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Theorem List for Metamath Proof Explorer - 301-400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxchnxbi 301 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
 |-  ( -.  ph  <->  ps )   &    |-  ( ph  <->  ch )   =>    |-  ( -.  ch  <->  ps )
 
Theoremxchnxbir 302 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
 |-  ( -.  ph  <->  ps )   &    |-  ( ch  <->  ph )   =>    |-  ( -.  ch  <->  ps )
 
Theoremxchbinx 303 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
 |-  ( ph  <->  -.  ps )   &    |-  ( ps 
 <->  ch )   =>    |-  ( ph  <->  -.  ch )
 
Theoremxchbinxr 304 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
 |-  ( ph  <->  -.  ps )   &    |-  ( ch 
 <->  ps )   =>    |-  ( ph  <->  -.  ch )
 
Theoremimbi2i 305 Introduce an antecedent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 6-Feb-2013.)
 |-  ( ph  <->  ps )   =>    |-  ( ( ch  ->  ph )  <->  ( ch  ->  ps ) )
 
Theorembibi2i 306 Inference adding a biconditional to the left in an equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 16-May-2013.)
 |-  ( ph  <->  ps )   =>    |-  ( ( ch  <->  ph )  <->  ( ch  <->  ps ) )
 
Theorembibi1i 307 Inference adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   =>    |-  ( ( ph  <->  ch )  <->  ( ps  <->  ch ) )
 
Theorembibi12i 308 The equivalence of two equivalences. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ( ph  <->  ch )  <->  ( ps  <->  th ) )
 
Theoremimbi2d 309 Deduction adding an antecedent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( th  ->  ps )  <->  ( th  ->  ch )
 ) )
 
Theoremimbi1d 310 Deduction adding a consequent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 17-Sep-2013.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( ps  ->  th )  <->  ( ch  ->  th )
 ) )
 
Theorembibi2d 311 Deduction adding a biconditional to the left in an equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( th  <->  ps )  <->  ( th  <->  ch ) ) )
 
Theorembibi1d 312 Deduction adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( ps  <->  th )  <->  ( ch  <->  th ) ) )
 
Theoremimbi12d 313 Deduction joining two equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ( ps  ->  th )  <->  ( ch  ->  ta )
 ) )
 
Theorembibi12d 314 Deduction joining two equivalences to form equivalence of biconditionals. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ( ps  <->  th )  <->  ( ch  <->  ta ) ) )
 
Theoremimbi1 315 Theorem *4.84 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  <->  ps )  ->  (
 ( ph  ->  ch )  <->  ( ps  ->  ch )
 ) )
 
Theoremimbi2 316 Theorem *4.85 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.)
 |-  ( ( ph  <->  ps )  ->  (
 ( ch  ->  ph )  <->  ( ch  ->  ps )
 ) )
 
Theoremimbi1i 317 Introduce a consequent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 17-Sep-2013.)
 |-  ( ph  <->  ps )   =>    |-  ( ( ph  ->  ch )  <->  ( ps  ->  ch ) )
 
Theoremimbi12i 318 Join two logical equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ( ph  ->  ch )  <->  ( ps  ->  th ) )
 
Theorembibi1 319 Theorem *4.86 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  <->  ps )  ->  (
 ( ph  <->  ch )  <->  ( ps  <->  ch ) ) )
 
Theoremcon2bi 320 Contraposition. Theorem *4.12 of [WhiteheadRussell] p. 117. (Contributed by NM, 15-Apr-1995.) (Proof shortened by Wolf Lammen, 3-Jan-2013.)
 |-  ( ( ph  <->  -.  ps )  <->  ( ps  <->  -.  ph ) )
 
Theoremcon2bid 321 A contraposition deduction. (Contributed by NM, 15-Apr-1995.)
 |-  ( ph  ->  ( ps 
 <->  -.  ch ) )   =>    |-  ( ph  ->  ( ch  <->  -.  ps ) )
 
Theoremcon1bid 322 A contraposition deduction. (Contributed by NM, 9-Oct-1999.)
 |-  ( ph  ->  ( -.  ps  <->  ch ) )   =>    |-  ( ph  ->  ( -.  ch  <->  ps ) )
 
Theoremcon1bii 323 A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.)
 |-  ( -.  ph  <->  ps )   =>    |-  ( -.  ps  <->  ph )
 
Theoremcon2bii 324 A contraposition inference. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  -.  ps )   =>    |-  ( ps  <->  -.  ph )
 
Theoremcon1b 325 Contraposition. Bidirectional version of con1 122. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( -.  ph  ->  ps )  <->  ( -.  ps  -> 
 ph ) )
 
Theoremcon2b 326 Contraposition. Bidirectional version of con2 110. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( ph  ->  -. 
 ps )  <->  ( ps  ->  -.  ph ) )
 
Theorembiimt 327 A wff is equivalent to itself with true antecedent. (Contributed by NM, 28-Jan-1996.)
 |-  ( ph  ->  ( ps 
 <->  ( ph  ->  ps )
 ) )
 
Theorempm5.5 328 Theorem *5.5 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  ->  (
 ( ph  ->  ps )  <->  ps ) )
 
Theorema1bi 329 Inference rule introducing a theorem as an antecedent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)
 |-  ph   =>    |-  ( ps  <->  ( ph  ->  ps ) )
 
Theoremmt2bi 330 A false consequent falsifies an antecedent. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.)
 |-  ph   =>    |-  ( -.  ps  <->  ( ps  ->  -.  ph ) )
 
Theoremmtt 331 Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Proof shortened by Wolf Lammen, 12-Nov-2012.)
 |-  ( -.  ph  ->  ( -.  ps  <->  ( ps  ->  ph ) ) )
 
Theorempm5.501 332 Theorem *5.501 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  ->  ( ps 
 <->  ( ph  <->  ps ) ) )
 
Theoremibib 333 Implication in terms of implication and biconditional. (Contributed by NM, 31-Mar-1994.) (Proof shortened by Wolf Lammen, 24-Jan-2013.)
 |-  ( ( ph  ->  ps )  <->  ( ph  ->  (
 ph 
 <->  ps ) ) )
 
Theoremibibr 334 Implication in terms of implication and biconditional. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 21-Dec-2013.)
 |-  ( ( ph  ->  ps )  <->  ( ph  ->  ( ps  <->  ph ) ) )
 
Theoremtbt 335 A wff is equivalent to its equivalence with truth. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ph   =>    |-  ( ps  <->  ( ps  <->  ph ) )
 
Theoremnbn2 336 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.)
 |-  ( -.  ph  ->  ( -.  ps  <->  ( ph  <->  ps ) ) )
 
Theorembibif 337 Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)
 |-  ( -.  ps  ->  ( ( ph  <->  ps )  <->  -.  ph ) )
 
Theoremnbn 338 The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
 |- 
 -.  ph   =>    |-  ( -.  ps  <->  ( ps  <->  ph ) )
 
Theoremnbn3 339 Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.)
 |-  ph   =>    |-  ( -.  ps  <->  ( ps  <->  -.  ph ) )
 
Theorempm5.21im 340 Two propositions are equivalent if they are both false. Closed form of 2false 341. Equivalent to a bi2 191-like version of the xor-connective. (Contributed by Wolf Lammen, 13-May-2013.)
 |-  ( -.  ph  ->  ( -.  ps  ->  ( ph 
 <->  ps ) ) )
 
Theorem2false 341 Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.)
 |- 
 -.  ph   &    |-  -.  ps   =>    |-  ( ph  <->  ps )
 
Theorem2falsed 342 Two falsehoods are equivalent (deduction rule). (Contributed by NM, 11-Oct-2013.)
 |-  ( ph  ->  -.  ps )   &    |-  ( ph  ->  -.  ch )   =>    |-  ( ph  ->  ( ps 
 <->  ch ) )
 
Theorempm5.21ni 343 Two propositions implying a false one are equivalent. (Contributed by NM, 16-Feb-1996.) (Proof shortened by Wolf Lammen, 19-May-2013.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  ps )   =>    |-  ( -.  ps  ->  (
 ph 
 <->  ch ) )
 
Theorempm5.21nii 344 Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 21-May-1999.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  ps )   &    |-  ( ps  ->  (
 ph 
 <->  ch ) )   =>    |-  ( ph  <->  ch )
 
Theorempm5.21ndd 345 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.)
 |-  ( ph  ->  ( ch  ->  ps ) )   &    |-  ( ph  ->  ( th  ->  ps ) )   &    |-  ( ph  ->  ( ps  ->  ( ch  <->  th ) ) )   =>    |-  ( ph  ->  ( ch  <->  th ) )
 
Theorembija 346 Combine antecedents into a single bi-conditional. This inference, reminiscent of ja 155, is reversible: The hypotheses can be deduced from the conclusion alone (see pm5.1im 231 and pm5.21im 340). (Contributed by Wolf Lammen, 13-May-2013.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( -.  ph  ->  ( -.  ps 
 ->  ch ) )   =>    |-  ( ( ph  <->  ps )  ->  ch )
 
Theorempm5.18 347 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.)
 |-  ( ( ph  <->  ps )  <->  -.  ( ph  <->  -.  ps ) )
 
Theoremxor3 348 Two ways to express "exclusive or." (Contributed by NM, 1-Jan-2006.)
 |-  ( -.  ( ph  <->  ps ) 
 <->  ( ph  <->  -.  ps ) )
 
Theoremnbbn 349 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.)
 |-  ( ( -.  ph  <->  ps ) 
 <->  -.  ( ph  <->  ps ) )
 
Theorembiass 350 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.)
 |-  ( ( ( ph  <->  ps ) 
 <->  ch )  <->  ( ph  <->  ( ps  <->  ch ) ) )
 
Theorempm5.19 351 Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.)
 |- 
 -.  ( ph  <->  -.  ph )
 
Theorembi2.04 352 Logical equivalence of commuted antecedents. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 ) 
 <->  ( ps  ->  ( ph  ->  ch ) ) )
 
Theorempm5.4 353 Antecedent absorption implication. Theorem *5.4 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( ph  ->  (
 ph  ->  ps ) )  <->  ( ph  ->  ps ) )
 
Theoremimdi 354 Distributive law for implication. Compare Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 ) 
 <->  ( ( ph  ->  ps )  ->  ( ph  ->  ch ) ) )
 
Theorempm5.41 355 Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 12-Oct-2012.)
 |-  ( ( ( ph  ->  ps )  ->  ( ph  ->  ch ) )  <->  ( ph  ->  ( ps  ->  ch )
 ) )
 
Theorempm4.8 356 Theorem *4.8 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  ->  -.  ph )  <->  -.  ph )
 
Theorempm4.81 357 Theorem *4.81 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( -.  ph  -> 
 ph )  <->  ph )
 
Theoremimim21b 358 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.)
 |-  ( ( ps  ->  ph )  ->  ( (
 ( ph  ->  ch )  ->  ( ps  ->  th )
 ) 
 <->  ( ps  ->  ( ch  ->  th ) ) ) )
 
1.3.6  Logical disjunction and conjunction

Here we define disjunction (logical 'or')  \/ (df-or 361) and conjunction (logical 'and')  /\ (df-an 362). We also define various rules for simplifying and applying them, e.g., olc 375, orc 376, and orcom 378.

 
Syntaxwo 359 Extend wff definition to include disjunction ('or').
 wff  ( ph  \/  ps )
 
Syntaxwa 360 Extend wff definition to include conjunction ('and').
 wff  ( ph  /\  ps )
 
Definitiondf-or 361 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 20621). After we define the constant true  T. (df-tru 1315) and the constant false  F. (df-fal 1316), we will be able to prove these truth table values:  ( (  T.  \/  T.  )  <->  T.  ) (truortru 1336), 
( (  T.  \/  F.  )  <->  T.  ) (truorfal 1337), 
( (  F.  \/  T.  )  <->  T.  ) (falortru 1338), and  ( (  F.  \/  F.  )  <->  F.  ) (falorfal 1339).

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  ( -.  ph  ->  ps ) for  ( ph  \/  ps ), we end up with an instance of previously proved theorem biid 229. This is the justification for the definition, along with the fact that it introduces a new symbol  \/. Contrast with  /\ (df-an 362), 
-> (wi 6),  -/\ (df-nan 1293), and  \/_ (df-xor 1301) . (Contributed by NM, 5-Aug-1993.)

 |-  ( ( ph  \/  ps )  <->  ( -.  ph  ->  ps ) )
 
Definitiondf-an 362 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  T. (df-tru 1315) and the constant false  F. (df-fal 1316), we will be able to prove these truth table values:  ( (  T.  /\  T.  )  <->  T.  ) (truantru 1332), 
( (  T.  /\  F.  )  <->  F.  ) (truanfal 1333),  ( (  F.  /\  T.  )  <->  F.  ) (falantru 1334), and  ( (  F.  /\  F.  )  <->  F.  ) (falanfal 1335).

Contrast with  \/ (df-or 361), 
-> (wi 6),  -/\ (df-nan 1293), and  \/_ (df-xor 1301) . (Contributed by NM, 5-Aug-1993.)

 |-  ( ( ph  /\  ps ) 
 <->  -.  ( ph  ->  -. 
 ps ) )
 
Theorempm4.64 363 Theorem *4.64 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( -.  ph  ->  ps )  <->  ( ph  \/  ps ) )
 
Theorempm2.53 364 Theorem *2.53 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  \/  ps )  ->  ( -.  ph 
 ->  ps ) )
 
Theorempm2.54 365 Theorem *2.54 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( -.  ph  ->  ps )  ->  ( ph  \/  ps ) )
 
Theoremori 366 Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.)
 |-  ( ph  \/  ps )   =>    |-  ( -.  ph  ->  ps )
 
Theoremorri 367 Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.)
 |-  ( -.  ph  ->  ps )   =>    |-  ( ph  \/  ps )
 
Theoremord 368 Deduce implication from disjunction. (Contributed by NM, 18-May-1994.)
 |-  ( ph  ->  ( ps  \/  ch ) )   =>    |-  ( ph  ->  ( -.  ps 
 ->  ch ) )
 
Theoremorrd 369 Deduce implication from disjunction. (Contributed by NM, 27-Nov-1995.)
 |-  ( ph  ->  ( -.  ps  ->  ch )
 )   =>    |-  ( ph  ->  ( ps  \/  ch ) )
 
Theoremjaoi 370 Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  ps )   =>    |-  ( ( ph  \/  ch )  ->  ps )
 
Theoremjaod 371 Deduction disjoining the antecedents of two implications. (Contributed by NM, 18-Aug-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( th  ->  ch ) )   =>    |-  ( ph  ->  (
 ( ps  \/  th )  ->  ch ) )
 
Theoremmpjaod 372 Eliminate a disjunction in a deduction. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( th  ->  ch ) )   &    |-  ( ph  ->  ( ps  \/  th )
 )   =>    |-  ( ph  ->  ch )
 
Theoremorel1 373 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.)
 |-  ( -.  ph  ->  ( ( ph  \/  ps )  ->  ps ) )
 
Theoremorel2 374 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.)
 |-  ( -.  ph  ->  ( ( ps  \/  ph )  ->  ps ) )
 
Theoremolc 375 Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96. (Contributed by NM, 30-Aug-1993.)
 |-  ( ph  ->  ( ps  \/  ph ) )
 
Theoremorc 376 Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104. (Contributed by NM, 30-Aug-1993.)
 |-  ( ph  ->  ( ph  \/  ps ) )
 
Theorempm1.4 377 Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  \/  ps )  ->  ( ps  \/  ph ) )
 
Theoremorcom 378 Commutative law for disjunction. Theorem *4.31 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 15-Nov-2012.)
 |-  ( ( ph  \/  ps )  <->  ( ps  \/  ph ) )
 
Theoremorcomd 379 Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.)
 |-  ( ph  ->  ( ps  \/  ch ) )   =>    |-  ( ph  ->  ( ch  \/  ps ) )
 
Theoremorcoms 380 Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.)
 |-  ( ( ph  \/  ps )  ->  ch )   =>    |-  (
 ( ps  \/  ph )  ->  ch )
 
Theoremorci 381 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
 |-  ph   =>    |-  ( ph  \/  ps )
 
Theoremolci 382 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
 |-  ph   =>    |-  ( ps  \/  ph )
 
Theoremorcd 383 Deduction introducing a disjunct. A translation of natural deduction rule  \/ IR ( \/ insertion right), see natded 4. (Contributed by NM, 20-Sep-2007.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ps  \/  ch ) )
 
Theoremolcd 384 Deduction introducing a disjunct. A translation of natural deduction rule  \/ IL ( \/ insertion left), see natded 4. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ch  \/  ps ) )
 
Theoremorcs 385 Deduction eliminating disjunct. Notational convention: We sometimes suffix with "s" the label of an inference that manipulates an antecedent, leaving the consequent unchanged. The "s" means that the inference eliminates the need for a syllogism (syl 17) -type inference in a proof. (Contributed by NM, 21-Jun-1994.)
 |-  ( ( ph  \/  ps )  ->  ch )   =>    |-  ( ph  ->  ch )
 
Theoremolcs 386 Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
 |-  ( ( ph  \/  ps )  ->  ch )   =>    |-  ( ps  ->  ch )
 
Theorempm2.07 387 Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  ->  ( ph  \/  ph ) )
 
Theorempm2.45 388 Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  \/  ps )  ->  -.  ph )
 
Theorempm2.46 389 Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  \/  ps )  ->  -.  ps )
 
Theorempm2.47 390 Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  \/  ps )  ->  ( -.  ph  \/  ps )
 )
 
Theorempm2.48 391 Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  \/  ps )  ->  ( ph  \/  -.  ps )
 )
 
Theorempm2.49 392 Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  \/  ps )  ->  ( -.  ph  \/  -.  ps ) )
 
Theorempm2.67-2 393 Slight generalization of Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  \/  ch )  ->  ps )  ->  ( ph  ->  ps )
 )
 
Theorempm2.67 394 Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  \/  ps )  ->  ps )  ->  ( ph  ->  ps )
 )
 
Theorempm2.25 395 Theorem *2.25 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  \/  (
 ( ph  \/  ps )  ->  ps ) )
 
Theorembiorf 396 A wff is equivalent to its disjunction with falsehood. Theorem *4.74 of [WhiteheadRussell] p. 121. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2012.)
 |-  ( -.  ph  ->  ( ps  <->  ( ph  \/  ps ) ) )
 
Theorembiortn 397 A wff is equivalent to its negated disjunction with falsehood. (Contributed by NM, 9-Jul-2012.)
 |-  ( ph  ->  ( ps 
 <->  ( -.  ph  \/  ps ) ) )
 
Theorembiorfi 398 A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.)
 |- 
 -.  ph   =>    |-  ( ps  <->  ( ps  \/  ph ) )
 
Theorempm2.621 399 Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  ->  ps )  ->  ( ( ph  \/  ps )  ->  ps ) )
 
Theorempm2.62 400 Theorem *2.62 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Dec-2013.)
 |-  ( ( ph  \/  ps )  ->  ( ( ph  ->  ps )  ->  ps )
 )
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