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Theorem List for Intuitionistic Logic Explorer - 401-500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsylanr1 401 A syllogism inference. (Contributed by NM, 9-Apr-2005.)
 |-  ( ph  ->  ch )   &    |-  (
 ( ps  /\  ( ch  /\  th ) ) 
 ->  ta )   =>    |-  ( ( ps  /\  ( ph  /\  th )
 )  ->  ta )
 
Theoremsylanr2 402 A syllogism inference. (Contributed by NM, 9-Apr-2005.)
 |-  ( ph  ->  th )   &    |-  (
 ( ps  /\  ( ch  /\  th ) ) 
 ->  ta )   =>    |-  ( ( ps  /\  ( ch  /\  ph )
 )  ->  ta )
 
Theoremsylani 403 A syllogism inference. (Contributed by NM, 2-May-1996.)
 |-  ( ph  ->  ch )   &    |-  ( ps  ->  ( ( ch 
 /\  th )  ->  ta )
 )   =>    |-  ( ps  ->  (
 ( ph  /\  th )  ->  ta ) )
 
Theoremsylan2i 404 A syllogism inference. (Contributed by NM, 1-Aug-1994.)
 |-  ( ph  ->  th )   &    |-  ( ps  ->  ( ( ch 
 /\  th )  ->  ta )
 )   =>    |-  ( ps  ->  (
 ( ch  /\  ph )  ->  ta ) )
 
Theoremsyl2ani 405 A syllogism inference. (Contributed by NM, 3-Aug-1999.)
 |-  ( ph  ->  ch )   &    |-  ( et  ->  th )   &    |-  ( ps  ->  ( ( ch  /\  th )  ->  ta ) )   =>    |-  ( ps  ->  ( ( ph  /\  et )  ->  ta ) )
 
Theoremsylan9 406 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 407 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  ( ch  ->  ta ) )   =>    |-  ( ( th  /\  ph )  ->  ( ps  ->  ta ) )
 
Theoremsyl2anc 408 Syllogism inference combined with contraction. (Contributed by NM, 16-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  ( ph  ->  th )
 
Theoremsylancl 409 Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ph  ->  ps )   &    |-  ch   &    |-  (
 ( ps  /\  ch )  ->  th )   =>    |-  ( ph  ->  th )
 
Theoremsylancr 410 Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |- 
 ps   &    |-  ( ph  ->  ch )   &    |-  (
 ( ps  /\  ch )  ->  th )   =>    |-  ( ph  ->  th )
 
Theoremsylanblc 411 Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019.)
 |-  ( ph  ->  ps )   &    |-  ch   &    |-  (
 ( ps  /\  ch ) 
 <-> 
 th )   =>    |-  ( ph  ->  th )
 
Theoremsylanblrc 412 Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019.)
 |-  ( ph  ->  ps )   &    |-  ch   &    |-  ( th 
 <->  ( ps  /\  ch ) )   =>    |-  ( ph  ->  th )
 
Theoremsylanbrc 413 Syllogism inference. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( th  <->  ( ps  /\  ch ) )   =>    |-  ( ph  ->  th )
 
Theoremsylancb 414 A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)
 |-  ( ph  <->  ps )   &    |-  ( ph  <->  ch )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  ( ph  ->  th )
 
Theoremsylancbr 415 A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)
 |-  ( ps  <->  ph )   &    |-  ( ch  <->  ph )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  ( ph  ->  th )
 
Theoremsylancom 416 Syllogism inference with commutation of antecents. (Contributed by NM, 2-Jul-2008.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ch 
 /\  ps )  ->  th )   =>    |-  (
 ( ph  /\  ps )  ->  th )
 
Theoremmpdan 417 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 418 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 419 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 420 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 421 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 422 An inference based on modus ponens. (Contributed by NM, 13-Apr-1995.)
 |-  ph   &    |- 
 ps   &    |-  ( ( ph  /\  ps )  ->  ch )   =>    |- 
 ch
 
Theoremmp4an 423 An inference based on modus ponens. (Contributed by Jeff Madsen, 15-Jun-2011.)
 |-  ph   &    |- 
 ps   &    |- 
 ch   &    |- 
 th   &    |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  th ) ) 
 ->  ta )   =>    |- 
 ta
 
Theoremmpan2d 424 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
 |-  ( ph  ->  ch )   &    |-  ( ph  ->  ( ( ps 
 /\  ch )  ->  th )
 )   =>    |-  ( ph  ->  ( ps  ->  th ) )
 
Theoremmpand 425 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 426 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 427 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 428 An inference based on modus ponens. (Contributed by NM, 12-Dec-2004.)
 |- 
 ps   &    |- 
 ch   &    |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   =>    |-  ( ph  ->  th )
 
Theoremmp2and 429 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  ( ( ps  /\  ch )  ->  th ) )   =>    |-  ( ph  ->  th )
 
Theoremmpanl1 430 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 431 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 432 An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)
 |-  ph   &    |- 
 ps   &    |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ch  ->  th )
 
Theoremmpanr1 433 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 434 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 435 An inference based on modus ponens. (Contributed by NM, 24-Jul-2009.)
 |- 
 ps   &    |- 
 ch   &    |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ph  ->  th )
 
Theoremmpanlr1 436 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 )
 
Theoremmpbirand 437 Detach truth from conjunction in biconditional. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
 |-  ( ph  ->  ch )   &    |-  ( ph  ->  ( ps  <->  ( ch  /\  th ) ) )   =>    |-  ( ph  ->  ( ps  <->  th ) )
 
Theoremmpbiran2d 438 Detach truth from conjunction in biconditional. Deduction form. (Contributed by Peter Mazsa, 24-Sep-2022.)
 |-  ( ph  ->  th )   &    |-  ( ph  ->  ( ps  <->  ( ch  /\  th ) ) )   =>    |-  ( ph  ->  ( ps  <->  ch ) )
 
Theorempm5.74da 439 Distribution of implication over biconditional (deduction form). (Contributed by NM, 4-May-2007.)
 |-  ( ( ph  /\  ps )  ->  ( ch  <->  th ) )   =>    |-  ( ph  ->  ( ( ps  ->  ch )  <->  ( ps  ->  th )
 ) )
 
Theoremimdistan 440 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 441 Distribution of implication with conjunction. (Contributed by NM, 1-Aug-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ( ph  /\  ch ) )
 
Theoremimdistanri 442 Distribution of implication with conjunction. (Contributed by NM, 8-Jan-2002.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ( ps 
 /\  ph )  ->  ( ch  /\  ph ) )
 
Theoremimdistand 443 Distribution of implication with conjunction (deduction form). (Contributed by NM, 27-Aug-2004.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   =>    |-  ( ph  ->  ( ( ps  /\  ch )  ->  ( ps  /\  th ) ) )
 
Theoremimdistanda 444 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 445 Distribution of implication over biconditional (deduction form). (Contributed by NM, 29-Oct-1996.) (Revised by NM, 31-Jan-2015.)
 |-  ( ph  ->  ( ps  ->  ( ch  <->  th ) ) )   =>    |-  ( ph  ->  ( ( ps  /\  ch )  <->  ( ps  /\  th ) ) )
 
Theorempm5.32rd 446 Distribution of implication over biconditional (deduction form). (Contributed by NM, 25-Dec-2004.)
 |-  ( ph  ->  ( ps  ->  ( ch  <->  th ) ) )   =>    |-  ( ph  ->  ( ( ch  /\  ps )  <->  ( th  /\  ps ) ) )
 
Theorempm5.32da 447 Distribution of implication over biconditional (deduction form). (Contributed by NM, 9-Dec-2006.)
 |-  ( ( ph  /\  ps )  ->  ( ch  <->  th ) )   =>    |-  ( ph  ->  ( ( ps  /\  ch ) 
 <->  ( ps  /\  th ) ) )
 
Theorempm5.32 448 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 449 Distribution of implication over biconditional (inference form). (Contributed by NM, 1-Aug-1994.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  <->  ( ph  /\  ch ) )
 
Theorempm5.32ri 450 Distribution of implication over biconditional (inference form). (Contributed by NM, 12-Mar-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ( ps 
 /\  ph )  <->  ( ch  /\  ph ) )
 
Theorembiadan2 451 Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.)
 |-  ( ph  ->  ps )   &    |-  ( ps  ->  ( ph  <->  ch ) )   =>    |-  ( ph  <->  ( ps  /\  ch ) )
 
Theoremanbi2i 452 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 453 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 454 Variant of anbi2i 452 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 455 Conjoin both sides of two equivalences. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ( ph  /\  ch ) 
 <->  ( ps  /\  th ) )
 
Theoremanbi12ci 456 Variant of anbi12i 455 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ( ph  /\  ch ) 
 <->  ( th  /\  ps ) )
 
Theoremsylan9bb 457 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( th  ->  ( ch  <->  ta ) )   =>    |-  ( ( ph  /\ 
 th )  ->  ( ps 
 <->  ta ) )
 
Theoremsylan9bbr 458 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( th  ->  ( ch  <->  ta ) )   =>    |-  ( ( th  /\  ph )  ->  ( ps  <->  ta ) )
 
Theoremanbi2d 459 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 460 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 461 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 462 Introduce a left conjunct to both sides of a logical equivalence. (Contributed by NM, 16-Nov-2013.)
 |-  ( ( ph  <->  ps )  ->  (
 ( ch  /\  ph )  <->  ( ch  /\  ps )
 ) )
 
Theorembitr 463 Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  <->  ps )  /\  ( ps  <->  ch ) )  ->  ( ph  <->  ch ) )
 
Theoremanbi12d 464 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 465 Modus ponens mixed with several conjunctions. (Contributed by Jim Kingdon, 7-Jan-2018.)
 |-  ( ( ( (
 ph  ->  ps )  /\  ch )  /\  ph )  ->  ( ps  /\  ch ) )
 
Theorempm5.3 466 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 467 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 468 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 469 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 470 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 471 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 472 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 473 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 474 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 475 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 476 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 477 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 478 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 479 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 480 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 481 Deduction adding two conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ch  /\  ( ph  /\  th ) ) 
 ->  ps )
 
Theoremad2antll 482 Deduction adding conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ch  /\  ( th  /\  ph ) )  ->  ps )
 
Theoremad3antrrr 483 Deduction adding three conjuncts to antecedent. (Contributed by NM, 28-Jul-2012.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ( ph  /\ 
 ch )  /\  th )  /\  ta )  ->  ps )
 
Theoremad3antlr 484 Deduction adding three conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ( ch 
 /\  ph )  /\  th )  /\  ta )  ->  ps )
 
Theoremad4antr 485 Deduction adding 4 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ( (
 ph  /\  ch )  /\  th )  /\  ta )  /\  et )  ->  ps )
 
Theoremad4antlr 486 Deduction adding 4 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ( ( ch  /\  ph )  /\  th )  /\  ta )  /\  et )  ->  ps )
 
Theoremad5antr 487 Deduction adding 5 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ( ( ( ph  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  ->  ps )
 
Theoremad5antlr 488 Deduction adding 5 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ( ( ( ch  /\  ph )  /\  th )  /\  ta )  /\  et )  /\  ze )  ->  ps )
 
Theoremad6antr 489 Deduction adding 6 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ( ( ( ( ph  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  ->  ps )
 
Theoremad6antlr 490 Deduction adding 6 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ( ( ( ( ( ch  /\  ph )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  ->  ps )
 
Theoremad7antr 491 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 492 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 493 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 494 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 495 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 496 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 497 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 498 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 499 Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( th  /\  ph )  /\  ( ta 
 /\  ps ) )  ->  ch )
 
Theoremad2ant2r 500 Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ph  /\ 
 th )  /\  ( ps  /\  ta ) ) 
 ->  ch )
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