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Theorem List for Intuitionistic Logic Explorer - 301-400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremancl 301 Conjoin antecedent to left of consequent. (Contributed by NM, 15-Aug-1994.)
((𝜑𝜓) → (𝜑 → (𝜑𝜓)))

Theoremanclb 302 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)))

Theorempm5.42 303 Theorem *5.42 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((𝜑 → (𝜓𝜒)) ↔ (𝜑 → (𝜓 → (𝜑𝜒))))

Theoremancr 304 Conjoin antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)
((𝜑𝜓) → (𝜑 → (𝜓𝜑)))

Theoremancrb 305 Conjoin antecedent to right of consequent. (Contributed by NM, 25-Jul-1999.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
((𝜑𝜓) ↔ (𝜑 → (𝜓𝜑)))

Theoremancli 306 Deduction conjoining antecedent to left of consequent. (Contributed by NM, 12-Aug-1993.)
(𝜑𝜓)       (𝜑 → (𝜑𝜓))

Theoremancri 307 Deduction conjoining antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)
(𝜑𝜓)       (𝜑 → (𝜓𝜑))

Theoremancld 308 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 309 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 310 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 311 Conjoin antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.)
((𝜑 → (𝜓𝜒)) → (𝜑 → (𝜓 → (𝜒𝜑))))

Theoremanc2li 312 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 313 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)))

Theorempm3.41 314 Theorem *3.41 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜒) → ((𝜑𝜓) → 𝜒))

Theorempm3.42 315 Theorem *3.42 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
((𝜓𝜒) → ((𝜑𝜓) → 𝜒))

Theorempm3.4 316 Conjunction implies implication. Theorem *3.4 of [WhiteheadRussell] p. 113. (Contributed by NM, 31-Jul-1995.)
((𝜑𝜓) → (𝜑𝜓))

Theorempm4.45im 317 Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 17-May-1998.)
(𝜑 ↔ (𝜑 ∧ (𝜓𝜑)))

Theoremanim12d 318 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 319 Add a conjunct to right of antecedent and consequent in a deduction. (Contributed by NM, 3-Apr-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) → (𝜒𝜃)))

Theoremanim2d 320 Add a conjunct to left of antecedent and consequent in a deduction. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) → (𝜃𝜒)))

Theoremanim12i 321 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 322 Variant of anim12i 321 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) → (𝜃𝜓))

Theoremanim1i 323 Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)       ((𝜑𝜒) → (𝜓𝜒))

Theoremanim2i 324 Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)       ((𝜒𝜑) → (𝜒𝜓))

Theoremanim12ii 325 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 326 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)))

Theorempm3.33 327 Theorem *3.33 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜓𝜒)) → (𝜑𝜒))

Theorempm3.34 328 Theorem *3.34 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
(((𝜓𝜒) ∧ (𝜑𝜓)) → (𝜑𝜒))

Theorempm3.35 329 Conjunctive detachment. Theorem *3.35 of [WhiteheadRussell] p. 112. (Contributed by NM, 14-Dec-2002.)
((𝜑 ∧ (𝜑𝜓)) → 𝜓)

Theorempm5.31 330 Theorem *5.31 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((𝜒 ∧ (𝜑𝜓)) → (𝜑 → (𝜓𝜒)))

Theoremimp4a 331 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → (𝜓 → ((𝜒𝜃) → 𝜏)))

Theoremimp4b 332 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((𝜑𝜓) → ((𝜒𝜃) → 𝜏))

Theoremimp4c 333 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → (((𝜓𝜒) ∧ 𝜃) → 𝜏))

Theoremimp4d 334 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → ((𝜓 ∧ (𝜒𝜃)) → 𝜏))

Theoremimp41 335 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theoremimp42 336 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜏)

Theoremimp43 337 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)

Theoremimp44 338 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)) → 𝜏)

Theoremimp45 339 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜏)

Theoremimp5a 340 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (𝜑 → (𝜓 → (𝜒 → ((𝜃𝜏) → 𝜂))))

Theoremimp5d 341 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (((𝜑𝜓) ∧ 𝜒) → ((𝜃𝜏) → 𝜂))

Theoremimp5g 342 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       ((𝜑𝜓) → (((𝜒𝜃) ∧ 𝜏) → 𝜂))

Theoremimp55 343 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) ∧ 𝜏) → 𝜂)

Theoremimp511 344 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       ((𝜑 ∧ ((𝜓 ∧ (𝜒𝜃)) ∧ 𝜏)) → 𝜂)

Theoremexpimpd 345 Exportation followed by a deduction version of importation. (Contributed by NM, 6-Sep-2008.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → ((𝜓𝜒) → 𝜃))

Theoremexp31 346 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))

Theoremexp32 347 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))

Theoremexp4a 348 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → ((𝜒𝜃) → 𝜏)))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexp4b 349 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 350 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (((𝜓𝜒) ∧ 𝜃) → 𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexp4d 351 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → ((𝜓 ∧ (𝜒𝜃)) → 𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexp41 352 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexp42 353 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexp43 354 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexp44 355 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexp45 356 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexpr 357 Export a wff from a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑𝜓) → (𝜒𝜃))

Theoremexp5c 358 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → ((𝜓𝜒) → ((𝜃𝜏) → 𝜂)))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))

Theoremexp53 359 An exportation inference. (Contributed by Jeff Hankins, 30-Aug-2009.)
((((𝜑𝜓) ∧ (𝜒𝜃)) ∧ 𝜏) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))

Theoremexpl 360 Export a wff from a left conjunct. (Contributed by Jeff Hankins, 28-Aug-2009.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (𝜑 → ((𝜓𝜒) → 𝜃))

Theoremimpr 361 Import a wff into a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑 ∧ (𝜓𝜒)) → 𝜃)

Theoremimpl 362 Export a wff from a left conjunct. (Contributed by Mario Carneiro, 9-Jul-2014.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (((𝜑𝜓) ∧ 𝜒) → 𝜃)

Theoremimpac 363 Importation with conjunction in consequent. (Contributed by NM, 9-Aug-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) → (𝜒𝜓))

Theoremexbiri 364 Inference form of exbir 1325. (Contributed by Alan Sare, 31-Dec-2011.) (Proof shortened by Wolf Lammen, 27-Jan-2013.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → (𝜓 → (𝜃𝜒)))

Theoremsimprbda 365 Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜓) → 𝜒)

Theoremsimplbda 366 Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜓) → 𝜃)

Theoremsimplbi2 367 Deduction eliminating a conjunct. (Contributed by Alan Sare, 31-Dec-2011.)
(𝜑 ↔ (𝜓𝜒))       (𝜓 → (𝜒𝜑))

Theoremdfbi2 368 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)))

Theorempm4.71 369 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 370 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 371 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 372 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 373 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 374 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)))

Theorempm4.24 375 Theorem *4.24 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 14-Mar-2014.)
(𝜑 ↔ (𝜑𝜑))

Theoremanidm 376 Idempotent law for conjunction. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Mar-2014.)
((𝜑𝜑) ↔ 𝜑)

Theoremanidms 377 Inference from idempotent law for conjunction. (Contributed by NM, 15-Jun-1994.)
((𝜑𝜑) → 𝜓)       (𝜑𝜓)

Theoremanidmdbi 378 Conjunction idempotence with antecedent. (Contributed by Roy F. Longton, 8-Aug-2005.)
((𝜑 → (𝜓𝜓)) ↔ (𝜑𝜓))

Theoremanasss 379 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜒)) → 𝜃)

Theoremanassrs 380 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       (((𝜑𝜓) ∧ 𝜒) → 𝜃)

Theoremanass 381 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 382 A syllogism inference. (Contributed by NM, 10-Mar-2005.)
(𝜑𝜓)    &   (((𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜑𝜒) ∧ 𝜃) → 𝜏)

Theoremsylanl2 383 A syllogism inference. (Contributed by NM, 1-Jan-2005.)
(𝜑𝜒)    &   (((𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜓𝜑) ∧ 𝜃) → 𝜏)

Theoremsylanr1 384 A syllogism inference. (Contributed by NM, 9-Apr-2005.)
(𝜑𝜒)    &   ((𝜓 ∧ (𝜒𝜃)) → 𝜏)       ((𝜓 ∧ (𝜑𝜃)) → 𝜏)

Theoremsylanr2 385 A syllogism inference. (Contributed by NM, 9-Apr-2005.)
(𝜑𝜃)    &   ((𝜓 ∧ (𝜒𝜃)) → 𝜏)       ((𝜓 ∧ (𝜒𝜑)) → 𝜏)

Theoremsylani 386 A syllogism inference. (Contributed by NM, 2-May-1996.)
(𝜑𝜒)    &   (𝜓 → ((𝜒𝜃) → 𝜏))       (𝜓 → ((𝜑𝜃) → 𝜏))

Theoremsylan2i 387 A syllogism inference. (Contributed by NM, 1-Aug-1994.)
(𝜑𝜃)    &   (𝜓 → ((𝜒𝜃) → 𝜏))       (𝜓 → ((𝜒𝜑) → 𝜏))

Theoremsyl2ani 388 A syllogism inference. (Contributed by NM, 3-Aug-1999.)
(𝜑𝜒)    &   (𝜂𝜃)    &   (𝜓 → ((𝜒𝜃) → 𝜏))       (𝜓 → ((𝜑𝜂) → 𝜏))

Theoremsylan9 389 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 390 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))       ((𝜃𝜑) → (𝜓𝜏))

Theoremsyl2anc 391 Syllogism inference combined with contraction. (Contributed by NM, 16-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)

Theoremsylancl 392 Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝜑𝜓)    &   𝜒    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)

Theoremsylancr 393 Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝜓    &   (𝜑𝜒)    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)

Theoremsylanbrc 394 Syllogism inference. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜃 ↔ (𝜓𝜒))       (𝜑𝜃)

Theoremsylancb 395 A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)
(𝜑𝜓)    &   (𝜑𝜒)    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)

Theoremsylancbr 396 A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)
(𝜓𝜑)    &   (𝜒𝜑)    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)

Theoremsylancom 397 Syllogism inference with commutation of antecents. (Contributed by NM, 2-Jul-2008.)
((𝜑𝜓) → 𝜒)    &   ((𝜒𝜓) → 𝜃)       ((𝜑𝜓) → 𝜃)

Theoremmpdan 398 An inference based on modus ponens. (Contributed by NM, 23-May-1999.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)
(𝜑𝜓)    &   ((𝜑𝜓) → 𝜒)       (𝜑𝜒)

Theoremmpancom 399 An inference based on modus ponens with commutation of antecedents. (Contributed by NM, 28-Oct-2003.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
(𝜓𝜑)    &   ((𝜑𝜓) → 𝜒)       (𝜓𝜒)

Theoremmpan 400 An inference based on modus ponens. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
𝜑    &   ((𝜑𝜓) → 𝜒)       (𝜓𝜒)

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