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

Theorem3bitr4rd 301 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜓))    &   (𝜑 → (𝜏𝜒))       (𝜑 → (𝜏𝜃))

Theorem3bitr3g 302 More general version of 3bitr3i 290. Useful for converting definitions in a formula. (Contributed by NM, 4-Jun-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜓𝜃)    &   (𝜒𝜏)       (𝜑 → (𝜃𝜏))

Theorem3bitr4g 303 More general version of 3bitr4i 292. Useful for converting definitions in a formula. (Contributed by NM, 11-May-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜃𝜓)    &   (𝜏𝜒)       (𝜑 → (𝜃𝜏))

Theoremnotnotb 304 Double negation. Theorem *4.13 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-1993.)
(𝜑 ↔ ¬ ¬ 𝜑)

Theoremcon34b 305 A biconditional form of contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116. (Contributed by NM, 11-May-1993.)
((𝜑𝜓) ↔ (¬ 𝜓 → ¬ 𝜑))

Theoremcon4bid 306 A contraposition deduction. (Contributed by NM, 21-May-1994.)
(𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))       (𝜑 → (𝜓𝜒))

Theoremnotbid 307 Deduction negating both sides of a logical equivalence. (Contributed by NM, 21-May-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))

Theoremnotbi 308 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 309 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 310 A contraposition inference. (Contributed by NM, 21-May-1994.)
𝜑 ↔ ¬ 𝜓)       (𝜑𝜓)

Theoremmtbi 311 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 312 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 313 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.)
(𝜑 → ¬ 𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜒)

Theoremmtbird 314 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜓)

Theoremmtbii 315 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.)
¬ 𝜓    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜒)

Theoremmtbiri 316 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.)
¬ 𝜒    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜓)

Theoremsylnib 317 A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.)
(𝜑 → ¬ 𝜓)    &   (𝜓𝜒)       (𝜑 → ¬ 𝜒)

Theoremsylnibr 318 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 319 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 320 A mixed syllogism inference from a biconditional and an implication. (Contributed by Wolf Lammen, 16-Dec-2013.)
(𝜓𝜑)    &   𝜓𝜒)       𝜑𝜒)

Theoremxchnxbi 321 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
𝜑𝜓)    &   (𝜑𝜒)       𝜒𝜓)

Theoremxchnxbir 322 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
𝜑𝜓)    &   (𝜒𝜑)       𝜒𝜓)

Theoremxchbinx 323 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
(𝜑 ↔ ¬ 𝜓)    &   (𝜓𝜒)       (𝜑 ↔ ¬ 𝜒)

Theoremxchbinxr 324 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
(𝜑 ↔ ¬ 𝜓)    &   (𝜒𝜓)       (𝜑 ↔ ¬ 𝜒)

Theoremimbi2i 325 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))

Theorembibi2i 326 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 327 Inference adding a biconditional to the right in an equivalence. (Contributed by NM, 26-May-1993.)
(𝜑𝜓)       ((𝜑𝜒) ↔ (𝜓𝜒))

Theorembibi12i 328 The equivalence of two equivalences. (Contributed by NM, 26-May-1993.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))

Theoremimbi2d 329 Deduction adding an antecedent to both sides of a logical equivalence. (Contributed by NM, 11-May-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) ↔ (𝜃𝜒)))

Theoremimbi1d 330 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 331 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 332 Deduction adding a biconditional to the right in an equivalence. (Contributed by NM, 11-May-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))

Theoremimbi12d 333 Deduction joining two equivalences to form equivalence of implications. (Contributed by NM, 16-May-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))

Theorembibi12d 334 Deduction joining two equivalences to form equivalence of biconditionals. (Contributed by NM, 26-May-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))

Theoremimbi12 335 Closed form of imbi12i 339. Was automatically derived from its "Virtual Deduction" version and Metamath's "minimize" command. (Contributed by Alan Sare, 18-Mar-2012.)
((𝜑𝜓) → ((𝜒𝜃) → ((𝜑𝜒) ↔ (𝜓𝜃))))

Theoremimbi1 336 Theorem *4.84 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))

Theoremimbi2 337 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 338 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 339 Join two logical equivalences to form equivalence of implications. (Contributed by NM, 1-Aug-1993.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))

Theorembibi1 340 Theorem *4.86 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))

Theorembitr3 341 Closed nested implication form of bitr3i 266. Derived automatically from bitr3VD 39398. (Contributed by Alan Sare, 31-Dec-2011.)
((𝜑𝜓) → ((𝜑𝜒) → (𝜓𝜒)))

Theoremcon2bi 342 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 343 A contraposition deduction. (Contributed by NM, 15-Apr-1995.)
(𝜑 → (𝜓 ↔ ¬ 𝜒))       (𝜑 → (𝜒 ↔ ¬ 𝜓))

Theoremcon1bid 344 A contraposition deduction. (Contributed by NM, 9-Oct-1999.)
(𝜑 → (¬ 𝜓𝜒))       (𝜑 → (¬ 𝜒𝜓))

Theoremcon1bii 345 A contraposition inference. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.)
𝜑𝜓)       𝜓𝜑)

Theoremcon2bii 346 A contraposition inference. (Contributed by NM, 12-Mar-1993.)
(𝜑 ↔ ¬ 𝜓)       (𝜓 ↔ ¬ 𝜑)

Theoremcon1b 347 Contraposition. Bidirectional version of con1 143. (Contributed by NM, 3-Jan-1993.)
((¬ 𝜑𝜓) ↔ (¬ 𝜓𝜑))

Theoremcon2b 348 Contraposition. Bidirectional version of con2 130. (Contributed by NM, 12-Mar-1993.)
((𝜑 → ¬ 𝜓) ↔ (𝜓 → ¬ 𝜑))

Theorembiimt 349 A wff is equivalent to itself with true antecedent. (Contributed by NM, 28-Jan-1996.)
(𝜑 → (𝜓 ↔ (𝜑𝜓)))

Theorempm5.5 350 Theorem *5.5 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
(𝜑 → ((𝜑𝜓) ↔ 𝜓))

Theorema1bi 351 Inference rule introducing a theorem as an antecedent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)
𝜑       (𝜓 ↔ (𝜑𝜓))

Theoremmt2bi 352 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 353 Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Proof shortened by Wolf Lammen, 12-Nov-2012.)
𝜑 → (¬ 𝜓 ↔ (𝜓𝜑)))

Theoremimnot 354 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), pm2.21 120 (false antecedent), pm5.5 350 (true antecedent). (Contributed by Mario Carneiro, 26-Apr-2019.) (Proof shortened by Wolf Lammen, 26-May-2019.)
𝜓 → ((𝜑𝜓) ↔ ¬ 𝜑))

Theorempm5.501 355 Theorem *5.501 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
(𝜑 → (𝜓 ↔ (𝜑𝜓)))

Theoremibib 356 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 357 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 358 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 359 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 360 Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)
𝜓 → ((𝜑𝜓) ↔ ¬ 𝜑))

Theoremnbn 361 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 362 Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.)
𝜑       𝜓 ↔ (𝜓 ↔ ¬ 𝜑))

Theorempm5.21im 363 Two propositions are equivalent if they are both false. Closed form of 2false 364. Equivalent to a biimpr 210-like version of the xor-connective. (Contributed by Wolf Lammen, 13-May-2013.)
𝜑 → (¬ 𝜓 → (𝜑𝜓)))

Theorem2false 364 Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.)
¬ 𝜑    &    ¬ 𝜓       (𝜑𝜓)

Theorem2falsed 365 Two falsehoods are equivalent (deduction rule). (Contributed by NM, 11-Oct-2013.)
(𝜑 → ¬ 𝜓)    &   (𝜑 → ¬ 𝜒)       (𝜑 → (𝜓𝜒))

Theorempm5.21ni 366 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 367 Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 21-May-1999.)
(𝜑𝜓)    &   (𝜒𝜓)    &   (𝜓 → (𝜑𝜒))       (𝜑𝜒)

Theorempm5.21ndd 368 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 369 Combine antecedents into a single biconditional. This inference, reminiscent of ja 173, is reversible: The hypotheses can be deduced from the conclusion alone (see pm5.1im 253 and pm5.21im 363). (Contributed by Wolf Lammen, 13-May-2013.)
(𝜑 → (𝜓𝜒))    &   𝜑 → (¬ 𝜓𝜒))       ((𝜑𝜓) → 𝜒)

Theorempm5.18 370 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 371 Two ways to express "exclusive or." (Contributed by NM, 1-Jan-2006.)
(¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))

Theoremnbbn 372 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 373 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)))

Theorempm5.19 374 Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.)
¬ (𝜑 ↔ ¬ 𝜑)

Theorembi2.04 375 Logical equivalence of commuted antecedents. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 11-May-1993.)
((𝜑 → (𝜓𝜒)) ↔ (𝜓 → (𝜑𝜒)))

Theorempm5.4 376 Antecedent absorption implication. Theorem *5.4 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
((𝜑 → (𝜑𝜓)) ↔ (𝜑𝜓))

Theoremimdi 377 Distributive law for implication. Compare Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒)))

Theorempm5.41 378 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)))

Theorempm4.8 379 Theorem *4.8 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
((𝜑 → ¬ 𝜑) ↔ ¬ 𝜑)

Theorempm4.81 380 Theorem *4.81 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑𝜑) ↔ 𝜑)

Theoremimim21b 381 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 disjunction and conjunction

Here we define disjunction (logical 'or') (df-or 384) and conjunction (logical 'and') (df-an 385). We also define various rules for simplifying and applying them, e.g., olc 398, orc 399, and orcom 401.

Syntaxwo 382 Extend wff definition to include disjunction ('or').
wff (𝜑𝜓)

Syntaxwa 383 Extend wff definition to include conjunction ('and').
wff (𝜑𝜓)

Definitiondf-or 384 Define disjunction (logical 'or'). Definition of [Margaris] p. 49. When the left operand, right operand, or both are true, the result is true; when both sides are false, the result is false. For example, it is true that (2 = 3 ∨ 4 = 4) (ex-or 27408). After we define the constant true (df-tru 1526) and the constant false (df-fal 1529), we will be able to prove these truth table values: ((⊤ ∨ ⊤) ↔ ⊤) (truortru 1550), ((⊤ ∨ ⊥) ↔ ⊤) (truorfal 1551), ((⊥ ∨ ⊤) ↔ ⊤) (falortru 1552), and ((⊥ ∨ ⊥) ↔ ⊥) (falorfal 1553).

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 𝜑𝜓) for (𝜑𝜓), we end up with an instance of previously proved theorem biid 251. This is the justification for the definition, along with the fact that it introduces a new symbol . Contrast with (df-an 385), (wi 4), (df-nan 1488), and (df-xor 1505) . (Contributed by NM, 27-Dec-1992.)

((𝜑𝜓) ↔ (¬ 𝜑𝜓))

Definitiondf-an 385 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 1526) and the constant false (df-fal 1529), we will be able to prove these truth table values: ((⊤ ∧ ⊤) ↔ ⊤) (truantru 1546), ((⊤ ∧ ⊥) ↔ ⊥) (truanfal 1547), ((⊥ ∧ ⊤) ↔ ⊥) (falantru 1548), and ((⊥ ∧ ⊥) ↔ ⊥) (falanfal 1549).

Contrast with (df-or 384), (wi 4), (df-nan 1488), and (df-xor 1505) . (Contributed by NM, 5-Jan-1993.)

((𝜑𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))

Theorempm4.64 386 Theorem *4.64 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑𝜓) ↔ (𝜑𝜓))

Theorempm2.53 387 Theorem *2.53 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → (¬ 𝜑𝜓))

Theorempm2.54 388 Theorem *2.54 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑𝜓) → (𝜑𝜓))

Theoremori 389 Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.)
(𝜑𝜓)       𝜑𝜓)

Theoremorri 390 Infer disjunction from implication. (Contributed by NM, 11-Jun-1994.)
𝜑𝜓)       (𝜑𝜓)

Theoremord 391 Deduce implication from disjunction. (Contributed by NM, 18-May-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (¬ 𝜓𝜒))

Theoremorrd 392 Deduce disjunction from implication. (Contributed by NM, 27-Nov-1995.)
(𝜑 → (¬ 𝜓𝜒))       (𝜑 → (𝜓𝜒))

Theoremjaoi 393 Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.)
(𝜑𝜓)    &   (𝜒𝜓)       ((𝜑𝜒) → 𝜓)

Theoremjaod 394 Deduction disjoining the antecedents of two implications. (Contributed by NM, 18-Aug-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))       (𝜑 → ((𝜓𝜃) → 𝜒))

Theoremmpjaod 395 Eliminate a disjunction in a deduction. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))    &   (𝜑 → (𝜓𝜃))       (𝜑𝜒)

Theoremorel1 396 Elimination of disjunction by denial of a disjunct. Theorem *2.55 of [WhiteheadRussell] p. 107. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Wolf Lammen, 21-Jul-2012.)
𝜑 → ((𝜑𝜓) → 𝜓))

Theoremorel2 397 Elimination of disjunction by denial of a disjunct. Theorem *2.56 of [WhiteheadRussell] p. 107. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Wolf Lammen, 5-Apr-2013.)
𝜑 → ((𝜓𝜑) → 𝜓))

Theoremolc 398 Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96. (Contributed by NM, 30-Aug-1993.)
(𝜑 → (𝜓𝜑))

Theoremorc 399 Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104. (Contributed by NM, 30-Aug-1993.)
(𝜑 → (𝜑𝜓))

Theorempm1.4 400 Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → (𝜓𝜑))

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