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

Theorem3bitrri 301 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
(𝜑𝜓)    &   (𝜓𝜒)    &   (𝜒𝜃)       (𝜃𝜑)

Theorem3bitr2i 302 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
(𝜑𝜓)    &   (𝜒𝜓)    &   (𝜒𝜃)       (𝜑𝜃)

Theorem3bitr2ri 303 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
(𝜑𝜓)    &   (𝜒𝜓)    &   (𝜒𝜃)       (𝜃𝜑)

Theorem3bitr3i 304 A chained inference from transitive law for logical equivalence. (Contributed by NM, 19-Aug-1993.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜓𝜃)       (𝜒𝜃)

Theorem3bitr3ri 305 A chained inference from transitive law for logical equivalence. (Contributed by NM, 21-Jun-1993.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜓𝜃)       (𝜃𝜒)

Theorem3bitr4i 306 A chained inference from transitive law for logical equivalence. This inference is frequently used to apply a definition to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.)
(𝜑𝜓)    &   (𝜒𝜑)    &   (𝜃𝜓)       (𝜒𝜃)

Theorem3bitr4ri 307 A chained inference from transitive law for logical equivalence. (Contributed by NM, 2-Sep-1995.)
(𝜑𝜓)    &   (𝜒𝜑)    &   (𝜃𝜓)       (𝜃𝜒)

Theorem3bitrd 308 Deduction from transitivity of biconditional. (Contributed by NM, 13-Aug-1999.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))    &   (𝜑 → (𝜃𝜏))       (𝜑 → (𝜓𝜏))

Theorem3bitrrd 309 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))    &   (𝜑 → (𝜃𝜏))       (𝜑 → (𝜏𝜓))

Theorem3bitr2d 310 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → (𝜓𝜏))

Theorem3bitr2rd 311 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → (𝜏𝜓))

Theorem3bitr3d 312 Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 24-Apr-1996.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜒𝜏))       (𝜑 → (𝜃𝜏))

Theorem3bitr3rd 313 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜒𝜏))       (𝜑 → (𝜏𝜃))

Theorem3bitr4d 314 Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 18-Oct-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜓))    &   (𝜑 → (𝜏𝜒))       (𝜑 → (𝜃𝜏))

Theorem3bitr4rd 315 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜓))    &   (𝜑 → (𝜏𝜒))       (𝜑 → (𝜏𝜃))

Theorem3bitr3g 316 More general version of 3bitr3i 304. Useful for converting definitions in a formula. (Contributed by NM, 4-Jun-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜓𝜃)    &   (𝜒𝜏)       (𝜑 → (𝜃𝜏))

Theorem3bitr4g 317 More general version of 3bitr4i 306. Useful for converting definitions in a formula. (Contributed by NM, 11-May-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜃𝜓)    &   (𝜏𝜒)       (𝜑 → (𝜃𝜏))

Theoremnotnotb 318 Double negation. Theorem *4.13 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-1993.)
(𝜑 ↔ ¬ ¬ 𝜑)

Theoremcon34b 319 A biconditional form of contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116. (Contributed by NM, 11-May-1993.)
((𝜑𝜓) ↔ (¬ 𝜓 → ¬ 𝜑))

Theoremcon4bid 320 A contraposition deduction. (Contributed by NM, 21-May-1994.)
(𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))       (𝜑 → (𝜓𝜒))

Theoremnotbid 321 Deduction negating both sides of a logical equivalence. (Contributed by NM, 21-May-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))

Theoremnotbi 322 Contraposition. Theorem *4.11 of [WhiteheadRussell] p. 117. (Contributed by NM, 21-May-1994.) (Proof shortened by Wolf Lammen, 12-Jun-2013.)
((𝜑𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))

Theoremnotbii 323 Negate both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.)
(𝜑𝜓)       𝜑 ↔ ¬ 𝜓)

Theoremcon4bii 324 A contraposition inference. (Contributed by NM, 21-May-1994.)
𝜑 ↔ ¬ 𝜓)       (𝜑𝜓)

Theoremmtbi 325 An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
¬ 𝜑    &   (𝜑𝜓)        ¬ 𝜓

Theoremmtbir 326 An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 14-Oct-2012.)
¬ 𝜓    &   (𝜑𝜓)        ¬ 𝜑

Theoremmtbid 327 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.)
(𝜑 → ¬ 𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜒)

Theoremmtbird 328 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜓)

Theoremmtbii 329 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.)
¬ 𝜓    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜒)

Theoremmtbiri 330 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.)
¬ 𝜒    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜓)

Theoremsylnib 331 A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.)
(𝜑 → ¬ 𝜓)    &   (𝜓𝜒)       (𝜑 → ¬ 𝜒)

Theoremsylnibr 332 A mixed syllogism inference from an implication and a biconditional. Useful for substituting a consequent with a definition. (Contributed by Wolf Lammen, 16-Dec-2013.)
(𝜑 → ¬ 𝜓)    &   (𝜒𝜓)       (𝜑 → ¬ 𝜒)

Theoremsylnbi 333 A mixed syllogism inference from a biconditional and an implication. Useful for substituting an antecedent with a definition. (Contributed by Wolf Lammen, 16-Dec-2013.)
(𝜑𝜓)    &   𝜓𝜒)       𝜑𝜒)

Theoremsylnbir 334 A mixed syllogism inference from a biconditional and an implication. (Contributed by Wolf Lammen, 16-Dec-2013.)
(𝜓𝜑)    &   𝜓𝜒)       𝜑𝜒)

Theoremxchnxbi 335 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
𝜑𝜓)    &   (𝜑𝜒)       𝜒𝜓)

Theoremxchnxbir 336 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
𝜑𝜓)    &   (𝜒𝜑)       𝜒𝜓)

Theoremxchbinx 337 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
(𝜑 ↔ ¬ 𝜓)    &   (𝜓𝜒)       (𝜑 ↔ ¬ 𝜒)

Theoremxchbinxr 338 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
(𝜑 ↔ ¬ 𝜓)    &   (𝜒𝜓)       (𝜑 ↔ ¬ 𝜒)

Theoremimbi2i 339 Introduce an antecedent to both sides of a logical equivalence. This and the next three rules are useful for building up wff's around a definition, in order to make use of the definition. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 6-Feb-2013.)
(𝜑𝜓)       ((𝜒𝜑) ↔ (𝜒𝜓))

TheoremjcndOLD 340 Obsolete version of jcnd 166 as of 10-Apr-2024. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝜓)    &   (𝜑 → ¬ 𝜒)       (𝜑 → ¬ (𝜓𝜒))

Theorembibi2i 341 Inference adding a biconditional to the left in an equivalence. (Contributed by NM, 26-May-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 16-May-2013.)
(𝜑𝜓)       ((𝜒𝜑) ↔ (𝜒𝜓))

Theorembibi1i 342 Inference adding a biconditional to the right in an equivalence. (Contributed by NM, 26-May-1993.)
(𝜑𝜓)       ((𝜑𝜒) ↔ (𝜓𝜒))

Theorembibi12i 343 The equivalence of two equivalences. (Contributed by NM, 26-May-1993.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))

Theoremimbi2d 344 Deduction adding an antecedent to both sides of a logical equivalence. (Contributed by NM, 11-May-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) ↔ (𝜃𝜒)))

Theoremimbi1d 345 Deduction adding a consequent to both sides of a logical equivalence. (Contributed by NM, 11-May-1993.) (Proof shortened by Wolf Lammen, 17-Sep-2013.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))

Theorembibi2d 346 Deduction adding a biconditional to the left in an equivalence. (Contributed by NM, 11-May-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) ↔ (𝜃𝜒)))

Theorembibi1d 347 Deduction adding a biconditional to the right in an equivalence. (Contributed by NM, 11-May-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))

Theoremimbi12d 348 Deduction joining two equivalences to form equivalence of implications. (Contributed by NM, 16-May-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))

Theorembibi12d 349 Deduction joining two equivalences to form equivalence of biconditionals. (Contributed by NM, 26-May-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))

Theoremimbi12 350 Closed form of imbi12i 354. Was automatically derived from its "Virtual Deduction" version and the Metamath program "MM-PA> MINIMIZE_WITH *" command. (Contributed by Alan Sare, 18-Mar-2012.)
((𝜑𝜓) → ((𝜒𝜃) → ((𝜑𝜒) ↔ (𝜓𝜃))))

Theoremimbi1 351 Theorem *4.84 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))

Theoremimbi2 352 Theorem *4.85 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.)
((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))

Theoremimbi1i 353 Introduce a consequent to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 17-Sep-2013.)
(𝜑𝜓)       ((𝜑𝜒) ↔ (𝜓𝜒))

Theoremimbi12i 354 Join two logical equivalences to form equivalence of implications. (Contributed by NM, 1-Aug-1993.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))

Theorembibi1 355 Theorem *4.86 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))

Theorembitr3 356 Closed nested implication form of bitr3i 280. Derived automatically from bitr3VD 41570. (Contributed by Alan Sare, 31-Dec-2011.)
((𝜑𝜓) → ((𝜑𝜒) → (𝜓𝜒)))

Theoremcon2bi 357 Contraposition. Theorem *4.12 of [WhiteheadRussell] p. 117. (Contributed by NM, 15-Apr-1995.) (Proof shortened by Wolf Lammen, 3-Jan-2013.)
((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑))

Theoremcon2bid 358 A contraposition deduction. (Contributed by NM, 15-Apr-1995.)
(𝜑 → (𝜓 ↔ ¬ 𝜒))       (𝜑 → (𝜒 ↔ ¬ 𝜓))

Theoremcon1bid 359 A contraposition deduction. (Contributed by NM, 9-Oct-1999.)
(𝜑 → (¬ 𝜓𝜒))       (𝜑 → (¬ 𝜒𝜓))

Theoremcon1bii 360 A contraposition inference. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.)
𝜑𝜓)       𝜓𝜑)

Theoremcon2bii 361 A contraposition inference. (Contributed by NM, 12-Mar-1993.)
(𝜑 ↔ ¬ 𝜓)       (𝜓 ↔ ¬ 𝜑)

Theoremcon1b 362 Contraposition. Bidirectional version of con1 148. (Contributed by NM, 3-Jan-1993.)
((¬ 𝜑𝜓) ↔ (¬ 𝜓𝜑))

Theoremcon2b 363 Contraposition. Bidirectional version of con2 137. (Contributed by NM, 12-Mar-1993.)
((𝜑 → ¬ 𝜓) ↔ (𝜓 → ¬ 𝜑))

Theorembiimt 364 A wff is equivalent to itself with true antecedent. (Contributed by NM, 28-Jan-1996.)
(𝜑 → (𝜓 ↔ (𝜑𝜓)))

Theorempm5.5 365 Theorem *5.5 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
(𝜑 → ((𝜑𝜓) ↔ 𝜓))

Theorema1bi 366 Inference introducing a theorem as an antecedent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)
𝜑       (𝜓 ↔ (𝜑𝜓))

Theoremmt2bi 367 A false consequent falsifies an antecedent. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.)
𝜑       𝜓 ↔ (𝜓 → ¬ 𝜑))

Theoremmtt 368 Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Proof shortened by Wolf Lammen, 12-Nov-2012.)
𝜑 → (¬ 𝜓 ↔ (𝜓𝜑)))

Theoremimnot 369 If a proposition is false, then implying it is equivalent to being false. One of four theorems that can be used to simplify an implication (𝜑𝜓), the other ones being ax-1 6 (true consequent), pm2.21 123 (false antecedent), pm5.5 365 (true antecedent). (Contributed by Mario Carneiro, 26-Apr-2019.) (Proof shortened by Wolf Lammen, 26-May-2019.)
𝜓 → ((𝜑𝜓) ↔ ¬ 𝜑))

Theorempm5.501 370 Theorem *5.501 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
(𝜑 → (𝜓 ↔ (𝜑𝜓)))

Theoremibib 371 Implication in terms of implication and biconditional. (Contributed by NM, 31-Mar-1994.) (Proof shortened by Wolf Lammen, 24-Jan-2013.)
((𝜑𝜓) ↔ (𝜑 → (𝜑𝜓)))

Theoremibibr 372 Implication in terms of implication and biconditional. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 21-Dec-2013.)
((𝜑𝜓) ↔ (𝜑 → (𝜓𝜑)))

Theoremtbt 373 A wff is equivalent to its equivalence with a truth. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
𝜑       (𝜓 ↔ (𝜓𝜑))

Theoremnbn2 374 The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by Juha Arpiainen, 19-Jan-2006.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)
𝜑 → (¬ 𝜓 ↔ (𝜑𝜓)))

Theorembibif 375 Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)
𝜓 → ((𝜑𝜓) ↔ ¬ 𝜑))

Theoremnbn 376 The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
¬ 𝜑       𝜓 ↔ (𝜓𝜑))

Theoremnbn3 377 Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.)
𝜑       𝜓 ↔ (𝜓 ↔ ¬ 𝜑))

Theorempm5.21im 378 Two propositions are equivalent if they are both false. Closed form of 2false 379. Equivalent to a biimpr 223-like version of the xor-connective. (Contributed by Wolf Lammen, 13-May-2013.)
𝜑 → (¬ 𝜓 → (𝜑𝜓)))

Theorem2false 379 Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.)
¬ 𝜑    &    ¬ 𝜓       (𝜑𝜓)

Theorem2falsed 380 Two falsehoods are equivalent (deduction form). (Contributed by NM, 11-Oct-2013.) (Proof shortened by Wolf Lammen, 11-Apr-2024.)
(𝜑 → ¬ 𝜓)    &   (𝜑 → ¬ 𝜒)       (𝜑 → (𝜓𝜒))

Theorem2falsedOLD 381 Obsolete version of 2falsed 380 as of 11-Apr-2024. (Contributed by NM, 11-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ¬ 𝜓)    &   (𝜑 → ¬ 𝜒)       (𝜑 → (𝜓𝜒))

Theorempm5.21ni 382 Two propositions implying a false one are equivalent. (Contributed by NM, 16-Feb-1996.) (Proof shortened by Wolf Lammen, 19-May-2013.)
(𝜑𝜓)    &   (𝜒𝜓)       𝜓 → (𝜑𝜒))

Theorempm5.21nii 383 Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 21-May-1999.)
(𝜑𝜓)    &   (𝜒𝜓)    &   (𝜓 → (𝜑𝜒))       (𝜑𝜒)

Theorempm5.21ndd 384 Eliminate an antecedent implied by each side of a biconditional, deduction version. (Contributed by Paul Chapman, 21-Nov-2012.) (Proof shortened by Wolf Lammen, 6-Oct-2013.)
(𝜑 → (𝜒𝜓))    &   (𝜑 → (𝜃𝜓))    &   (𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → (𝜒𝜃))

Theorembija 385 Combine antecedents into a single biconditional. This inference, reminiscent of ja 189, is reversible: The hypotheses can be deduced from the conclusion alone (see pm5.1im 266 and pm5.21im 378). (Contributed by Wolf Lammen, 13-May-2013.)
(𝜑 → (𝜓𝜒))    &   𝜑 → (¬ 𝜓𝜒))       ((𝜑𝜓) → 𝜒)

Theorempm5.18 386 Theorem *5.18 of [WhiteheadRussell] p. 124. This theorem says that logical equivalence is the same as negated "exclusive or". (Contributed by NM, 28-Jun-2002.) (Proof shortened by Andrew Salmon, 20-Jun-2011.) (Proof shortened by Wolf Lammen, 15-Oct-2013.)
((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))

Theoremxor3 387 Two ways to express "exclusive or". (Contributed by NM, 1-Jan-2006.)
(¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))

Theoremnbbn 388 Move negation outside of biconditional. Compare Theorem *5.18 of [WhiteheadRussell] p. 124. (Contributed by NM, 27-Jun-2002.) (Proof shortened by Wolf Lammen, 20-Sep-2013.)
((¬ 𝜑𝜓) ↔ ¬ (𝜑𝜓))

Theorembiass 389 Associative law for the biconditional. An axiom of system DS in Vladimir Lifschitz, "On calculational proofs", Annals of Pure and Applied Logic, 113:207-224, 2002, http://www.cs.utexas.edu/users/ai-lab/pub-view.php?PubID=26805. Interestingly, this law was not included in Principia Mathematica but was apparently first noted by Jan Lukasiewicz circa 1923. (Contributed by NM, 8-Jan-2005.) (Proof shortened by Juha Arpiainen, 19-Jan-2006.) (Proof shortened by Wolf Lammen, 21-Sep-2013.)
(((𝜑𝜓) ↔ 𝜒) ↔ (𝜑 ↔ (𝜓𝜒)))

Theorembiluk 390 Lukasiewicz's shortest axiom for equivalential calculus. Storrs McCall, ed., Polish Logic 1920-1939 (Oxford, 1967), p. 96. (Contributed by NM, 10-Jan-2005.)
((𝜑𝜓) ↔ ((𝜒𝜓) ↔ (𝜑𝜒)))

Theorempm5.19 391 Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.)
¬ (𝜑 ↔ ¬ 𝜑)

Theorembi2.04 392 Logical equivalence of commuted antecedents. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 11-May-1993.)
((𝜑 → (𝜓𝜒)) ↔ (𝜓 → (𝜑𝜒)))

Theorempm5.4 393 Antecedent absorption implication. Theorem *5.4 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
((𝜑 → (𝜑𝜓)) ↔ (𝜑𝜓))

Theoremimdi 394 Distributive law for implication. Compare Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒)))

Theorempm5.41 395 Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 12-Oct-2012.)
(((𝜑𝜓) → (𝜑𝜒)) ↔ (𝜑 → (𝜓𝜒)))

Theorempm4.8 396 Theorem *4.8 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
((𝜑 → ¬ 𝜑) ↔ ¬ 𝜑)

Theorempm4.81 397 A formula is equivalent to its negation implying it. Theorem *4.81 of [WhiteheadRussell] p. 122. Note that the second step, using pm2.24 124, could also use ax-1 6. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑𝜑) ↔ 𝜑)

Theoremimim21b 398 Simplify an implication between two implications when the antecedent of the first is a consequence of the antecedent of the second. The reverse form is useful in producing the successor step in induction proofs. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Wolf Lammen, 14-Sep-2013.)
((𝜓𝜑) → (((𝜑𝜒) → (𝜓𝜃)) ↔ (𝜓 → (𝜒𝜃))))

1.2.6  Logical conjunction

This section defines conjunction of two formulas, denoted by infix " " and read "and". It is defined in terms of implication and negation, which is possible in classical logic (but not in intuitionistic logic: see iset.mm).

After the definition, we briefly introduce conversion of simple expressions to and from conjunction. Two simple operations called importation (imp 410) and exportation (ex 416) follow. In the propositions-as-types interpretation, they correspond to uncurrying and currying respectively. They are foundational for this section. Most of the theorems proved here trace back to them, mostly indirectly, in a layered fashion, where more complex expressions are built from simpler ones. Here are some of these successive layers: importation and exportation, commutativity and associativity laws, adding antecedents and simplifying, conjunction of consequents, syllogisms, etc.

As indicated in the "note on definitions" in the section comment for logical equivalence, some theorems containing only implication, negation and conjunction are placed in the section after disjunction since theirs proofs use disjunction (although this is not required since definitions are conservative, see said section comment).

Syntaxwa 399 Extend wff definition to include conjunction ("and").
wff (𝜑𝜓)

Definitiondf-an 400 Define conjunction (logical "and"). Definition of [Margaris] p. 49. When both the left and right operand are true, the result is true; when either is false, the result is false. For example, it is true that (2 = 2 ∧ 3 = 3). After we define the constant true (df-tru 1541) and the constant false (df-fal 1551), we will be able to prove these truth table values: ((⊤ ∧ ⊤) ↔ ⊤) (truantru 1571), ((⊤ ∧ ⊥) ↔ ⊥) (truanfal 1572), ((⊥ ∧ ⊤) ↔ ⊥) (falantru 1573), and ((⊥ ∧ ⊥) ↔ ⊥) (falanfal 1574).

This is our first use of the biconditional connective in a definition; we use the biconditional connective in place of the traditional "<=def=>", which means the same thing, except that we can manipulate the biconditional connective directly in proofs rather than having to rely on an informal definition substitution rule. Note that if we mechanically substitute ¬ (𝜑 → ¬ 𝜓) for (𝜑𝜓), we end up with an instance of previously proved theorem biid 264. This is the justification for the definition, along with the fact that it introduces a new symbol . Contrast with (df-or 845), (wi 4), (df-nan 1483), and (df-xor 1503). (Contributed by NM, 5-Jan-1993.)

((𝜑𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))

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