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Theorem List for Intuitionistic Logic Explorer - 301-400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremjctl 301 Inference conjoining a theorem to the left of a consequent. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 24-Oct-2012.)
𝜓       (𝜑 → (𝜓𝜑))
 
Theoremjctr 302 Inference conjoining a theorem to the right of a consequent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 24-Oct-2012.)
𝜓       (𝜑 → (𝜑𝜓))
 
Theoremjctild 303 Deduction conjoining a theorem to left of consequent in an implication. (Contributed by NM, 21-Apr-2005.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝜃)       (𝜑 → (𝜓 → (𝜃𝜒)))
 
Theoremjctird 304 Deduction conjoining a theorem to right of consequent in an implication. (Contributed by NM, 21-Apr-2005.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))
 
Theoremancl 305 Conjoin antecedent to left of consequent. (Contributed by NM, 15-Aug-1994.)
((𝜑𝜓) → (𝜑 → (𝜑𝜓)))
 
Theoremanclb 306 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.)
((𝜑𝜓) ↔ (𝜑 → (𝜑𝜓)))
 
Theorempm5.42 307 Theorem *5.42 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((𝜑 → (𝜓𝜒)) ↔ (𝜑 → (𝜓 → (𝜑𝜒))))
 
Theoremancr 308 Conjoin antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)
((𝜑𝜓) → (𝜑 → (𝜓𝜑)))
 
Theoremancrb 309 Conjoin antecedent to right of consequent. (Contributed by NM, 25-Jul-1999.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
((𝜑𝜓) ↔ (𝜑 → (𝜓𝜑)))
 
Theoremancli 310 Deduction conjoining antecedent to left of consequent. (Contributed by NM, 12-Aug-1993.)
(𝜑𝜓)       (𝜑 → (𝜑𝜓))
 
Theoremancri 311 Deduction conjoining antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)
(𝜑𝜓)       (𝜑 → (𝜓𝜑))
 
Theoremancld 312 Deduction conjoining antecedent to left of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 1-Nov-2012.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → (𝜓𝜒)))
 
Theoremancrd 313 Deduction conjoining antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 1-Nov-2012.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → (𝜒𝜓)))
 
Theoremanc2l 314 Conjoin antecedent to left of consequent in nested implication. (Contributed by NM, 10-Aug-1994.) (Proof shortened by Wolf Lammen, 14-Jul-2013.)
((𝜑 → (𝜓𝜒)) → (𝜑 → (𝜓 → (𝜑𝜒))))
 
Theoremanc2r 315 Conjoin antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.)
((𝜑 → (𝜓𝜒)) → (𝜑 → (𝜓 → (𝜒𝜑))))
 
Theoremanc2li 316 Deduction conjoining antecedent to left of consequent in nested implication. (Contributed by NM, 10-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → (𝜑𝜒)))
 
Theoremanc2ri 317 Deduction conjoining antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → (𝜒𝜑)))
 
Theorempm3.41 318 Theorem *3.41 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜒) → ((𝜑𝜓) → 𝜒))
 
Theorempm3.42 319 Theorem *3.42 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
((𝜓𝜒) → ((𝜑𝜓) → 𝜒))
 
Theorempm3.4 320 Conjunction implies implication. Theorem *3.4 of [WhiteheadRussell] p. 113. (Contributed by NM, 31-Jul-1995.)
((𝜑𝜓) → (𝜑𝜓))
 
Theorempm4.45im 321 Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 17-May-1998.)
(𝜑 ↔ (𝜑 ∧ (𝜓𝜑)))
 
Theoremanim12d 322 Conjoin antecedents and consequents in a deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 18-Dec-2013.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) → (𝜒𝜏)))
 
Theoremanim1d 323 Add a conjunct to right of antecedent and consequent in a deduction. (Contributed by NM, 3-Apr-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) → (𝜒𝜃)))
 
Theoremanim2d 324 Add a conjunct to left of antecedent and consequent in a deduction. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) → (𝜃𝜒)))
 
Theoremanim12i 325 Conjoin antecedents and consequents of two premises. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Dec-2013.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) → (𝜓𝜃))
 
Theoremanim12ci 326 Variant of anim12i 325 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) → (𝜃𝜓))
 
Theoremanim1i 327 Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)       ((𝜑𝜒) → (𝜓𝜒))
 
Theoremanim2i 328 Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)       ((𝜒𝜑) → (𝜒𝜓))
 
Theoremanim12ii 329 Conjoin antecedents and consequents in a deduction. (Contributed by NM, 11-Nov-2007.) (Proof shortened by Wolf Lammen, 19-Jul-2013.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜓𝜏))       ((𝜑𝜃) → (𝜓 → (𝜒𝜏)))
 
Theoremprth 330 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.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜒) → (𝜓𝜃)))
 
Theorempm3.33 331 Theorem *3.33 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜓𝜒)) → (𝜑𝜒))
 
Theorempm3.34 332 Theorem *3.34 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
(((𝜓𝜒) ∧ (𝜑𝜓)) → (𝜑𝜒))
 
Theorempm3.35 333 Conjunctive detachment. Theorem *3.35 of [WhiteheadRussell] p. 112. (Contributed by NM, 14-Dec-2002.)
((𝜑 ∧ (𝜑𝜓)) → 𝜓)
 
Theorempm5.31 334 Theorem *5.31 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((𝜒 ∧ (𝜑𝜓)) → (𝜑 → (𝜓𝜒)))
 
Theoremimp4a 335 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → (𝜓 → ((𝜒𝜃) → 𝜏)))
 
Theoremimp4b 336 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((𝜑𝜓) → ((𝜒𝜃) → 𝜏))
 
Theoremimp4c 337 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → (((𝜓𝜒) ∧ 𝜃) → 𝜏))
 
Theoremimp4d 338 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → ((𝜓 ∧ (𝜒𝜃)) → 𝜏))
 
Theoremimp41 339 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
 
Theoremimp42 340 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜏)
 
Theoremimp43 341 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)
 
Theoremimp44 342 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)) → 𝜏)
 
Theoremimp45 343 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜏)
 
Theoremimp5a 344 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (𝜑 → (𝜓 → (𝜒 → ((𝜃𝜏) → 𝜂))))
 
Theoremimp5d 345 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (((𝜑𝜓) ∧ 𝜒) → ((𝜃𝜏) → 𝜂))
 
Theoremimp5g 346 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       ((𝜑𝜓) → (((𝜒𝜃) ∧ 𝜏) → 𝜂))
 
Theoremimp55 347 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) ∧ 𝜏) → 𝜂)
 
Theoremimp511 348 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       ((𝜑 ∧ ((𝜓 ∧ (𝜒𝜃)) ∧ 𝜏)) → 𝜂)
 
Theoremexpimpd 349 Exportation followed by a deduction version of importation. (Contributed by NM, 6-Sep-2008.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → ((𝜓𝜒) → 𝜃))
 
Theoremexp31 350 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))
 
Theoremexp32 351 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))
 
Theoremexp4a 352 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → ((𝜒𝜃) → 𝜏)))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp4b 353 An exportation inference. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)
((𝜑𝜓) → ((𝜒𝜃) → 𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp4c 354 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (((𝜓𝜒) ∧ 𝜃) → 𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp4d 355 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → ((𝜓 ∧ (𝜒𝜃)) → 𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp41 356 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp42 357 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp43 358 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp44 359 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp45 360 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexpr 361 Export a wff from a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑𝜓) → (𝜒𝜃))
 
Theoremexp5c 362 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → ((𝜓𝜒) → ((𝜃𝜏) → 𝜂)))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp53 363 An exportation inference. (Contributed by Jeff Hankins, 30-Aug-2009.)
((((𝜑𝜓) ∧ (𝜒𝜃)) ∧ 𝜏) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexpl 364 Export a wff from a left conjunct. (Contributed by Jeff Hankins, 28-Aug-2009.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (𝜑 → ((𝜓𝜒) → 𝜃))
 
Theoremimpr 365 Import a wff into a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑 ∧ (𝜓𝜒)) → 𝜃)
 
Theoremimpl 366 Export a wff from a left conjunct. (Contributed by Mario Carneiro, 9-Jul-2014.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (((𝜑𝜓) ∧ 𝜒) → 𝜃)
 
Theoremimpac 367 Importation with conjunction in consequent. (Contributed by NM, 9-Aug-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) → (𝜒𝜓))
 
Theoremexbiri 368 Inference form of exbir 1341. (Contributed by Alan Sare, 31-Dec-2011.) (Proof shortened by Wolf Lammen, 27-Jan-2013.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → (𝜓 → (𝜃𝜒)))
 
Theoremsimprbda 369 Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜓) → 𝜒)
 
Theoremsimplbda 370 Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜓) → 𝜃)
 
Theoremsimplbi2 371 Deduction eliminating a conjunct. (Contributed by Alan Sare, 31-Dec-2011.)
(𝜑 ↔ (𝜓𝜒))       (𝜓 → (𝜒𝜑))
 
Theoremsimpl2im 372 Implication from an eliminated conjunct implied by the antecedent. (Contributed by BJ/AV, 5-Apr-2021.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜃)       (𝜑𝜃)
 
Theoremsimplbiim 373 Implication from an eliminated conjunct equivalent to the antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑 ↔ (𝜓𝜒))    &   (𝜒𝜃)       (𝜑𝜃)
 
Theoremdfbi2 374 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.)
((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
 
Theorempm4.71 375 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.)
((𝜑𝜓) ↔ (𝜑 ↔ (𝜑𝜓)))
 
Theorempm4.71r 376 Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct reversed). (Contributed by NM, 25-Jul-1999.)
((𝜑𝜓) ↔ (𝜑 ↔ (𝜓𝜑)))
 
Theorempm4.71i 377 Inference converting an implication to a biconditional with conjunction. Inference from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 4-Jan-2004.)
(𝜑𝜓)       (𝜑 ↔ (𝜑𝜓))
 
Theorempm4.71ri 378 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.)
(𝜑𝜓)       (𝜑 ↔ (𝜓𝜑))
 
Theorempm4.71d 379 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.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 ↔ (𝜓𝜒)))
 
Theorempm4.71rd 380 Deduction converting an implication to a biconditional with conjunction. Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 10-Feb-2005.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 ↔ (𝜒𝜓)))
 
Theorempm4.24 381 Theorem *4.24 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 14-Mar-2014.)
(𝜑 ↔ (𝜑𝜑))
 
Theoremanidm 382 Idempotent law for conjunction. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Mar-2014.)
((𝜑𝜑) ↔ 𝜑)
 
Theoremanidms 383 Inference from idempotent law for conjunction. (Contributed by NM, 15-Jun-1994.)
((𝜑𝜑) → 𝜓)       (𝜑𝜓)
 
Theoremanidmdbi 384 Conjunction idempotence with antecedent. (Contributed by Roy F. Longton, 8-Aug-2005.)
((𝜑 → (𝜓𝜓)) ↔ (𝜑𝜓))
 
Theoremanasss 385 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜒)) → 𝜃)
 
Theoremanassrs 386 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       (((𝜑𝜓) ∧ 𝜒) → 𝜃)
 
Theoremanass 387 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.)
(((𝜑𝜓) ∧ 𝜒) ↔ (𝜑 ∧ (𝜓𝜒)))
 
Theoremsylanl1 388 A syllogism inference. (Contributed by NM, 10-Mar-2005.)
(𝜑𝜓)    &   (((𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜑𝜒) ∧ 𝜃) → 𝜏)
 
Theoremsylanl2 389 A syllogism inference. (Contributed by NM, 1-Jan-2005.)
(𝜑𝜒)    &   (((𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜓𝜑) ∧ 𝜃) → 𝜏)
 
Theoremsylanr1 390 A syllogism inference. (Contributed by NM, 9-Apr-2005.)
(𝜑𝜒)    &   ((𝜓 ∧ (𝜒𝜃)) → 𝜏)       ((𝜓 ∧ (𝜑𝜃)) → 𝜏)
 
Theoremsylanr2 391 A syllogism inference. (Contributed by NM, 9-Apr-2005.)
(𝜑𝜃)    &   ((𝜓 ∧ (𝜒𝜃)) → 𝜏)       ((𝜓 ∧ (𝜒𝜑)) → 𝜏)
 
Theoremsylani 392 A syllogism inference. (Contributed by NM, 2-May-1996.)
(𝜑𝜒)    &   (𝜓 → ((𝜒𝜃) → 𝜏))       (𝜓 → ((𝜑𝜃) → 𝜏))
 
Theoremsylan2i 393 A syllogism inference. (Contributed by NM, 1-Aug-1994.)
(𝜑𝜃)    &   (𝜓 → ((𝜒𝜃) → 𝜏))       (𝜓 → ((𝜒𝜑) → 𝜏))
 
Theoremsyl2ani 394 A syllogism inference. (Contributed by NM, 3-Aug-1999.)
(𝜑𝜒)    &   (𝜂𝜃)    &   (𝜓 → ((𝜒𝜃) → 𝜏))       (𝜓 → ((𝜑𝜂) → 𝜏))
 
Theoremsylan9 395 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))       ((𝜑𝜃) → (𝜓𝜏))
 
Theoremsylan9r 396 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))       ((𝜃𝜑) → (𝜓𝜏))
 
Theoremsyl2anc 397 Syllogism inference combined with contraction. (Contributed by NM, 16-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
Theoremsylancl 398 Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝜑𝜓)    &   𝜒    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
Theoremsylancr 399 Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝜓    &   (𝜑𝜒)    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
Theoremsylanblc 400 Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019.)
(𝜑𝜓)    &   𝜒    &   ((𝜓𝜒) ↔ 𝜃)       (𝜑𝜃)
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