P. 4 of Theorem List - Metamath Proof Explorer

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

Type	Label	Description
Statement
Theorem	3bitr3g 301	More general version of 3bitr3i 289. Useful for converting definitions in a formula. (Contributed by NM, 4-Jun-1995.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜓 ↔ 𝜃)    &   ⊢ (𝜒 ↔ 𝜏)    ⇒   ⊢ (𝜑 → (𝜃 ↔ 𝜏))
Theorem	3bitr4g 302	More general version of 3bitr4i 291. Useful for converting definitions in a formula. (Contributed by NM, 11-May-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜃 ↔ 𝜓)    &   ⊢ (𝜏 ↔ 𝜒)    ⇒   ⊢ (𝜑 → (𝜃 ↔ 𝜏))
Theorem	notnotb 303	Double negation. Theorem *4.13 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-1993.)
⊢ (𝜑 ↔ ¬ ¬ 𝜑)
Theorem	notnotdOLD 304	Obsolete proof of notnotd 137 as of 27-Mar-2021. (Contributed by Jarvin Udandy, 2-Sep-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ¬ ¬ 𝜓)
Theorem	con34b 305	A biconditional form of contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116. (Contributed by NM, 11-May-1993.)
⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜓 → ¬ 𝜑))
Theorem	con4bid 306	A contraposition deduction. (Contributed by NM, 21-May-1994.)
⊢ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	notbid 307	Deduction negating both sides of a logical equivalence. (Contributed by NM, 21-May-1994.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))
Theorem	notbi 308	Contraposition. Theorem *4.11 of [WhiteheadRussell] p. 117. (Contributed by NM, 21-May-1994.) (Proof shortened by Wolf Lammen, 12-Jun-2013.)
⊢ ((𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))
Theorem	notbii 309	Negate both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (¬ 𝜑 ↔ ¬ 𝜓)
Theorem	con4bii 310	A contraposition inference. (Contributed by NM, 21-May-1994.)
⊢ (¬ 𝜑 ↔ ¬ 𝜓)    ⇒   ⊢ (𝜑 ↔ 𝜓)
Theorem	mtbi 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    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ¬ 𝜓
Theorem	mtbir 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    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ¬ 𝜑
Theorem	mtbid 313	A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.)
⊢ (𝜑 → ¬ 𝜓)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜒)
Theorem	mtbird 314	A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.)
⊢ (𝜑 → ¬ 𝜒)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	mtbii 315	An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.)
⊢ ¬ 𝜓    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜒)
Theorem	mtbiri 316	An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.)
⊢ ¬ 𝜒    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	sylnib 317	A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.)
⊢ (𝜑 → ¬ 𝜓)    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (𝜑 → ¬ 𝜒)
Theorem	sylnibr 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)    &   ⊢ (𝜒 ↔ 𝜓)    ⇒   ⊢ (𝜑 → ¬ 𝜒)
Theorem	sylnbi 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.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (¬ 𝜓 → 𝜒)    ⇒   ⊢ (¬ 𝜑 → 𝜒)
Theorem	sylnbir 320	A mixed syllogism inference from a biconditional and an implication. (Contributed by Wolf Lammen, 16-Dec-2013.)
⊢ (𝜓 ↔ 𝜑)    &   ⊢ (¬ 𝜓 → 𝜒)    ⇒   ⊢ (¬ 𝜑 → 𝜒)
Theorem	xchnxbi 321	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
⊢ (¬ 𝜑 ↔ 𝜓)    &   ⊢ (𝜑 ↔ 𝜒)    ⇒   ⊢ (¬ 𝜒 ↔ 𝜓)
Theorem	xchnxbir 322	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
⊢ (¬ 𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜑)    ⇒   ⊢ (¬ 𝜒 ↔ 𝜓)
Theorem	xchbinx 323	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
⊢ (𝜑 ↔ ¬ 𝜓)    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (𝜑 ↔ ¬ 𝜒)
Theorem	xchbinxr 324	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
⊢ (𝜑 ↔ ¬ 𝜓)    &   ⊢ (𝜒 ↔ 𝜓)    ⇒   ⊢ (𝜑 ↔ ¬ 𝜒)
Theorem	imbi2i 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.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ((𝜒 → 𝜑) ↔ (𝜒 → 𝜓))
Theorem	bibi2i 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.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ((𝜒 ↔ 𝜑) ↔ (𝜒 ↔ 𝜓))
Theorem	bibi1i 327	Inference adding a biconditional to the right in an equivalence. (Contributed by NM, 26-May-1993.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒))
Theorem	bibi12i 328	The equivalence of two equivalences. (Contributed by NM, 26-May-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜃))
Theorem	imbi2d 329	Deduction adding an antecedent to both sides of a logical equivalence. (Contributed by NM, 11-May-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ((𝜃 → 𝜓) ↔ (𝜃 → 𝜒)))
Theorem	imbi1d 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 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜃)))
Theorem	bibi2d 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 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ((𝜃 ↔ 𝜓) ↔ (𝜃 ↔ 𝜒)))
Theorem	bibi1d 332	Deduction adding a biconditional to the right in an equivalence. (Contributed by NM, 11-May-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ((𝜓 ↔ 𝜃) ↔ (𝜒 ↔ 𝜃)))
Theorem	imbi12d 333	Deduction joining two equivalences to form equivalence of implications. (Contributed by NM, 16-May-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏)))
Theorem	bibi12d 334	Deduction joining two equivalences to form equivalence of biconditionals. (Contributed by NM, 26-May-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → ((𝜓 ↔ 𝜃) ↔ (𝜒 ↔ 𝜏)))
Theorem	imbi12 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.)
⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))))
Theorem	imbi1 336	Theorem *4.84 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ↔ 𝜓) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜒)))
Theorem	imbi2 337	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 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.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ((𝜑 → 𝜒) ↔ (𝜓 → 𝜒))
Theorem	imbi12i 339	Join two logical equivalences to form equivalence of implications. (Contributed by NM, 1-Aug-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))
Theorem	bibi1 340	Theorem *4.86 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒)))
Theorem	bitr3 341	Closed nested implication form of bitr3i 265. Derived automatically from bitr3VD 38106. (Contributed by Alan Sare, 31-Dec-2011.)
⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ 𝜒) → (𝜓 ↔ 𝜒)))
Theorem	con2bi 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))
Theorem	con2bid 343	A contraposition deduction. (Contributed by NM, 15-Apr-1995.)
⊢ (𝜑 → (𝜓 ↔ ¬ 𝜒))    ⇒   ⊢ (𝜑 → (𝜒 ↔ ¬ 𝜓))
Theorem	con1bid 344	A contraposition deduction. (Contributed by NM, 9-Oct-1999.)
⊢ (𝜑 → (¬ 𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (¬ 𝜒 ↔ 𝜓))
Theorem	con1bii 345	A contraposition inference. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.)
⊢ (¬ 𝜑 ↔ 𝜓)    ⇒   ⊢ (¬ 𝜓 ↔ 𝜑)
Theorem	con2bii 346	A contraposition inference. (Contributed by NM, 12-Mar-1993.)
⊢ (𝜑 ↔ ¬ 𝜓)    ⇒   ⊢ (𝜓 ↔ ¬ 𝜑)
Theorem	con1b 347	Contraposition. Bidirectional version of con1 142. (Contributed by NM, 3-Jan-1993.)
⊢ ((¬ 𝜑 → 𝜓) ↔ (¬ 𝜓 → 𝜑))
Theorem	con2b 348	Contraposition. Bidirectional version of con2 129. (Contributed by NM, 12-Mar-1993.)
⊢ ((𝜑 → ¬ 𝜓) ↔ (𝜓 → ¬ 𝜑))
Theorem	biimt 349	A wff is equivalent to itself with true antecedent. (Contributed by NM, 28-Jan-1996.)
⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓)))
Theorem	pm5.5 350	Theorem *5.5 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ (𝜑 → ((𝜑 → 𝜓) ↔ 𝜓))
Theorem	a1bi 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 ↔ (𝜑 → 𝜓))
Theorem	mt2bi 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))
Theorem	mtt 353	Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Proof shortened by Wolf Lammen, 12-Nov-2012.)
⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (𝜓 → 𝜑)))
Theorem	imnot 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 (𝜑 → 𝜓), the other ones being ax-1 6 (true consequent), pm2.21 119 (false antecedent), pm5.5 350 (true antecedent). (Contributed by Mario Carneiro, 26-Apr-2019.) (Proof shortened by Wolf Lammen, 26-May-2019.)
⊢ (¬ 𝜓 → ((𝜑 → 𝜓) ↔ ¬ 𝜑))
Theorem	pm5.501 355	Theorem *5.501 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ (𝜑 → (𝜓 ↔ (𝜑 ↔ 𝜓)))
Theorem	ibib 356	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 357	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 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 ↔ (𝜓 ↔ 𝜑))
Theorem	nbn2 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 ↔ (𝜑 ↔ 𝜓)))
Theorem	bibif 360	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 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 ↔ (𝜓 ↔ 𝜑))
Theorem	nbn3 362	Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.)
⊢ 𝜑    ⇒   ⊢ (¬ 𝜓 ↔ (𝜓 ↔ ¬ 𝜑))
Theorem	pm5.21im 363	Two propositions are equivalent if they are both false. Closed form of 2false 364. Equivalent to a biimpr 209-like version of the xor-connective. (Contributed by Wolf Lammen, 13-May-2013.)
⊢ (¬ 𝜑 → (¬ 𝜓 → (𝜑 ↔ 𝜓)))
Theorem	2false 364	Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.)
⊢ ¬ 𝜑    &   ⊢ ¬ 𝜓    ⇒   ⊢ (𝜑 ↔ 𝜓)
Theorem	2falsed 365	Two falsehoods are equivalent (deduction rule). (Contributed by NM, 11-Oct-2013.)
⊢ (𝜑 → ¬ 𝜓)    &   ⊢ (𝜑 → ¬ 𝜒)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	pm5.21ni 366	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 367	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 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 → (𝜒 → 𝜓))    &   ⊢ (𝜑 → (𝜃 → 𝜓))    &   ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))    ⇒   ⊢ (𝜑 → (𝜒 ↔ 𝜃))
Theorem	bija 369	Combine antecedents into a single biconditional. This inference, reminiscent of ja 172, is reversible: The hypotheses can be deduced from the conclusion alone (see pm5.1im 252 and pm5.21im 363). (Contributed by Wolf Lammen, 13-May-2013.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (¬ 𝜑 → (¬ 𝜓 → 𝜒))    ⇒   ⊢ ((𝜑 ↔ 𝜓) → 𝜒)
Theorem	pm5.18 370	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.)
⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
Theorem	xor3 371	Two ways to express "exclusive or." (Contributed by NM, 1-Jan-2006.)
⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
Theorem	nbbn 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.)
⊢ ((¬ 𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
Theorem	biass 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.)
⊢ (((𝜑 ↔ 𝜓) ↔ 𝜒) ↔ (𝜑 ↔ (𝜓 ↔ 𝜒)))
Theorem	pm5.19 374	Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.)
⊢ ¬ (𝜑 ↔ ¬ 𝜑)
Theorem	bi2.04 375	Logical equivalence of commuted antecedents. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 11-May-1993.)
⊢ ((𝜑 → (𝜓 → 𝜒)) ↔ (𝜓 → (𝜑 → 𝜒)))
Theorem	pm5.4 376	Antecedent absorption implication. Theorem *5.4 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
⊢ ((𝜑 → (𝜑 → 𝜓)) ↔ (𝜑 → 𝜓))
Theorem	imdi 377	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	pm5.41 378	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	pm4.8 379	Theorem *4.8 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 → ¬ 𝜑) ↔ ¬ 𝜑)
Theorem	pm4.81 380	Theorem *4.81 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
⊢ ((¬ 𝜑 → 𝜑) ↔ 𝜑)
Theorem	imim21b 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.)
⊢ ((𝜓 → 𝜑) → (((𝜑 → 𝜒) → (𝜓 → 𝜃)) ↔ (𝜓 → (𝜒 → 𝜃))))
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.
Syntax	wo 382	Extend wff definition to include disjunction ('or').
wff (𝜑 ∨ 𝜓)
Syntax	wa 383	Extend wff definition to include conjunction ('and').
wff (𝜑 ∧ 𝜓)
Definition	df-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 26670). After we define the constant true ⊤ (df-tru 1478) and the constant false ⊥ (df-fal 1481), we will be able to prove these truth table values: ((⊤ ∨ ⊤) ↔ ⊤) (truortru 1501), ((⊤ ∨ ⊥) ↔ ⊤) (truorfal 1502), ((⊥ ∨ ⊤) ↔ ⊤) (falortru 1503), and ((⊥ ∨ ⊥) ↔ ⊥) (falorfal 1504). 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 250. 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 1440), and ⊻ (df-xor 1457) . (Contributed by NM, 27-Dec-1992.)
⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))
Definition	df-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 1478) and the constant false ⊥ (df-fal 1481), we will be able to prove these truth table values: ((⊤ ∧ ⊤) ↔ ⊤) (truantru 1497), ((⊤ ∧ ⊥) ↔ ⊥) (truanfal 1498), ((⊥ ∧ ⊤) ↔ ⊥) (falantru 1499), and ((⊥ ∧ ⊥) ↔ ⊥) (falanfal 1500). Contrast with ∨ (df-or 384), → (wi 4), ⊼ (df-nan 1440), and ⊻ (df-xor 1457) . (Contributed by NM, 5-Jan-1993.)
⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))
Theorem	pm4.64 386	Theorem *4.64 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
⊢ ((¬ 𝜑 → 𝜓) ↔ (𝜑 ∨ 𝜓))
Theorem	pm2.53 387	Theorem *2.53 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓))
Theorem	pm2.54 388	Theorem *2.54 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ ((¬ 𝜑 → 𝜓) → (𝜑 ∨ 𝜓))
Theorem	ori 389	Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.)
⊢ (𝜑 ∨ 𝜓)    ⇒   ⊢ (¬ 𝜑 → 𝜓)
Theorem	orri 390	Infer disjunction from implication. (Contributed by NM, 11-Jun-1994.)
⊢ (¬ 𝜑 → 𝜓)    ⇒   ⊢ (𝜑 ∨ 𝜓)
Theorem	ord 391	Deduce implication from disjunction. (Contributed by NM, 18-May-1994.)
⊢ (𝜑 → (𝜓 ∨ 𝜒))    ⇒   ⊢ (𝜑 → (¬ 𝜓 → 𝜒))
Theorem	orrd 392	Deduce disjunction from implication. (Contributed by NM, 27-Nov-1995.)
⊢ (𝜑 → (¬ 𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 ∨ 𝜒))
Theorem	jaoi 393	Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 → 𝜓)    ⇒   ⊢ ((𝜑 ∨ 𝜒) → 𝜓)
Theorem	jaod 394	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 395	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 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 → ((𝜑 ∨ 𝜓) → 𝜓))
Theorem	orel2 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 → ((𝜓 ∨ 𝜑) → 𝜓))
Theorem	olc 398	Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96. (Contributed by NM, 30-Aug-1993.)
⊢ (𝜑 → (𝜓 ∨ 𝜑))
Theorem	orc 399	Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104. (Contributed by NM, 30-Aug-1993.)
⊢ (𝜑 → (𝜑 ∨ 𝜓))
Theorem	pm1.4 400	Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∨ 𝜓) → (𝜓 ∨ 𝜑))