    |-  ( ph  ->  th )
 
Theoremsylanblrc 407 Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019.)
 |-  ( ph  ->  ps )   &    |-  ch   &    |-  ( th 
 <->  ( ps  /\  ch ) )   =>    |-  ( ph  ->  th )
 
Theoremsylanbrc 408 Syllogism inference. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( th  <->  ( ps  /\  ch ) )   =>    |-  ( ph  ->  th )
 
Theoremsylancb 409 A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)
 |-  ( ph  <->  ps )   &    |-  ( ph  <->  ch )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  ( ph  ->  th )
 
Theoremsylancbr 410 A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)
 |-  ( ps  <->  ph )   &    |-  ( ch  <->  ph )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  ( ph  ->  th )
 
Theoremsylancom 411 Syllogism inference with commutation of antecents. (Contributed by NM, 2-Jul-2008.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ch 
 /\  ps )  ->  th )   =>    |-  (
 ( ph  /\  ps )  ->  th )
 
Theoremmpdan 412 An inference based on modus ponens. (Contributed by NM, 23-May-1999.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)
 |-  ( ph  ->  ps )   &    |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ph  ->  ch )
 
Theoremmpancom 413 An inference based on modus ponens with commutation of antecedents. (Contributed by NM, 28-Oct-2003.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
 |-  ( ps  ->  ph )   &    |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ps  ->  ch )
 
Theoremmpidan 414 A deduction which "stacks" a hypothesis. (Contributed by Stanislas Polu, 9-Mar-2020.) (Proof shortened by Wolf Lammen, 28-Mar-2021.)
 |-  ( ph  ->  ch )   &    |-  (
 ( ( ph  /\  ps )  /\  ch )  ->  th )   =>    |-  ( ( ph  /\  ps )  ->  th )
 
Theoremmpan 415 An inference based on modus ponens. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
 |-  ph   &    |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ps  ->  ch )
 
Theoremmpan2 416 An inference based on modus ponens. (Contributed by NM, 16-Sep-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)
 |- 
 ps   &    |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ph  ->  ch )
 
Theoremmp2an 417 An inference based on modus ponens. (Contributed by NM, 13-Apr-1995.)
 |-  ph   &    |- 
 ps   &    |-  ( ( ph  /\  ps )  ->  ch )   =>    |- 
 ch
 
Theoremmp4an 418 An inference based on modus ponens. (Contributed by Jeff Madsen, 15-Jun-2011.)
 |-  ph   &    |- 
 ps   &    |- 
 ch   &    |- 
 th   &    |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  th ) ) 
 ->  ta )   =>    |- 
 ta
 
Theoremmpan2d 419 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
 |-  ( ph  ->  ch )   &    |-  ( ph  ->  ( ( ps 
 /\  ch )  ->  th )
 )   =>    |-  ( ph  ->  ( ps  ->  th ) )
 
Theoremmpand 420 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ( ps 
 /\  ch )  ->  th )
 )   =>    |-  ( ph  ->  ( ch  ->  th ) )
 
Theoremmpani 421 An inference based on modus ponens. (Contributed by NM, 10-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)
 |- 
 ps   &    |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   =>    |-  ( ph  ->  ( ch  ->  th )
 )
 
Theoremmpan2i 422 An inference based on modus ponens. (Contributed by NM, 10-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)
 |- 
 ch   &    |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   =>    |-  ( ph  ->  ( ps  ->  th )
 )
 
Theoremmp2ani 423 An inference based on modus ponens. (Contributed by NM, 12-Dec-2004.)
 |- 
 ps   &    |- 
 ch   &    |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   =>    |-  ( ph  ->  th )
 
Theoremmp2and 424 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  ( ( ps  /\  ch )  ->  th ) )   =>    |-  ( ph  ->  th )
 
Theoremmpanl1 425 An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
 |-  ph   &    |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ps  /\  ch )  ->  th )
 
Theoremmpanl2 426 An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 |- 
 ps   &    |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ph  /\  ch )  ->  th )
 
Theoremmpanl12 427 An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)
 |-  ph   &    |- 
 ps   &    |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ch  ->  th )
 
Theoremmpanr1 428 An inference based on modus ponens. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 |- 
 ps   &    |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ch )  ->  th )
 
Theoremmpanr2 429 An inference based on modus ponens. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
 |- 
 ch   &    |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ps )  ->  th )
 
Theoremmpanr12 430 An inference based on modus ponens. (Contributed by NM, 24-Jul-2009.)
 |- 
 ps   &    |- 
 ch   &    |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ph  ->  th )
 
Theoremmpanlr1 431 An inference based on modus ponens. (Contributed by NM, 30-Dec-2004.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
 |- 
 ps   &    |-  ( ( ( ph  /\  ( ps  /\  ch ) )  /\  th )  ->  ta )   =>    |-  ( ( ( ph  /\ 
 ch )  /\  th )  ->  ta )
 
Theorempm5.74da 432 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 4-May-2007.)
 |-  ( ( ph  /\  ps )  ->  ( ch  <->  th ) )   =>    |-  ( ph  ->  ( ( ps  ->  ch )  <->  ( ps  ->  th )
 ) )
 
Theoremimdistan 433 Distribution of implication with conjunction. (Contributed by NM, 31-May-1999.) (Proof shortened by Wolf Lammen, 6-Dec-2012.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 ) 
 <->  ( ( ph  /\  ps )  ->  ( ph  /\  ch ) ) )
 
Theoremimdistani 434 Distribution of implication with conjunction. (Contributed by NM, 1-Aug-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ( ph  /\  ch ) )
 
Theoremimdistanri 435 Distribution of implication with conjunction. (Contributed by NM, 8-Jan-2002.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ( ps 
 /\  ph )  ->  ( ch  /\  ph ) )
 
Theoremimdistand 436 Distribution of implication with conjunction (deduction rule). (Contributed by NM, 27-Aug-2004.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   =>    |-  ( ph  ->  ( ( ps  /\  ch )  ->  ( ps  /\  th ) ) )
 
Theoremimdistanda 437 Distribution of implication with conjunction (deduction version with conjoined antecedent). (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  ( ( ph  /\  ps )  ->  ( ch  ->  th ) )   =>    |-  ( ph  ->  (
 ( ps  /\  ch )  ->  ( ps  /\  th ) ) )
 
Theorempm5.32d 438 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 29-Oct-1996.) (Revised by NM, 31-Jan-2015.)
 |-  ( ph  ->  ( ps  ->  ( ch  <->  th ) ) )   =>    |-  ( ph  ->  ( ( ps  /\  ch )  <->  ( ps  /\  th ) ) )
 
Theorempm5.32rd 439 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 25-Dec-2004.)
 |-  ( ph  ->  ( ps  ->  ( ch  <->  th ) ) )   =>    |-  ( ph  ->  ( ( ch  /\  ps )  <->  ( th  /\  ps ) ) )
 
Theorempm5.32da 440 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 9-Dec-2006.)
 |-  ( ( ph  /\  ps )  ->  ( ch  <->  th ) )   =>    |-  ( ph  ->  ( ( ps  /\  ch ) 
 <->  ( ps  /\  th ) ) )
 
Theorempm5.32 441 Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.) (Revised by NM, 31-Jan-2015.)
 |-  ( ( ph  ->  ( ps  <->  ch ) )  <->  ( ( ph  /\ 
 ps )  <->  ( ph  /\  ch ) ) )
 
Theorempm5.32i 442 Distribution of implication over biconditional (inference rule). (Contributed by NM, 1-Aug-1994.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  <->  ( ph  /\  ch ) )
 
Theorempm5.32ri 443 Distribution of implication over biconditional (inference rule). (Contributed by NM, 12-Mar-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ( ps 
 /\  ph )  <->  ( ch  /\  ph ) )
 
Theorembiadan2 444 Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.)
 |-  ( ph  ->  ps )   &    |-  ( ps  ->  ( ph  <->  ch ) )   =>    |-  ( ph  <->  ( ps  /\  ch ) )
 
Theoremanbi2i 445 Introduce a left conjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
 |-  ( ph  <->  ps )   =>    |-  ( ( ch  /\  ph )  <->  ( ch  /\  ps ) )
 
Theoremanbi1i 446 Introduce a right conjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
 |-  ( ph  <->  ps )   =>    |-  ( ( ph  /\  ch ) 
 <->  ( ps  /\  ch ) )
 
Theoremanbi2ci 447 Variant of anbi2i 445 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( ph  <->  ps )   =>    |-  ( ( ph  /\  ch ) 
 <->  ( ch  /\  ps ) )
 
Theoremanbi12i 448 Conjoin both sides of two equivalences. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ( ph  /\  ch ) 
 <->  ( ps  /\  th ) )
 
Theoremanbi12ci 449 Variant of anbi12i 448 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ( ph  /\  ch ) 
 <->  ( th  /\  ps ) )
 
Theoremsylan9bb 450 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( th  ->  ( ch  <->  ta ) )   =>    |-  ( ( ph  /\ 
 th )  ->  ( ps 
 <->  ta ) )
 
Theoremsylan9bbr 451 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( th  ->  ( ch  <->  ta ) )   =>    |-  ( ( th  /\  ph )  ->  ( ps  <->  ta ) )
 
Theoremanbi2d 452 Deduction adding a left conjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( th  /\  ps ) 
 <->  ( th  /\  ch ) ) )
 
Theoremanbi1d 453 Deduction adding a right conjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( ps  /\  th ) 
 <->  ( ch  /\  th ) ) )
 
Theoremanbi1 454 Introduce a right conjunct to both sides of a logical equivalence. Theorem *4.36 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  <->  ps )  ->  (
 ( ph  /\  ch )  <->  ( ps  /\  ch )
 ) )
 
Theoremanbi2 455 Introduce a left conjunct to both sides of a logical equivalence. (Contributed by NM, 16-Nov-2013.)
 |-  ( ( ph  <->  ps )  ->  (
 ( ch  /\  ph )  <->  ( ch  /\  ps )
 ) )
 
Theorembitr 456 Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  <->  ps )  /\  ( ps  <->  ch ) )  ->  ( ph  <->  ch ) )
 
Theoremanbi12d 457 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ( ps  /\  th ) 
 <->  ( ch  /\  ta ) ) )
 
Theoremmpan10 458 Modus ponens mixed with several conjunctions. (Contributed by Jim Kingdon, 7-Jan-2018.)
 |-  ( ( ( (
 ph  ->  ps )  /\  ch )  /\  ph )  ->  ( ps  /\  ch ) )
 
Theorempm5.3 459 Theorem *5.3 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 |-  ( ( ( ph  /\ 
 ps )  ->  ch )  <->  ( ( ph  /\  ps )  ->  ( ph  /\  ch ) ) )
 
Theoremadantll 460 Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( th  /\  ph )  /\  ps )  ->  ch )
 
Theoremadantlr 461 Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ph  /\ 
 th )  /\  ps )  ->  ch )
 
Theoremadantrl 462 Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ph  /\  ( th  /\  ps ) ) 
 ->  ch )
 
Theoremadantrr 463 Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ph  /\  ( ps  /\  th ) ) 
 ->  ch )
 
Theoremadantlll 464 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 2-Dec-2012.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ( ( ta  /\  ph )  /\  ps )  /\  ch )  ->  th )
 
Theoremadantllr 465 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ( (
 ph  /\  ta )  /\  ps )  /\  ch )  ->  th )
 
Theoremadantlrl 466 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ( ph  /\  ( ta  /\  ps ) )  /\  ch )  ->  th )
 
Theoremadantlrr 467 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ( ph  /\  ( ps  /\  ta ) )  /\  ch )  ->  th )
 
Theoremadantrll 468 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ph  /\  (
 ( ta  /\  ps )  /\  ch ) ) 
 ->  th )
 
Theoremadantrlr 469 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ph  /\  (
 ( ps  /\  ta )  /\  ch ) ) 
 ->  th )
 
Theoremadantrrl 470 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ( ps  /\  ( ta  /\  ch ) ) )  ->  th )
 
Theoremadantrrr 471 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ( ps  /\  ( ch  /\  ta ) ) )  ->  th )
 
Theoremad2antrr 472 Deduction adding two conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.) (Proof shortened by Wolf Lammen, 20-Nov-2012.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ph  /\  ch )  /\  th )  ->  ps )
 
Theoremad2antlr 473 Deduction adding two conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.) (Proof shortened by Wolf Lammen, 20-Nov-2012.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ch  /\  ph )  /\  th )  ->  ps )
 
Theoremad2antrl 474 Deduction adding two conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ch  /\  ( ph  /\  th ) ) 
 ->  ps )
 
Theoremad2antll 475 Deduction adding conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ch  /\  ( th  /\  ph ) )  ->  ps )
 
Theoremad3antrrr 476 Deduction adding three conjuncts to antecedent. (Contributed by NM, 28-Jul-2012.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ( ph  /\ 
 ch )  /\  th )  /\  ta )  ->  ps )
 
Theoremad3antlr 477 Deduction adding three conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ( ch 
 /\  ph )  /\  th )  /\  ta )  ->  ps )
 
Theoremad4antr 478 Deduction adding 4 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ( (
 ph  /\  ch )  /\  th )  /\  ta )  /\  et )  ->  ps )
 
Theoremad4antlr 479 Deduction adding 4 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ( ( ch  /\  ph )  /\  th )  /\  ta )  /\  et )  ->  ps )
 
Theoremad5antr 480 Deduction adding 5 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ( ( ( ph  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  ->  ps )
 
Theoremad5antlr 481 Deduction adding 5 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ( ( ( ch  /\  ph )  /\  th )  /\  ta )  /\  et )  /\  ze )  ->  ps )
 
Theoremad6antr 482 Deduction adding 6 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ( ( ( ( ph  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  ->  ps )
 
Theoremad6antlr 483 Deduction adding 6 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ( ( ( ( ch  /\  ph )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  ->  ps )
 
Theoremad7antr 484 Deduction adding 7 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ( ( ( ( ( ph  /\ 
 ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  ->  ps )
 
Theoremad7antlr 485 Deduction adding 7 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ( ( ( ( ( ch 
 /\  ph )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  ->  ps )
 
Theoremad8antr 486 Deduction adding 8 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ( ( ( ( ( (
 ph  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  ->  ps )
 
Theoremad8antlr 487 Deduction adding 8 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ( ( ( ( ( ( ch  /\  ph )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  ->  ps )
 
Theoremad9antr 488 Deduction adding 9 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ( ( ( ( ( ( ( ph  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  /\  la )  ->  ps )
 
Theoremad9antlr 489 Deduction adding 9 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ( ( ( ( ( ( ( ch  /\  ph )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  /\  la )  ->  ps )
 
Theoremad10antr 490 Deduction adding 10 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ( ( ( ( ( ( ( ( ph  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  /\  la )  /\  ka )  ->  ps )
 
Theoremad10antlr 491 Deduction adding 10 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ( ( ( ( ( ( ( ( ch  /\  ph )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  /\  la )  /\  ka )  ->  ps )
 
Theoremad2ant2l 492 Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( th  /\  ph )  /\  ( ta 
 /\  ps ) )  ->  ch )
 
Theoremad2ant2r 493 Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ph  /\ 
 th )  /\  ( ps  /\  ta ) ) 
 ->  ch )
 
Theoremad2ant2lr 494 Deduction adding two conjuncts to antecedent. (Contributed by NM, 23-Nov-2007.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( th  /\  ph )  /\  ( ps 
 /\  ta ) )  ->  ch )
 
Theoremad2ant2rl 495 Deduction adding two conjuncts to antecedent. (Contributed by NM, 24-Nov-2007.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ph  /\ 
 th )  /\  ( ta  /\  ps ) ) 
 ->  ch )
 
Theoremsimpll 496 Simplification of a conjunction. (Contributed by NM, 18-Mar-2007.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  ph )
 
Theoremsimplr 497 Simplification of a conjunction. (Contributed by NM, 20-Mar-2007.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  ps )
 
Theoremsimprl 498 Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  ps )
 
Theoremsimprr 499 Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  ch )
 
Theoremsimplll 500 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
 |-  ( ( ( (
 ph  /\  ps )  /\  ch )  /\  th )  ->  ph )
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