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Theorem List for Metamath Proof Explorer - 301-400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem3bitr4g 301 More general version of 3bitr4i 290. Useful for converting definitions in a formula. (Contributed by NM, 11-May-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜃𝜓)    &   (𝜏𝜒)       (𝜑 → (𝜃𝜏))
 
Theoremnotnotb 302 Double negation. Theorem *4.13 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-1993.)
(𝜑 ↔ ¬ ¬ 𝜑)
 
TheoremnotnotdOLD 303 Obsolete proof of notnotd 136 as of 27-Mar-2021. (Contributed by Jarvin Udandy, 2-Sep-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝜓)       (𝜑 → ¬ ¬ 𝜓)
 
Theoremcon34b 304 A biconditional form of contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116. (Contributed by NM, 11-May-1993.)
((𝜑𝜓) ↔ (¬ 𝜓 → ¬ 𝜑))
 
Theoremcon4bid 305 A contraposition deduction. (Contributed by NM, 21-May-1994.)
(𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))       (𝜑 → (𝜓𝜒))
 
Theoremnotbid 306 Deduction negating both sides of a logical equivalence. (Contributed by NM, 21-May-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))
 
Theoremnotbi 307 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 308 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 309 A contraposition inference. (Contributed by NM, 21-May-1994.)
𝜑 ↔ ¬ 𝜓)       (𝜑𝜓)
 
Theoremmtbi 310 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 311 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 312 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.)
(𝜑 → ¬ 𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜒)
 
Theoremmtbird 313 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜓)
 
Theoremmtbii 314 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.)
¬ 𝜓    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜒)
 
Theoremmtbiri 315 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.)
¬ 𝜒    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜓)
 
Theoremsylnib 316 A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.)
(𝜑 → ¬ 𝜓)    &   (𝜓𝜒)       (𝜑 → ¬ 𝜒)
 
Theoremsylnibr 317 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 318 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 319 A mixed syllogism inference from a biconditional and an implication. (Contributed by Wolf Lammen, 16-Dec-2013.)
(𝜓𝜑)    &   𝜓𝜒)       𝜑𝜒)
 
Theoremxchnxbi 320 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
𝜑𝜓)    &   (𝜑𝜒)       𝜒𝜓)
 
Theoremxchnxbir 321 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
𝜑𝜓)    &   (𝜒𝜑)       𝜒𝜓)
 
Theoremxchbinx 322 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
(𝜑 ↔ ¬ 𝜓)    &   (𝜓𝜒)       (𝜑 ↔ ¬ 𝜒)
 
Theoremxchbinxr 323 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
(𝜑 ↔ ¬ 𝜓)    &   (𝜒𝜓)       (𝜑 ↔ ¬ 𝜒)
 
Theoremimbi2i 324 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 325 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 326 Inference adding a biconditional to the right in an equivalence. (Contributed by NM, 26-May-1993.)
(𝜑𝜓)       ((𝜑𝜒) ↔ (𝜓𝜒))
 
Theorembibi12i 327 The equivalence of two equivalences. (Contributed by NM, 26-May-1993.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))
 
Theoremimbi2d 328 Deduction adding an antecedent to both sides of a logical equivalence. (Contributed by NM, 11-May-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) ↔ (𝜃𝜒)))
 
Theoremimbi1d 329 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 330 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 331 Deduction adding a biconditional to the right in an equivalence. (Contributed by NM, 11-May-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))
 
Theoremimbi12d 332 Deduction joining two equivalences to form equivalence of implications. (Contributed by NM, 16-May-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
 
Theorembibi12d 333 Deduction joining two equivalences to form equivalence of biconditionals. (Contributed by NM, 26-May-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
 
Theoremimbi12 334 Closed form of imbi12i 338. 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 335 Theorem *4.84 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))
 
Theoremimbi2 336 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 337 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 338 Join two logical equivalences to form equivalence of implications. (Contributed by NM, 1-Aug-1993.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))
 
Theorembibi1 339 Theorem *4.86 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))
 
Theorembitr3 340 Closed nested implication form of bitr3i 264. Derived automatically from bitr3VD 38003. (Contributed by Alan Sare, 31-Dec-2011.)
((𝜑𝜓) → ((𝜑𝜒) → (𝜓𝜒)))
 
Theoremcon2bi 341 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 342 A contraposition deduction. (Contributed by NM, 15-Apr-1995.)
(𝜑 → (𝜓 ↔ ¬ 𝜒))       (𝜑 → (𝜒 ↔ ¬ 𝜓))
 
Theoremcon1bid 343 A contraposition deduction. (Contributed by NM, 9-Oct-1999.)
(𝜑 → (¬ 𝜓𝜒))       (𝜑 → (¬ 𝜒𝜓))
 
Theoremcon1bii 344 A contraposition inference. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.)
𝜑𝜓)       𝜓𝜑)
 
Theoremcon2bii 345 A contraposition inference. (Contributed by NM, 12-Mar-1993.)
(𝜑 ↔ ¬ 𝜓)       (𝜓 ↔ ¬ 𝜑)
 
Theoremcon1b 346 Contraposition. Bidirectional version of con1 141. (Contributed by NM, 3-Jan-1993.)
((¬ 𝜑𝜓) ↔ (¬ 𝜓𝜑))
 
Theoremcon2b 347 Contraposition. Bidirectional version of con2 128. (Contributed by NM, 12-Mar-1993.)
((𝜑 → ¬ 𝜓) ↔ (𝜓 → ¬ 𝜑))
 
Theorembiimt 348 A wff is equivalent to itself with true antecedent. (Contributed by NM, 28-Jan-1996.)
(𝜑 → (𝜓 ↔ (𝜑𝜓)))
 
Theorempm5.5 349 Theorem *5.5 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
(𝜑 → ((𝜑𝜓) ↔ 𝜓))
 
Theorema1bi 350 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 351 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 352 Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Proof shortened by Wolf Lammen, 12-Nov-2012.)
𝜑 → (¬ 𝜓 ↔ (𝜓𝜑)))
 
Theoremimnot 353 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), pm2.21 118 (false antecedent), pm5.5 349 (true antecedent). (Contributed by Mario Carneiro, 26-Apr-2019.) (Proof shortened by Wolf Lammen, 26-May-2019.)
𝜓 → ((𝜑𝜓) ↔ ¬ 𝜑))
 
Theorempm5.501 354 Theorem *5.501 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
(𝜑 → (𝜓 ↔ (𝜑𝜓)))
 
Theoremibib 355 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 356 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 357 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 358 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 359 Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)
𝜓 → ((𝜑𝜓) ↔ ¬ 𝜑))
 
Theoremnbn 360 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 361 Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.)
𝜑       𝜓 ↔ (𝜓 ↔ ¬ 𝜑))
 
Theorempm5.21im 362 Two propositions are equivalent if they are both false. Closed form of 2false 363. Equivalent to a biimpr 208-like version of the xor-connective. (Contributed by Wolf Lammen, 13-May-2013.)
𝜑 → (¬ 𝜓 → (𝜑𝜓)))
 
Theorem2false 363 Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.)
¬ 𝜑    &    ¬ 𝜓       (𝜑𝜓)
 
Theorem2falsed 364 Two falsehoods are equivalent (deduction rule). (Contributed by NM, 11-Oct-2013.)
(𝜑 → ¬ 𝜓)    &   (𝜑 → ¬ 𝜒)       (𝜑 → (𝜓𝜒))
 
Theorempm5.21ni 365 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 366 Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 21-May-1999.)
(𝜑𝜓)    &   (𝜒𝜓)    &   (𝜓 → (𝜑𝜒))       (𝜑𝜒)
 
Theorempm5.21ndd 367 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 368 Combine antecedents into a single biconditional. This inference, reminiscent of ja 171, is reversible: The hypotheses can be deduced from the conclusion alone (see pm5.1im 251 and pm5.21im 362). (Contributed by Wolf Lammen, 13-May-2013.)
(𝜑 → (𝜓𝜒))    &   𝜑 → (¬ 𝜓𝜒))       ((𝜑𝜓) → 𝜒)
 
Theorempm5.18 369 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 370 Two ways to express "exclusive or." (Contributed by NM, 1-Jan-2006.)
(¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
 
Theoremnbbn 371 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 372 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)))
 
Theorempm5.19 373 Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.)
¬ (𝜑 ↔ ¬ 𝜑)
 
Theorembi2.04 374 Logical equivalence of commuted antecedents. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 11-May-1993.)
((𝜑 → (𝜓𝜒)) ↔ (𝜓 → (𝜑𝜒)))
 
Theorempm5.4 375 Antecedent absorption implication. Theorem *5.4 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
((𝜑 → (𝜑𝜓)) ↔ (𝜑𝜓))
 
Theoremimdi 376 Distributive law for implication. Compare Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒)))
 
Theorempm5.41 377 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)))
 
Theorempm4.8 378 Theorem *4.8 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
((𝜑 → ¬ 𝜑) ↔ ¬ 𝜑)
 
Theorempm4.81 379 Theorem *4.81 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑𝜑) ↔ 𝜑)
 
Theoremimim21b 380 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 383) and conjunction (logical 'and') (df-an 384). We also define various rules for simplifying and applying them, e.g., olc 397, orc 398, and orcom 400.

 
Syntaxwo 381 Extend wff definition to include disjunction ('or').
wff (𝜑𝜓)
 
Syntaxwa 382 Extend wff definition to include conjunction ('and').
wff (𝜑𝜓)
 
Definitiondf-or 383 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 26408). After we define the constant true (df-tru 1477) and the constant false (df-fal 1480), we will be able to prove these truth table values: ((⊤ ∨ ⊤) ↔ ⊤) (truortru 1500), ((⊤ ∨ ⊥) ↔ ⊤) (truorfal 1501), ((⊥ ∨ ⊤) ↔ ⊤) (falortru 1502), and ((⊥ ∨ ⊥) ↔ ⊥) (falorfal 1503).

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 249. This is the justification for the definition, along with the fact that it introduces a new symbol . Contrast with (df-an 384), (wi 4), (df-nan 1439), and (df-xor 1456) . (Contributed by NM, 27-Dec-1992.)

((𝜑𝜓) ↔ (¬ 𝜑𝜓))
 
Definitiondf-an 384 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 1477) and the constant false (df-fal 1480), we will be able to prove these truth table values: ((⊤ ∧ ⊤) ↔ ⊤) (truantru 1496), ((⊤ ∧ ⊥) ↔ ⊥) (truanfal 1497), ((⊥ ∧ ⊤) ↔ ⊥) (falantru 1498), and ((⊥ ∧ ⊥) ↔ ⊥) (falanfal 1499).

Contrast with (df-or 383), (wi 4), (df-nan 1439), and (df-xor 1456) . (Contributed by NM, 5-Jan-1993.)

((𝜑𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))
 
Theorempm4.64 385 Theorem *4.64 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑𝜓) ↔ (𝜑𝜓))
 
Theorempm2.53 386 Theorem *2.53 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → (¬ 𝜑𝜓))
 
Theorempm2.54 387 Theorem *2.54 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑𝜓) → (𝜑𝜓))
 
Theoremori 388 Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.)
(𝜑𝜓)       𝜑𝜓)
 
Theoremorri 389 Infer disjunction from implication. (Contributed by NM, 11-Jun-1994.)
𝜑𝜓)       (𝜑𝜓)
 
Theoremord 390 Deduce implication from disjunction. (Contributed by NM, 18-May-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (¬ 𝜓𝜒))
 
Theoremorrd 391 Deduce disjunction from implication. (Contributed by NM, 27-Nov-1995.)
(𝜑 → (¬ 𝜓𝜒))       (𝜑 → (𝜓𝜒))
 
Theoremjaoi 392 Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.)
(𝜑𝜓)    &   (𝜒𝜓)       ((𝜑𝜒) → 𝜓)
 
Theoremjaod 393 Deduction disjoining the antecedents of two implications. (Contributed by NM, 18-Aug-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))       (𝜑 → ((𝜓𝜃) → 𝜒))
 
Theoremmpjaod 394 Eliminate a disjunction in a deduction. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))    &   (𝜑 → (𝜓𝜃))       (𝜑𝜒)
 
Theoremorel1 395 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 396 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 397 Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96. (Contributed by NM, 30-Aug-1993.)
(𝜑 → (𝜓𝜑))
 
Theoremorc 398 Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104. (Contributed by NM, 30-Aug-1993.)
(𝜑 → (𝜑𝜓))
 
Theorempm1.4 399 Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → (𝜓𝜑))
 
Theoremorcom 400 Commutative law for disjunction. Theorem *4.31 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 15-Nov-2012.)
((𝜑𝜓) ↔ (𝜓𝜑))
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