P. 4 of Theorem List - New Foundations Explorer

Theorem List for New Foundations Explorer - 301-400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	xchbinx 301	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
⊢ (φ ↔ ¬ ψ)    &   ⊢ (ψ ↔ χ)    ⇒   ⊢ (φ ↔ ¬ χ)
Theorem	xchbinxr 302	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
⊢ (φ ↔ ¬ ψ)    &   ⊢ (χ ↔ ψ)    ⇒   ⊢ (φ ↔ ¬ χ)
Theorem	imbi2i 303	Introduce an antecedent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 6-Feb-2013.)
⊢ (φ ↔ ψ)    ⇒   ⊢ ((χ → φ) ↔ (χ → ψ))
Theorem	bibi2i 304	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.)
⊢ (φ ↔ ψ)    ⇒   ⊢ ((χ ↔ φ) ↔ (χ ↔ ψ))
Theorem	bibi1i 305	Inference adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)
⊢ (φ ↔ ψ)    ⇒   ⊢ ((φ ↔ χ) ↔ (ψ ↔ χ))
Theorem	bibi12i 306	The equivalence of two equivalences. (Contributed by NM, 5-Aug-1993.)
⊢ (φ ↔ ψ)    &   ⊢ (χ ↔ θ)    ⇒   ⊢ ((φ ↔ χ) ↔ (ψ ↔ θ))
Theorem	imbi2d 307	Deduction adding an antecedent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → ((θ → ψ) ↔ (θ → χ)))
Theorem	imbi1d 308	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.)
⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → ((ψ → θ) ↔ (χ → θ)))
Theorem	bibi2d 309	Deduction adding a biconditional to the left in an equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.)
⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → ((θ ↔ ψ) ↔ (θ ↔ χ)))
Theorem	bibi1d 310	Deduction adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)
⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → ((ψ ↔ θ) ↔ (χ ↔ θ)))
Theorem	imbi12d 311	Deduction joining two equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.)
⊢ (φ → (ψ ↔ χ))    &   ⊢ (φ → (θ ↔ τ))    ⇒   ⊢ (φ → ((ψ → θ) ↔ (χ → τ)))
Theorem	bibi12d 312	Deduction joining two equivalences to form equivalence of biconditionals. (Contributed by NM, 5-Aug-1993.)
⊢ (φ → (ψ ↔ χ))    &   ⊢ (φ → (θ ↔ τ))    ⇒   ⊢ (φ → ((ψ ↔ θ) ↔ (χ ↔ τ)))
Theorem	imbi1 313	Theorem *4.84 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
⊢ ((φ ↔ ψ) → ((φ → χ) ↔ (ψ → χ)))
Theorem	imbi2 314	Theorem *4.85 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.)
⊢ ((φ ↔ ψ) → ((χ → φ) ↔ (χ → ψ)))
Theorem	imbi1i 315	Introduce a consequent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 17-Sep-2013.)
⊢ (φ ↔ ψ)    ⇒   ⊢ ((φ → χ) ↔ (ψ → χ))
Theorem	imbi12i 316	Join two logical equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.)
⊢ (φ ↔ ψ)    &   ⊢ (χ ↔ θ)    ⇒   ⊢ ((φ → χ) ↔ (ψ → θ))
Theorem	bibi1 317	Theorem *4.86 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
⊢ ((φ ↔ ψ) → ((φ ↔ χ) ↔ (ψ ↔ χ)))
Theorem	con2bi 318	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))
Theorem	con2bid 319	A contraposition deduction. (Contributed by NM, 15-Apr-1995.)
⊢ (φ → (ψ ↔ ¬ χ))    ⇒   ⊢ (φ → (χ ↔ ¬ ψ))
Theorem	con1bid 320	A contraposition deduction. (Contributed by NM, 9-Oct-1999.)
⊢ (φ → (¬ ψ ↔ χ))    ⇒   ⊢ (φ → (¬ χ ↔ ψ))
Theorem	con1bii 321	A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.)
⊢ (¬ φ ↔ ψ)    ⇒   ⊢ (¬ ψ ↔ φ)
Theorem	con2bii 322	A contraposition inference. (Contributed by NM, 5-Aug-1993.)
⊢ (φ ↔ ¬ ψ)    ⇒   ⊢ (ψ ↔ ¬ φ)
Theorem	con1b 323	Contraposition. Bidirectional version of con1 120. (Contributed by NM, 5-Aug-1993.)
⊢ ((¬ φ → ψ) ↔ (¬ ψ → φ))
Theorem	con2b 324	Contraposition. Bidirectional version of con2 108. (Contributed by NM, 5-Aug-1993.)
⊢ ((φ → ¬ ψ) ↔ (ψ → ¬ φ))
Theorem	biimt 325	A wff is equivalent to itself with true antecedent. (Contributed by NM, 28-Jan-1996.)
⊢ (φ → (ψ ↔ (φ → ψ)))
Theorem	pm5.5 326	Theorem *5.5 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ (φ → ((φ → ψ) ↔ ψ))
Theorem	a1bi 327	Inference rule introducing a theorem as an antecedent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)
⊢ φ    ⇒   ⊢ (ψ ↔ (φ → ψ))
Theorem	mt2bi 328	A false consequent falsifies an antecedent. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.)
⊢ φ    ⇒   ⊢ (¬ ψ ↔ (ψ → ¬ φ))
Theorem	mtt 329	Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Proof shortened by Wolf Lammen, 12-Nov-2012.)
⊢ (¬ φ → (¬ ψ ↔ (ψ → φ)))
Theorem	pm5.501 330	Theorem *5.501 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ (φ → (ψ ↔ (φ ↔ ψ)))
Theorem	ibib 331	Implication in terms of implication and biconditional. (Contributed by NM, 31-Mar-1994.) (Proof shortened by Wolf Lammen, 24-Jan-2013.)
⊢ ((φ → ψ) ↔ (φ → (φ ↔ ψ)))
Theorem	ibibr 332	Implication in terms of implication and biconditional. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 21-Dec-2013.)
⊢ ((φ → ψ) ↔ (φ → (ψ ↔ φ)))
Theorem	tbt 333	A wff is equivalent to its equivalence with truth. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ φ    ⇒   ⊢ (ψ ↔ (ψ ↔ φ))
Theorem	nbn2 334	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 ↔ (φ ↔ ψ)))
Theorem	bibif 335	Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)
⊢ (¬ ψ → ((φ ↔ ψ) ↔ ¬ φ))
Theorem	nbn 336	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.)
⊢ ¬ φ    ⇒   ⊢ (¬ ψ ↔ (ψ ↔ φ))
Theorem	nbn3 337	Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.)
⊢ φ    ⇒   ⊢ (¬ ψ ↔ (ψ ↔ ¬ φ))
Theorem	pm5.21im 338	Two propositions are equivalent if they are both false. Closed form of 2false 339. Equivalent to a bi2 189-like version of the xor-connective. (Contributed by Wolf Lammen, 13-May-2013.)
⊢ (¬ φ → (¬ ψ → (φ ↔ ψ)))
Theorem	2false 339	Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.)
⊢ ¬ φ    &   ⊢ ¬ ψ    ⇒   ⊢ (φ ↔ ψ)
Theorem	2falsed 340	Two falsehoods are equivalent (deduction rule). (Contributed by NM, 11-Oct-2013.)
⊢ (φ → ¬ ψ)    &   ⊢ (φ → ¬ χ)    ⇒   ⊢ (φ → (ψ ↔ χ))
Theorem	pm5.21ni 341	Two propositions implying a false one are equivalent. (Contributed by NM, 16-Feb-1996.) (Proof shortened by Wolf Lammen, 19-May-2013.)
⊢ (φ → ψ)    &   ⊢ (χ → ψ)    ⇒   ⊢ (¬ ψ → (φ ↔ χ))
Theorem	pm5.21nii 342	Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 21-May-1999.)
⊢ (φ → ψ)    &   ⊢ (χ → ψ)    &   ⊢ (ψ → (φ ↔ χ))    ⇒   ⊢ (φ ↔ χ)
Theorem	pm5.21ndd 343	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 → (χ → ψ))    &   ⊢ (φ → (θ → ψ))    &   ⊢ (φ → (ψ → (χ ↔ θ)))    ⇒   ⊢ (φ → (χ ↔ θ))
Theorem	bija 344	Combine antecedents into a single bi-conditional. This inference, reminiscent of ja 153, is reversible: The hypotheses can be deduced from the conclusion alone (see pm5.1im 229 and pm5.21im 338). (Contributed by Wolf Lammen, 13-May-2013.)
⊢ (φ → (ψ → χ))    &   ⊢ (¬ φ → (¬ ψ → χ))    ⇒   ⊢ ((φ ↔ ψ) → χ)
Theorem	pm5.18 345	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.)
⊢ ((φ ↔ ψ) ↔ ¬ (φ ↔ ¬ ψ))
Theorem	xor3 346	Two ways to express "exclusive or." (Contributed by NM, 1-Jan-2006.)
⊢ (¬ (φ ↔ ψ) ↔ (φ ↔ ¬ ψ))
Theorem	nbbn 347	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.)
⊢ ((¬ φ ↔ ψ) ↔ ¬ (φ ↔ ψ))
Theorem	biass 348	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.)
⊢ (((φ ↔ ψ) ↔ χ) ↔ (φ ↔ (ψ ↔ χ)))
Theorem	pm5.19 349	Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.)
⊢ ¬ (φ ↔ ¬ φ)
Theorem	bi2.04 350	Logical equivalence of commuted antecedents. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 5-Aug-1993.)
⊢ ((φ → (ψ → χ)) ↔ (ψ → (φ → χ)))
Theorem	pm5.4 351	Antecedent absorption implication. Theorem *5.4 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
⊢ ((φ → (φ → ψ)) ↔ (φ → ψ))
Theorem	imdi 352	Distributive law for implication. Compare Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
⊢ ((φ → (ψ → χ)) ↔ ((φ → ψ) → (φ → χ)))
Theorem	pm5.41 353	Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 12-Oct-2012.)
⊢ (((φ → ψ) → (φ → χ)) ↔ (φ → (ψ → χ)))
Theorem	pm4.8 354	Theorem *4.8 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
⊢ ((φ → ¬ φ) ↔ ¬ φ)
Theorem	pm4.81 355	Theorem *4.81 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
⊢ ((¬ φ → φ) ↔ φ)
Theorem	imim21b 356	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.)
⊢ ((ψ → φ) → (((φ → χ) → (ψ → θ)) ↔ (ψ → (χ → θ))))
1.2.6  Logical disjunction and conjunction Here we define disjunction (logical 'or') ∨ (df-or 359) and conjunction (logical 'and') ∧ (df-an 360). We also define various rules for simplifying and applying them, e.g., olc 373, orc 374, and orcom 376.
Syntax	wo 357	Extend wff definition to include disjunction ('or').
wff (φ ∨ ψ)
Syntax	wa 358	Extend wff definition to include conjunction ('and').
wff (φ ∧ ψ)
Definition	df-or 359	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) (see ex-or in set.mm). After we define the constant true ⊤ (df-tru 1319) and the constant false ⊥ (df-fal 1320), we will be able to prove these truth table values: (( ⊤ ∨ ⊤ ) ↔ ⊤ ) (truortru 1340), (( ⊤ ∨ ⊥ ) ↔ ⊤ ) (truorfal 1341), (( ⊥ ∨ ⊤ ) ↔ ⊤ ) (falortru 1342), and (( ⊥ ∨ ⊥ ) ↔ ⊥ ) (falorfal 1343). 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 (φ ∨ ψ), we end up with an instance of previously proved theorem biid 227. This is the justification for the definition, along with the fact that it introduces a new symbol ∨. Contrast with ∧ (df-an 360), → (wi 4), ⊼ (df-nan 1288), and ⊻ (df-xor 1305) . (Contributed by NM, 5-Aug-1993.)
⊢ ((φ ∨ ψ) ↔ (¬ φ → ψ))
Definition	df-an 360	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 1319) and the constant false ⊥ (df-fal 1320), we will be able to prove these truth table values: (( ⊤ ∧ ⊤ ) ↔ ⊤ ) (truantru 1336), (( ⊤ ∧ ⊥ ) ↔ ⊥ ) (truanfal 1337), (( ⊥ ∧ ⊤ ) ↔ ⊥ ) (falantru 1338), and (( ⊥ ∧ ⊥ ) ↔ ⊥ ) (falanfal 1339). Contrast with ∨ (df-or 359), → (wi 4), ⊼ (df-nan 1288), and ⊻ (df-xor 1305) . (Contributed by NM, 5-Aug-1993.)
⊢ ((φ ∧ ψ) ↔ ¬ (φ → ¬ ψ))
Theorem	pm4.64 361	Theorem *4.64 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
⊢ ((¬ φ → ψ) ↔ (φ ∨ ψ))
Theorem	pm2.53 362	Theorem *2.53 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ ((φ ∨ ψ) → (¬ φ → ψ))
Theorem	pm2.54 363	Theorem *2.54 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ ((¬ φ → ψ) → (φ ∨ ψ))
Theorem	ori 364	Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.)
⊢ (φ ∨ ψ)    ⇒   ⊢ (¬ φ → ψ)
Theorem	orri 365	Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.)
⊢ (¬ φ → ψ)    ⇒   ⊢ (φ ∨ ψ)
Theorem	ord 366	Deduce implication from disjunction. (Contributed by NM, 18-May-1994.)
⊢ (φ → (ψ ∨ χ))    ⇒   ⊢ (φ → (¬ ψ → χ))
Theorem	orrd 367	Deduce implication from disjunction. (Contributed by NM, 27-Nov-1995.)
⊢ (φ → (¬ ψ → χ))    ⇒   ⊢ (φ → (ψ ∨ χ))
Theorem	jaoi 368	Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.)
⊢ (φ → ψ)    &   ⊢ (χ → ψ)    ⇒   ⊢ ((φ ∨ χ) → ψ)
Theorem	jaod 369	Deduction disjoining the antecedents of two implications. (Contributed by NM, 18-Aug-1994.)
⊢ (φ → (ψ → χ))    &   ⊢ (φ → (θ → χ))    ⇒   ⊢ (φ → ((ψ ∨ θ) → χ))
Theorem	mpjaod 370	Eliminate a disjunction in a deduction. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (φ → (ψ → χ))    &   ⊢ (φ → (θ → χ))    &   ⊢ (φ → (ψ ∨ θ))    ⇒   ⊢ (φ → χ)
Theorem	orel1 371	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 → ((φ ∨ ψ) → ψ))
Theorem	orel2 372	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 → ((ψ ∨ φ) → ψ))
Theorem	olc 373	Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96. (Contributed by NM, 30-Aug-1993.)
⊢ (φ → (ψ ∨ φ))
Theorem	orc 374	Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104. (Contributed by NM, 30-Aug-1993.)
⊢ (φ → (φ ∨ ψ))
Theorem	pm1.4 375	Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
⊢ ((φ ∨ ψ) → (ψ ∨ φ))
Theorem	orcom 376	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.)
⊢ ((φ ∨ ψ) ↔ (ψ ∨ φ))
Theorem	orcomd 377	Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.)
⊢ (φ → (ψ ∨ χ))    ⇒   ⊢ (φ → (χ ∨ ψ))
Theorem	orcoms 378	Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.)
⊢ ((φ ∨ ψ) → χ)    ⇒   ⊢ ((ψ ∨ φ) → χ)
Theorem	orci 379	Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
⊢ φ    ⇒   ⊢ (φ ∨ ψ)
Theorem	olci 380	Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
⊢ φ    ⇒   ⊢ (ψ ∨ φ)
Theorem	orcd 381	Deduction introducing a disjunct. A translation of natural deduction rule ∨ IR ( ∨ insertion right), see natded in set.mm. (Contributed by NM, 20-Sep-2007.)
⊢ (φ → ψ)    ⇒   ⊢ (φ → (ψ ∨ χ))
Theorem	olcd 382	Deduction introducing a disjunct. A translation of natural deduction rule ∨ IL ( ∨ insertion left), see natded in set.mm. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
⊢ (φ → ψ)    ⇒   ⊢ (φ → (χ ∨ ψ))
Theorem	orcs 383	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 15) -type inference in a proof. (Contributed by NM, 21-Jun-1994.)
⊢ ((φ ∨ ψ) → χ)    ⇒   ⊢ (φ → χ)
Theorem	olcs 384	Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
⊢ ((φ ∨ ψ) → χ)    ⇒   ⊢ (ψ → χ)
Theorem	pm2.07 385	Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)
⊢ (φ → (φ ∨ φ))
Theorem	pm2.45 386	Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ (φ ∨ ψ) → ¬ φ)
Theorem	pm2.46 387	Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ (φ ∨ ψ) → ¬ ψ)
Theorem	pm2.47 388	Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ (φ ∨ ψ) → (¬ φ ∨ ψ))
Theorem	pm2.48 389	Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ (φ ∨ ψ) → (φ ∨ ¬ ψ))
Theorem	pm2.49 390	Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ (φ ∨ ψ) → (¬ φ ∨ ¬ ψ))
Theorem	pm2.67-2 391	Slight generalization of Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ (((φ ∨ χ) → ψ) → (φ → ψ))
Theorem	pm2.67 392	Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ (((φ ∨ ψ) → ψ) → (φ → ψ))
Theorem	pm2.25 393	Theorem *2.25 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
⊢ (φ ∨ ((φ ∨ ψ) → ψ))
Theorem	biorf 394	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.)
⊢ (¬ φ → (ψ ↔ (φ ∨ ψ)))
Theorem	biortn 395	A wff is equivalent to its negated disjunction with falsehood. (Contributed by NM, 9-Jul-2012.)
⊢ (φ → (ψ ↔ (¬ φ ∨ ψ)))
Theorem	biorfi 396	A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.)
⊢ ¬ φ    ⇒   ⊢ (ψ ↔ (ψ ∨ φ))
Theorem	pm2.621 397	Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ ((φ → ψ) → ((φ ∨ ψ) → ψ))
Theorem	pm2.62 398	Theorem *2.62 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Dec-2013.)
⊢ ((φ ∨ ψ) → ((φ → ψ) → ψ))
Theorem	pm2.68 399	Theorem *2.68 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
⊢ (((φ → ψ) → ψ) → (φ ∨ ψ))
Theorem	dfor2 400	Logical 'or' expressed in terms of implication only. Theorem *5.25 of [WhiteheadRussell] p. 124. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Wolf Lammen, 20-Oct-2012.)
⊢ ((φ ∨ ψ) ↔ ((φ → ψ) → ψ))