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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.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))       (𝜑 → (𝜓 → (𝜒𝜃)))

Theoremjca31 302 Join three consequents. (Contributed by Jeff Hankins, 1-Aug-2009.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)       (𝜑 → ((𝜓𝜒) ∧ 𝜃))

Theoremjca32 303 Join three consequents. (Contributed by FL, 1-Aug-2009.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)       (𝜑 → (𝜓 ∧ (𝜒𝜃)))

Theoremjcai 304 Deduction replacing implication with conjunction. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝜓𝜒))

Theoremjctil 305 Inference conjoining a theorem to left of consequent in an implication. (Contributed by NM, 31-Dec-1993.)
(𝜑𝜓)    &   𝜒       (𝜑 → (𝜒𝜓))

Theoremjctir 306 Inference conjoining a theorem to right of consequent in an implication. (Contributed by NM, 31-Dec-1993.)
(𝜑𝜓)    &   𝜒       (𝜑 → (𝜓𝜒))

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.)
𝜓       (𝜑 → (𝜓𝜑))

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.)
𝜓       (𝜑 → (𝜑𝜓))

Theoremjctild 309 Deduction conjoining a theorem to left of consequent in an implication. (Contributed by NM, 21-Apr-2005.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝜃)       (𝜑 → (𝜓 → (𝜃𝜒)))

Theoremjctird 310 Deduction conjoining a theorem to right of consequent in an implication. (Contributed by NM, 21-Apr-2005.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))

Theoremancl 311 Conjoin antecedent to left of consequent. (Contributed by NM, 15-Aug-1994.)
((𝜑𝜓) → (𝜑 → (𝜑𝜓)))

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.)
((𝜑𝜓) ↔ (𝜑 → (𝜑𝜓)))

Theorempm5.42 313 Theorem *5.42 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((𝜑 → (𝜓𝜒)) ↔ (𝜑 → (𝜓 → (𝜑𝜒))))

Theoremancr 314 Conjoin antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)
((𝜑𝜓) → (𝜑 → (𝜓𝜑)))

Theoremancrb 315 Conjoin antecedent to right of consequent. (Contributed by NM, 25-Jul-1999.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
((𝜑𝜓) ↔ (𝜑 → (𝜓𝜑)))

Theoremancli 316 Deduction conjoining antecedent to left of consequent. (Contributed by NM, 12-Aug-1993.)
(𝜑𝜓)       (𝜑 → (𝜑𝜓))

Theoremancri 317 Deduction conjoining antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)
(𝜑𝜓)       (𝜑 → (𝜓𝜑))

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.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → (𝜓𝜒)))

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.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → (𝜒𝜓)))

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.)
((𝜑 → (𝜓𝜒)) → (𝜑 → (𝜓 → (𝜑𝜒))))

Theoremanc2r 321 Conjoin antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.)
((𝜑 → (𝜓𝜒)) → (𝜑 → (𝜓 → (𝜒𝜑))))

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.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → (𝜑𝜒)))

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.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → (𝜒𝜑)))

Theorempm3.41 324 Theorem *3.41 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜒) → ((𝜑𝜓) → 𝜒))

Theorempm3.42 325 Theorem *3.42 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
((𝜓𝜒) → ((𝜑𝜓) → 𝜒))

Theorempm3.4 326 Conjunction implies implication. Theorem *3.4 of [WhiteheadRussell] p. 113. (Contributed by NM, 31-Jul-1995.)
((𝜑𝜓) → (𝜑𝜓))

Theorempm4.45im 327 Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 17-May-1998.)
(𝜑 ↔ (𝜑 ∧ (𝜓𝜑)))

Theoremanim12d 328 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 329 Add a conjunct to right of antecedent and consequent in a deduction. (Contributed by NM, 3-Apr-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) → (𝜒𝜃)))

Theoremanim2d 330 Add a conjunct to left of antecedent and consequent in a deduction. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) → (𝜃𝜒)))

Theoremanim12i 331 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 332 Variant of anim12i 331 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) → (𝜃𝜓))

Theoremanim1i 333 Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)       ((𝜑𝜒) → (𝜓𝜒))

Theoremanim2i 334 Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)       ((𝜒𝜑) → (𝜒𝜓))

Theoremanim12ii 335 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 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.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜒) → (𝜓𝜃)))

Theorempm3.33 337 Theorem *3.33 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜓𝜒)) → (𝜑𝜒))

Theorempm3.34 338 Theorem *3.34 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
(((𝜓𝜒) ∧ (𝜑𝜓)) → (𝜑𝜒))

Theorempm3.35 339 Conjunctive detachment. Theorem *3.35 of [WhiteheadRussell] p. 112. (Contributed by NM, 14-Dec-2002.)
((𝜑 ∧ (𝜑𝜓)) → 𝜓)

Theorempm5.31 340 Theorem *5.31 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((𝜒 ∧ (𝜑𝜓)) → (𝜑 → (𝜓𝜒)))

Theoremimp4a 341 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → (𝜓 → ((𝜒𝜃) → 𝜏)))

Theoremimp4b 342 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((𝜑𝜓) → ((𝜒𝜃) → 𝜏))

Theoremimp4c 343 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → (((𝜓𝜒) ∧ 𝜃) → 𝜏))

Theoremimp4d 344 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → ((𝜓 ∧ (𝜒𝜃)) → 𝜏))

Theoremimp41 345 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theoremimp42 346 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜏)

Theoremimp43 347 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)

Theoremimp44 348 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)) → 𝜏)

Theoremimp45 349 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜏)

Theoremimp5a 350 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (𝜑 → (𝜓 → (𝜒 → ((𝜃𝜏) → 𝜂))))

Theoremimp5d 351 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (((𝜑𝜓) ∧ 𝜒) → ((𝜃𝜏) → 𝜂))

Theoremimp5g 352 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       ((𝜑𝜓) → (((𝜒𝜃) ∧ 𝜏) → 𝜂))

Theoremimp55 353 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) ∧ 𝜏) → 𝜂)

Theoremimp511 354 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       ((𝜑 ∧ ((𝜓 ∧ (𝜒𝜃)) ∧ 𝜏)) → 𝜂)

Theoremexpimpd 355 Exportation followed by a deduction version of importation. (Contributed by NM, 6-Sep-2008.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → ((𝜓𝜒) → 𝜃))

Theoremexp31 356 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))

Theoremexp32 357 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))

Theoremexp4a 358 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → ((𝜒𝜃) → 𝜏)))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexp4b 359 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 360 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (((𝜓𝜒) ∧ 𝜃) → 𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexp4d 361 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → ((𝜓 ∧ (𝜒𝜃)) → 𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexp41 362 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexp42 363 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexp43 364 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexp44 365 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexp45 366 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexpr 367 Export a wff from a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑𝜓) → (𝜒𝜃))

Theoremexp5c 368 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → ((𝜓𝜒) → ((𝜃𝜏) → 𝜂)))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))

Theoremexp53 369 An exportation inference. (Contributed by Jeff Hankins, 30-Aug-2009.)
((((𝜑𝜓) ∧ (𝜒𝜃)) ∧ 𝜏) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))

Theoremexpl 370 Export a wff from a left conjunct. (Contributed by Jeff Hankins, 28-Aug-2009.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (𝜑 → ((𝜓𝜒) → 𝜃))

Theoremimpr 371 Import a wff into a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑 ∧ (𝜓𝜒)) → 𝜃)

Theoremimpl 372 Export a wff from a left conjunct. (Contributed by Mario Carneiro, 9-Jul-2014.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (((𝜑𝜓) ∧ 𝜒) → 𝜃)

Theoremimpac 373 Importation with conjunction in consequent. (Contributed by NM, 9-Aug-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) → (𝜒𝜓))

Theoremexbiri 374 Inference form of exbir 1366. (Contributed by Alan Sare, 31-Dec-2011.) (Proof shortened by Wolf Lammen, 27-Jan-2013.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → (𝜓 → (𝜃𝜒)))

Theoremsimprbda 375 Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜓) → 𝜒)

Theoremsimplbda 376 Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜓) → 𝜃)

Theoremsimplbi2 377 Deduction eliminating a conjunct. (Contributed by Alan Sare, 31-Dec-2011.)
(𝜑 ↔ (𝜓𝜒))       (𝜓 → (𝜒𝜑))

Theoremsimpl2im 378 Implication from an eliminated conjunct implied by the antecedent. (Contributed by BJ/AV, 5-Apr-2021.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜃)       (𝜑𝜃)

Theoremsimplbiim 379 Implication from an eliminated conjunct equivalent to the antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑 ↔ (𝜓𝜒))    &   (𝜒𝜃)       (𝜑𝜃)

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.)
((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))

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.)
((𝜑𝜓) ↔ (𝜑 ↔ (𝜑𝜓)))

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.)
((𝜑𝜓) ↔ (𝜑 ↔ (𝜓𝜑)))

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.)
(𝜑𝜓)       (𝜑 ↔ (𝜑𝜓))

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.)
(𝜑𝜓)       (𝜑 ↔ (𝜓𝜑))

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.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 ↔ (𝜓𝜒)))

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.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 ↔ (𝜒𝜓)))

Theorempm4.24 387 Theorem *4.24 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 14-Mar-2014.)
(𝜑 ↔ (𝜑𝜑))

Theoremanidm 388 Idempotent law for conjunction. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Mar-2014.)
((𝜑𝜑) ↔ 𝜑)

Theoremanidms 389 Inference from idempotent law for conjunction. (Contributed by NM, 15-Jun-1994.)
((𝜑𝜑) → 𝜓)       (𝜑𝜓)

Theoremanidmdbi 390 Conjunction idempotence with antecedent. (Contributed by Roy F. Longton, 8-Aug-2005.)
((𝜑 → (𝜓𝜓)) ↔ (𝜑𝜓))

Theoremanasss 391 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜒)) → 𝜃)

Theoremanassrs 392 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       (((𝜑𝜓) ∧ 𝜒) → 𝜃)

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.)
(((𝜑𝜓) ∧ 𝜒) ↔ (𝜑 ∧ (𝜓𝜒)))

Theoremsylanl1 394 A syllogism inference. (Contributed by NM, 10-Mar-2005.)
(𝜑𝜓)    &   (((𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜑𝜒) ∧ 𝜃) → 𝜏)

Theoremsylanl2 395 A syllogism inference. (Contributed by NM, 1-Jan-2005.)
(𝜑𝜒)    &   (((𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜓𝜑) ∧ 𝜃) → 𝜏)

Theoremsylanr1 396 A syllogism inference. (Contributed by NM, 9-Apr-2005.)
(𝜑𝜒)    &   ((𝜓 ∧ (𝜒𝜃)) → 𝜏)       ((𝜓 ∧ (𝜑𝜃)) → 𝜏)

Theoremsylanr2 397 A syllogism inference. (Contributed by NM, 9-Apr-2005.)
(𝜑𝜃)    &   ((𝜓 ∧ (𝜒𝜃)) → 𝜏)       ((𝜓 ∧ (𝜒𝜑)) → 𝜏)

Theoremsylani 398 A syllogism inference. (Contributed by NM, 2-May-1996.)
(𝜑𝜒)    &   (𝜓 → ((𝜒𝜃) → 𝜏))       (𝜓 → ((𝜑𝜃) → 𝜏))

Theoremsylan2i 399 A syllogism inference. (Contributed by NM, 1-Aug-1994.)
(𝜑𝜃)    &   (𝜓 → ((𝜒𝜃) → 𝜏))       (𝜓 → ((𝜒𝜑) → 𝜏))

Theoremsyl2ani 400 A syllogism inference. (Contributed by NM, 3-Aug-1999.)
(𝜑𝜒)    &   (𝜂𝜃)    &   (𝜓 → ((𝜒𝜃) → 𝜏))       (𝜓 → ((𝜑𝜂) → 𝜏))

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