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Theorem List for Intuitionistic Logic Explorer - 301-400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremjcad 301 Deduction conjoining the consequents of two implications. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 23-Jul-2013.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  th ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  /\  th ) ) )
 
Theoremjca31 302 Join three consequents. (Contributed by Jeff Hankins, 1-Aug-2009.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   =>    |-  ( ph  ->  (
 ( ps  /\  ch )  /\  th ) )
 
Theoremjca32 303 Join three consequents. (Contributed by FL, 1-Aug-2009.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   =>    |-  ( ph  ->  ( ps  /\  ( ch  /\  th ) ) )
 
Theoremjcai 304 Deduction replacing implication with conjunction. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  /\  ch ) )
 
Theoremjctil 305 Inference conjoining a theorem to left of consequent in an implication. (Contributed by NM, 31-Dec-1993.)
 |-  ( ph  ->  ps )   &    |-  ch   =>    |-  ( ph  ->  ( ch  /\  ps ) )
 
Theoremjctir 306 Inference conjoining a theorem to right of consequent in an implication. (Contributed by NM, 31-Dec-1993.)
 |-  ( ph  ->  ps )   &    |-  ch   =>    |-  ( ph  ->  ( ps  /\  ch ) )
 
Theoremjctl 307 Inference conjoining a theorem to the left of a consequent. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 24-Oct-2012.)
 |- 
 ps   =>    |-  ( ph  ->  ( ps  /\  ph ) )
 
Theoremjctr 308 Inference conjoining a theorem to the right of a consequent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 24-Oct-2012.)
 |- 
 ps   =>    |-  ( ph  ->  ( ph  /\  ps ) )
 
Theoremjctild 309 Deduction conjoining a theorem to left of consequent in an implication. (Contributed by NM, 21-Apr-2005.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  th )   =>    |-  ( ph  ->  ( ps  ->  ( th  /\  ch ) ) )
 
Theoremjctird 310 Deduction conjoining a theorem to right of consequent in an implication. (Contributed by NM, 21-Apr-2005.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  th )   =>    |-  ( ph  ->  ( ps  ->  ( ch  /\  th ) ) )
 
Theoremancl 311 Conjoin antecedent to left of consequent. (Contributed by NM, 15-Aug-1994.)
 |-  ( ( ph  ->  ps )  ->  ( ph  ->  ( ph  /\  ps ) ) )
 
Theoremanclb 312 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.)
 |-  ( ( ph  ->  ps )  <->  ( ph  ->  (
 ph  /\  ps )
 ) )
 
Theorempm5.42 313 Theorem *5.42 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 ) 
 <->  ( ph  ->  ( ps  ->  ( ph  /\  ch ) ) ) )
 
Theoremancr 314 Conjoin antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)
 |-  ( ( ph  ->  ps )  ->  ( ph  ->  ( ps  /\  ph )
 ) )
 
Theoremancrb 315 Conjoin antecedent to right of consequent. (Contributed by NM, 25-Jul-1999.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
 |-  ( ( ph  ->  ps )  <->  ( ph  ->  ( ps  /\  ph )
 ) )
 
Theoremancli 316 Deduction conjoining antecedent to left of consequent. (Contributed by NM, 12-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ph  /\  ps ) )
 
Theoremancri 317 Deduction conjoining antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ps  /\  ph ) )
 
Theoremancld 318 Deduction conjoining antecedent to left of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 1-Nov-2012.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  ( ps  /\ 
 ch ) ) )
 
Theoremancrd 319 Deduction conjoining antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 1-Nov-2012.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  /\ 
 ps ) ) )
 
Theoremanc2l 320 Conjoin antecedent to left of consequent in nested implication. (Contributed by NM, 10-Aug-1994.) (Proof shortened by Wolf Lammen, 14-Jul-2013.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 )  ->  ( ph  ->  ( ps  ->  ( ph  /\  ch ) ) ) )
 
Theoremanc2r 321 Conjoin antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 )  ->  ( ph  ->  ( ps  ->  ( ch  /\  ph ) ) ) )
 
Theoremanc2li 322 Deduction conjoining antecedent to left of consequent in nested implication. (Contributed by NM, 10-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  ( ph  /\ 
 ch ) ) )
 
Theoremanc2ri 323 Deduction conjoining antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  /\  ph ) ) )
 
Theorempm3.41 324 Theorem *3.41 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  ->  ch )  ->  ( ( ph  /\  ps )  ->  ch ) )
 
Theorempm3.42 325 Theorem *3.42 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ps  ->  ch )  ->  ( ( ph  /\  ps )  ->  ch ) )
 
Theorempm3.4 326 Conjunction implies implication. Theorem *3.4 of [WhiteheadRussell] p. 113. (Contributed by NM, 31-Jul-1995.)
 |-  ( ( ph  /\  ps )  ->  ( ph  ->  ps ) )
 
Theorempm4.45im 327 Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 17-May-1998.)
 |-  ( ph  <->  ( ph  /\  ( ps  ->  ph ) ) )
 
Theoremanim12d 328 Conjoin antecedents and consequents in a deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 18-Dec-2013.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( th  ->  ta ) )   =>    |-  ( ph  ->  (
 ( ps  /\  th )  ->  ( ch  /\  ta ) ) )
 
Theoremanim1d 329 Add a conjunct to right of antecedent and consequent in a deduction. (Contributed by NM, 3-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ( ps  /\  th )  ->  ( ch  /\  th ) ) )
 
Theoremanim2d 330 Add a conjunct to left of antecedent and consequent in a deduction. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ( th  /\  ps )  ->  ( th  /\  ch ) ) )
 
Theoremanim12i 331 Conjoin antecedents and consequents of two premises. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Dec-2013.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  th )   =>    |-  ( ( ph  /\  ch )  ->  ( ps  /\  th ) )
 
Theoremanim12ci 332 Variant of anim12i 331 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  th )   =>    |-  ( ( ph  /\  ch )  ->  ( th  /\  ps ) )
 
Theoremanim1i 333 Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ph  /\  ch )  ->  ( ps  /\  ch ) )
 
Theoremanim2i 334 Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ch  /\  ph )  ->  ( ch  /\  ps ) )
 
Theoremanim12ii 335 Conjoin antecedents and consequents in a deduction. (Contributed by NM, 11-Nov-2007.) (Proof shortened by Wolf Lammen, 19-Jul-2013.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  ( ps  ->  ta ) )   =>    |-  ( ( ph  /\  th )  ->  ( ps  ->  ( ch  /\  ta )
 ) )
 
Theoremprth 336 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.)
 |-  ( ( ( ph  ->  ps )  /\  ( ch  ->  th ) )  ->  ( ( ph  /\  ch )  ->  ( ps  /\  th ) ) )
 
Theorempm3.33 337 Theorem *3.33 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  ->  ps )  /\  ( ps  ->  ch ) )  ->  ( ph  ->  ch )
 )
 
Theorempm3.34 338 Theorem *3.34 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ps 
 ->  ch )  /\  ( ph  ->  ps ) )  ->  ( ph  ->  ch )
 )
 
Theorempm3.35 339 Conjunctive detachment. Theorem *3.35 of [WhiteheadRussell] p. 112. (Contributed by NM, 14-Dec-2002.)
 |-  ( ( ph  /\  ( ph  ->  ps ) )  ->  ps )
 
Theorempm5.31 340 Theorem *5.31 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ch  /\  ( ph  ->  ps )
 )  ->  ( ph  ->  ( ps  /\  ch ) ) )
 
Theoremimp4a 341 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ph  ->  ( ps  ->  ( ( ch  /\  th )  ->  ta ) ) )
 
Theoremimp4b 342 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  (
 ( ch  /\  th )  ->  ta ) )
 
Theoremimp4c 343 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ph  ->  ( ( ( ps  /\  ch )  /\  th )  ->  ta ) )
 
Theoremimp4d 344 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ph  ->  ( ( ps  /\  ( ch  /\  th ) ) 
 ->  ta ) )
 
Theoremimp41 345 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ( ( ( ph  /\  ps )  /\  ch )  /\  th )  ->  ta )
 
Theoremimp42 346 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ( (
 ph  /\  ( ps  /\ 
 ch ) )  /\  th )  ->  ta )
 
Theoremimp43 347 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ( (
 ph  /\  ps )  /\  ( ch  /\  th ) )  ->  ta )
 
Theoremimp44 348 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ( ph  /\  ( ( ps  /\  ch )  /\  th )
 )  ->  ta )
 
Theoremimp45 349 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ( ph  /\  ( ps  /\  ( ch  /\  th ) ) )  ->  ta )
 
Theoremimp5a 350 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ( th  /\  ta )  ->  et ) ) ) )
 
Theoremimp5d 351 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )   =>    |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  ( ( th  /\ 
 ta )  ->  et )
 )
 
Theoremimp5g 352 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )   =>    |-  ( ( ph  /\  ps )  ->  ( ( ( ch  /\  th )  /\  ta )  ->  et )
 )
 
Theoremimp55 353 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )   =>    |-  ( ( ( ph  /\  ( ps  /\  ( ch  /\  th ) ) )  /\  ta )  ->  et )
 
Theoremimp511 354 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )   =>    |-  ( ( ph  /\  (
 ( ps  /\  ( ch  /\  th ) ) 
 /\  ta ) )  ->  et )
 
Theoremexpimpd 355 Exportation followed by a deduction version of importation. (Contributed by NM, 6-Sep-2008.)
 |-  ( ( ph  /\  ps )  ->  ( ch  ->  th ) )   =>    |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )
 
Theoremexp31 356 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
 
Theoremexp32 357 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
 
Theoremexp4a 358 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ( ch 
 /\  th )  ->  ta )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexp4b 359 An exportation inference. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)
 |-  ( ( ph  /\  ps )  ->  ( ( ch 
 /\  th )  ->  ta )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexp4c 360 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  (
 ( ( ps  /\  ch )  /\  th )  ->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexp4d 361 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  (
 ( ps  /\  ( ch  /\  th ) ) 
 ->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexp41 362 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ( ( (
 ph  /\  ps )  /\  ch )  /\  th )  ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexp42 363 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ( ( ph  /\  ( ps  /\  ch ) )  /\  th )  ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexp43 364 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  th ) ) 
 ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexp44 365 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ( ph  /\  (
 ( ps  /\  ch )  /\  th ) ) 
 ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexp45 366 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ( ph  /\  ( ps  /\  ( ch  /\  th ) ) )  ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexpr 367 Export a wff from a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ps )  ->  ( ch  ->  th ) )
 
Theoremexp5c 368 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  ( ( th  /\ 
 ta )  ->  et )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp53 369 An exportation inference. (Contributed by Jeff Hankins, 30-Aug-2009.)
 |-  ( ( ( (
 ph  /\  ps )  /\  ( ch  /\  th ) )  /\  ta )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexpl 370 Export a wff from a left conjunct. (Contributed by Jeff Hankins, 28-Aug-2009.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )
 
Theoremimpr 371 Import a wff into a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)
 |-  ( ( ph  /\  ps )  ->  ( ch  ->  th ) )   =>    |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )
 
Theoremimpl 372 Export a wff from a left conjunct. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   =>    |-  ( ( (
 ph  /\  ps )  /\  ch )  ->  th )
 
Theoremimpac 373 Importation with conjunction in consequent. (Contributed by NM, 9-Aug-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ( ch  /\  ps ) )
 
Theoremexbiri 374 Inference form of exbir 1368. (Contributed by Alan Sare, 31-Dec-2011.) (Proof shortened by Wolf Lammen, 27-Jan-2013.)
 |-  ( ( ph  /\  ps )  ->  ( ch  <->  th ) )   =>    |-  ( ph  ->  ( ps  ->  ( th  ->  ch ) ) )
 
Theoremsimprbda 375 Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)
 |-  ( ph  ->  ( ps 
 <->  ( ch  /\  th ) ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ch )
 
Theoremsimplbda 376 Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)
 |-  ( ph  ->  ( ps 
 <->  ( ch  /\  th ) ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  th )
 
Theoremsimplbi2 377 Deduction eliminating a conjunct. (Contributed by Alan Sare, 31-Dec-2011.)
 |-  ( ph  <->  ( ps  /\  ch ) )   =>    |-  ( ps  ->  ( ch  ->  ph ) )
 
Theoremsimpl2im 378 Implication from an eliminated conjunct implied by the antecedent. (Contributed by BJ/AV, 5-Apr-2021.)
 |-  ( ph  ->  ( ps  /\  ch ) )   &    |-  ( ch  ->  th )   =>    |-  ( ph  ->  th )
 
Theoremsimplbiim 379 Implication from an eliminated conjunct equivalent to the antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  <->  ( ps  /\  ch ) )   &    |-  ( ch  ->  th )   =>    |-  ( ph  ->  th )
 
Theoremdfbi2 380 A theorem similar to the standard definition of the biconditional. Definition of [Margaris] p. 49. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 31-Jan-2015.)
 |-  ( ( ph  <->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph ) ) )
 
Theorempm4.71 381 Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 2-Dec-2012.)
 |-  ( ( ph  ->  ps )  <->  ( ph  <->  ( ph  /\  ps ) ) )
 
Theorempm4.71r 382 Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct reversed). (Contributed by NM, 25-Jul-1999.)
 |-  ( ( ph  ->  ps )  <->  ( ph  <->  ( ps  /\  ph ) ) )
 
Theorempm4.71i 383 Inference converting an implication to a biconditional with conjunction. Inference from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 4-Jan-2004.)
 |-  ( ph  ->  ps )   =>    |-  ( ph 
 <->  ( ph  /\  ps ) )
 
Theorempm4.71ri 384 Inference converting an implication to a biconditional with conjunction. Inference from Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct reversed). (Contributed by NM, 1-Dec-2003.)
 |-  ( ph  ->  ps )   =>    |-  ( ph 
 <->  ( ps  /\  ph )
 )
 
Theorempm4.71d 385 Deduction converting an implication to a biconditional with conjunction. Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  <->  ( ps  /\  ch ) ) )
 
Theorempm4.71rd 386 Deduction converting an implication to a biconditional with conjunction. Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 10-Feb-2005.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  <->  ( ch  /\  ps ) ) )
 
Theorempm4.24 387 Theorem *4.24 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 14-Mar-2014.)
 |-  ( ph  <->  ( ph  /\  ph )
 )
 
Theoremanidm 388 Idempotent law for conjunction. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Mar-2014.)
 |-  ( ( ph  /\  ph )  <->  ph )
 
Theoremanidms 389 Inference from idempotent law for conjunction. (Contributed by NM, 15-Jun-1994.)
 |-  ( ( ph  /\  ph )  ->  ps )   =>    |-  ( ph  ->  ps )
 
Theoremanidmdbi 390 Conjunction idempotence with antecedent. (Contributed by Roy F. Longton, 8-Aug-2005.)
 |-  ( ( ph  ->  ( ps  /\  ps )
 ) 
 <->  ( ph  ->  ps )
 )
 
Theoremanasss 391 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )
 
Theoremanassrs 392 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )
 
Theoremanass 393 Associative law for conjunction. Theorem *4.32 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch ) 
 <->  ( ph  /\  ( ps  /\  ch ) ) )
 
Theoremsylanl1 394 A syllogism inference. (Contributed by NM, 10-Mar-2005.)
 |-  ( ph  ->  ps )   &    |-  (
 ( ( ps  /\  ch )  /\  th )  ->  ta )   =>    |-  ( ( ( ph  /\ 
 ch )  /\  th )  ->  ta )
 
Theoremsylanl2 395 A syllogism inference. (Contributed by NM, 1-Jan-2005.)
 |-  ( ph  ->  ch )   &    |-  (
 ( ( ps  /\  ch )  /\  th )  ->  ta )   =>    |-  ( ( ( ps 
 /\  ph )  /\  th )  ->  ta )
 
Theoremsylanr1 396 A syllogism inference. (Contributed by NM, 9-Apr-2005.)
 |-  ( ph  ->  ch )   &    |-  (
 ( ps  /\  ( ch  /\  th ) ) 
 ->  ta )   =>    |-  ( ( ps  /\  ( ph  /\  th )
 )  ->  ta )
 
Theoremsylanr2 397 A syllogism inference. (Contributed by NM, 9-Apr-2005.)
 |-  ( ph  ->  th )   &    |-  (
 ( ps  /\  ( ch  /\  th ) ) 
 ->  ta )   =>    |-  ( ( ps  /\  ( ch  /\  ph )
 )  ->  ta )
 
Theoremsylani 398 A syllogism inference. (Contributed by NM, 2-May-1996.)
 |-  ( ph  ->  ch )   &    |-  ( ps  ->  ( ( ch 
 /\  th )  ->  ta )
 )   =>    |-  ( ps  ->  (
 ( ph  /\  th )  ->  ta ) )
 
Theoremsylan2i 399 A syllogism inference. (Contributed by NM, 1-Aug-1994.)
 |-  ( ph  ->  th )   &    |-  ( ps  ->  ( ( ch 
 /\  th )  ->  ta )
 )   =>    |-  ( ps  ->  (
 ( ch  /\  ph )  ->  ta ) )
 
Theoremsyl2ani 400 A syllogism inference. (Contributed by NM, 3-Aug-1999.)
 |-  ( ph  ->  ch )   &    |-  ( et  ->  th )   &    |-  ( ps  ->  ( ( ch  /\  th )  ->  ta ) )   =>    |-  ( ps  ->  ( ( ph  /\  et )  ->  ta ) )
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