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Theorem List for Intuitionistic Logic Explorer - 301-400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembiantrur 301 A wff is equivalent to its conjunction with truth. (Contributed by NM, 3-Aug-1994.)
 |-  ph   =>    |-  ( ps  <->  ( ph  /\  ps ) )
 
Theorembiantrud 302 A wff is equivalent to its conjunction with truth. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Wolf Lammen, 23-Oct-2013.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ch  <->  ( ch  /\  ps ) ) )
 
Theorembiantrurd 303 A wff is equivalent to its conjunction with truth. (Contributed by NM, 1-May-1995.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ch  <->  ( ps  /\  ch ) ) )
 
Theoremjca 304 Deduce conjunction of the consequents of two implications ("join consequents with 'and'"). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   =>    |-  ( ph  ->  ( ps  /\  ch ) )
 
Theoremjcad 305 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 ) ) )
 
Theoremjca2 306 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 12-Oct-2010.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ps  ->  th )   =>    |-  ( ph  ->  ( ps  ->  ( ch  /\  th ) ) )
 
Theoremjca31 307 Join three consequents. (Contributed by Jeff Hankins, 1-Aug-2009.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   =>    |-  ( ph  ->  (
 ( ps  /\  ch )  /\  th ) )
 
Theoremjca32 308 Join three consequents. (Contributed by FL, 1-Aug-2009.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   =>    |-  ( ph  ->  ( ps  /\  ( ch  /\  th ) ) )
 
Theoremjcai 309 Deduction replacing implication with conjunction. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  /\  ch ) )
 
Theoremjctil 310 Inference conjoining a theorem to left of consequent in an implication. (Contributed by NM, 31-Dec-1993.)
 |-  ( ph  ->  ps )   &    |-  ch   =>    |-  ( ph  ->  ( ch  /\  ps ) )
 
Theoremjctir 311 Inference conjoining a theorem to right of consequent in an implication. (Contributed by NM, 31-Dec-1993.)
 |-  ( ph  ->  ps )   &    |-  ch   =>    |-  ( ph  ->  ( ps  /\  ch ) )
 
Theoremjctl 312 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 313 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 314 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 315 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 316 Conjoin antecedent to left of consequent. (Contributed by NM, 15-Aug-1994.)
 |-  ( ( ph  ->  ps )  ->  ( ph  ->  ( ph  /\  ps ) ) )
 
Theoremanclb 317 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 )
 ) )
 
Theorempm5.42 318 Theorem *5.42 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 ) 
 <->  ( ph  ->  ( ps  ->  ( ph  /\  ch ) ) ) )
 
Theoremancr 319 Conjoin antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)
 |-  ( ( ph  ->  ps )  ->  ( ph  ->  ( ps  /\  ph )
 ) )
 
Theoremancrb 320 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 321 Deduction conjoining antecedent to left of consequent. (Contributed by NM, 12-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ph  /\  ps ) )
 
Theoremancri 322 Deduction conjoining antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ps  /\  ph ) )
 
Theoremancld 323 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 324 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 325 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 326 Conjoin antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 )  ->  ( ph  ->  ( ps  ->  ( ch  /\  ph ) ) ) )
 
Theoremanc2li 327 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 328 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 ) ) )
 
Theorempm3.41 329 Theorem *3.41 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  ->  ch )  ->  ( ( ph  /\  ps )  ->  ch ) )
 
Theorempm3.42 330 Theorem *3.42 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ps  ->  ch )  ->  ( ( ph  /\  ps )  ->  ch ) )
 
Theorempm3.4 331 Conjunction implies implication. Theorem *3.4 of [WhiteheadRussell] p. 113. (Contributed by NM, 31-Jul-1995.)
 |-  ( ( ph  /\  ps )  ->  ( ph  ->  ps ) )
 
Theorempm4.45im 332 Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 17-May-1998.)
 |-  ( ph  <->  ( ph  /\  ( ps  ->  ph ) ) )
 
Theoremanim12d 333 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 334 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 335 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 336 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 337 Variant of anim12i 336 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  th )   =>    |-  ( ( ph  /\  ch )  ->  ( th  /\  ps ) )
 
Theoremanim1i 338 Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ph  /\  ch )  ->  ( ps  /\  ch ) )
 
Theoremanim2i 339 Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ch  /\  ph )  ->  ( ch  /\  ps ) )
 
Theoremanim12ii 340 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 )
 ) )
 
Theoremanim12 341 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 ) ) )
 
Theorempm3.33 342 Theorem *3.33 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  ->  ps )  /\  ( ps  ->  ch ) )  ->  ( ph  ->  ch )
 )
 
Theorempm3.34 343 Theorem *3.34 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ps 
 ->  ch )  /\  ( ph  ->  ps ) )  ->  ( ph  ->  ch )
 )
 
Theorempm3.35 344 Conjunctive detachment. Theorem *3.35 of [WhiteheadRussell] p. 112. (Contributed by NM, 14-Dec-2002.)
 |-  ( ( ph  /\  ( ph  ->  ps ) )  ->  ps )
 
Theorempm5.31 345 Theorem *5.31 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ch  /\  ( ph  ->  ps )
 )  ->  ( ph  ->  ( ps  /\  ch ) ) )
 
Theoremimp4a 346 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ph  ->  ( ps  ->  ( ( ch  /\  th )  ->  ta ) ) )
 
Theoremimp4b 347 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  (
 ( ch  /\  th )  ->  ta ) )
 
Theoremimp4c 348 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ph  ->  ( ( ( ps  /\  ch )  /\  th )  ->  ta ) )
 
Theoremimp4d 349 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ph  ->  ( ( ps  /\  ( ch  /\  th ) ) 
 ->  ta ) )
 
Theoremimp41 350 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ( ( ( ph  /\  ps )  /\  ch )  /\  th )  ->  ta )
 
Theoremimp42 351 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ( (
 ph  /\  ( ps  /\ 
 ch ) )  /\  th )  ->  ta )
 
Theoremimp43 352 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ( (
 ph  /\  ps )  /\  ( ch  /\  th ) )  ->  ta )
 
Theoremimp44 353 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ( ph  /\  ( ( ps  /\  ch )  /\  th )
 )  ->  ta )
 
Theoremimp45 354 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ( ph  /\  ( ps  /\  ( ch  /\  th ) ) )  ->  ta )
 
Theoremimp5a 355 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ( th  /\  ta )  ->  et ) ) ) )
 
Theoremimp5d 356 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )   =>    |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  ( ( th  /\ 
 ta )  ->  et )
 )
 
Theoremimp5g 357 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )   =>    |-  ( ( ph  /\  ps )  ->  ( ( ( ch  /\  th )  /\  ta )  ->  et )
 )
 
Theoremimp55 358 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )   =>    |-  ( ( ( ph  /\  ( ps  /\  ( ch  /\  th ) ) )  /\  ta )  ->  et )
 
Theoremimp511 359 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )   =>    |-  ( ( ph  /\  (
 ( ps  /\  ( ch  /\  th ) ) 
 /\  ta ) )  ->  et )
 
Theoremexpimpd 360 Exportation followed by a deduction version of importation. (Contributed by NM, 6-Sep-2008.)
 |-  ( ( ph  /\  ps )  ->  ( ch  ->  th ) )   =>    |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )
 
Theoremexp31 361 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
 
Theoremexp32 362 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
 
Theoremexp4a 363 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ( ch 
 /\  th )  ->  ta )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexp4b 364 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 365 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  (
 ( ( ps  /\  ch )  /\  th )  ->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexp4d 366 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  (
 ( ps  /\  ( ch  /\  th ) ) 
 ->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexp41 367 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ( ( (
 ph  /\  ps )  /\  ch )  /\  th )  ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexp42 368 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ( ( ph  /\  ( ps  /\  ch ) )  /\  th )  ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexp43 369 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  th ) ) 
 ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexp44 370 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ( ph  /\  (
 ( ps  /\  ch )  /\  th ) ) 
 ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexp45 371 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ( ph  /\  ( ps  /\  ( ch  /\  th ) ) )  ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexpr 372 Export a wff from a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ps )  ->  ( ch  ->  th ) )
 
Theoremexp5c 373 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  ( ( th  /\ 
 ta )  ->  et )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp53 374 An exportation inference. (Contributed by Jeff Hankins, 30-Aug-2009.)
 |-  ( ( ( (
 ph  /\  ps )  /\  ( ch  /\  th ) )  /\  ta )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexpl 375 Export a wff from a left conjunct. (Contributed by Jeff Hankins, 28-Aug-2009.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )
 
Theoremimpr 376 Import a wff into a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)
 |-  ( ( ph  /\  ps )  ->  ( ch  ->  th ) )   =>    |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )
 
Theoremimpl 377 Export a wff from a left conjunct. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   =>    |-  ( ( (
 ph  /\  ps )  /\  ch )  ->  th )
 
Theoremimpac 378 Importation with conjunction in consequent. (Contributed by NM, 9-Aug-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ( ch  /\  ps ) )
 
Theoremexbiri 379 Inference form of exbir 1397. (Contributed by Alan Sare, 31-Dec-2011.) (Proof shortened by Wolf Lammen, 27-Jan-2013.)
 |-  ( ( ph  /\  ps )  ->  ( ch  <->  th ) )   =>    |-  ( ph  ->  ( ps  ->  ( th  ->  ch ) ) )
 
Theoremsimprbda 380 Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)
 |-  ( ph  ->  ( ps 
 <->  ( ch  /\  th ) ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ch )
 
Theoremsimplbda 381 Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)
 |-  ( ph  ->  ( ps 
 <->  ( ch  /\  th ) ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  th )
 
Theoremsimplbi2 382 Deduction eliminating a conjunct. (Contributed by Alan Sare, 31-Dec-2011.)
 |-  ( ph  <->  ( ps  /\  ch ) )   =>    |-  ( ps  ->  ( ch  ->  ph ) )
 
Theoremsimpl2im 383 Implication from an eliminated conjunct implied by the antecedent. (Contributed by BJ/AV, 5-Apr-2021.)
 |-  ( ph  ->  ( ps  /\  ch ) )   &    |-  ( ch  ->  th )   =>    |-  ( ph  ->  th )
 
Theoremsimplbiim 384 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 385 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 ) ) )
 
Theorempm4.71 386 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 387 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 388 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 389 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 390 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 391 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 ) ) )
 
Theorempm4.24 392 Theorem *4.24 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 14-Mar-2014.)
 |-  ( ph  <->  ( ph  /\  ph )
 )
 
Theoremanidm 393 Idempotent law for conjunction. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Mar-2014.)
 |-  ( ( ph  /\  ph )  <->  ph )
 
Theoremanidms 394 Inference from idempotent law for conjunction. (Contributed by NM, 15-Jun-1994.)
 |-  ( ( ph  /\  ph )  ->  ps )   =>    |-  ( ph  ->  ps )
 
Theoremanidmdbi 395 Conjunction idempotence with antecedent. (Contributed by Roy F. Longton, 8-Aug-2005.)
 |-  ( ( ph  ->  ( ps  /\  ps )
 ) 
 <->  ( ph  ->  ps )
 )
 
Theoremanasss 396 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )
 
Theoremanassrs 397 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )
 
Theoremanass 398 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 399 A syllogism inference. (Contributed by NM, 10-Mar-2005.)
 |-  ( ph  ->  ps )   &    |-  (
 ( ( ps  /\  ch )  /\  th )  ->  ta )   =>    |-  ( ( ( ph  /\ 
 ch )  /\  th )  ->  ta )
 
Theoremsylanl2 400 A syllogism inference. (Contributed by NM, 1-Jan-2005.)
 |-  ( ph  ->  ch )   &    |-  (
 ( ( ps  /\  ch )  /\  th )  ->  ta )   =>    |-  ( ( ( ps 
 /\  ph )  /\  th )  ->  ta )
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