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	3bitr3i 301	A chained inference from transitive law for logical equivalence. (Contributed by NM, 19-Aug-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜑 ↔ 𝜒)    &   ⊢ (𝜓 ↔ 𝜃)    ⇒   ⊢ (𝜒 ↔ 𝜃)
Theorem	3bitr3ri 302	A chained inference from transitive law for logical equivalence. (Contributed by NM, 21-Jun-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜑 ↔ 𝜒)    &   ⊢ (𝜓 ↔ 𝜃)    ⇒   ⊢ (𝜃 ↔ 𝜒)
Theorem	3bitr4i 303	A chained inference from transitive law for logical equivalence. This inference is frequently used to apply a definition to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜑)    &   ⊢ (𝜃 ↔ 𝜓)    ⇒   ⊢ (𝜒 ↔ 𝜃)
Theorem	3bitr4ri 304	A chained inference from transitive law for logical equivalence. (Contributed by NM, 2-Sep-1995.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜑)    &   ⊢ (𝜃 ↔ 𝜓)    ⇒   ⊢ (𝜃 ↔ 𝜒)
Theorem	3bitrd 305	Deduction from transitivity of biconditional. (Contributed by NM, 13-Aug-1999.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜒 ↔ 𝜃))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜏))
Theorem	3bitrrd 306	Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜒 ↔ 𝜃))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → (𝜏 ↔ 𝜓))
Theorem	3bitr2d 307	Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜏))
Theorem	3bitr2rd 308	Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → (𝜏 ↔ 𝜓))
Theorem	3bitr3d 309	Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 24-Apr-1996.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜓 ↔ 𝜃))    &   ⊢ (𝜑 → (𝜒 ↔ 𝜏))    ⇒   ⊢ (𝜑 → (𝜃 ↔ 𝜏))
Theorem	3bitr3rd 310	Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜓 ↔ 𝜃))    &   ⊢ (𝜑 → (𝜒 ↔ 𝜏))    ⇒   ⊢ (𝜑 → (𝜏 ↔ 𝜃))
Theorem	3bitr4d 311	Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 18-Oct-1995.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜓))    &   ⊢ (𝜑 → (𝜏 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜃 ↔ 𝜏))
Theorem	3bitr4rd 312	Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜓))    &   ⊢ (𝜑 → (𝜏 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜏 ↔ 𝜃))
Theorem	3bitr3g 313	More general version of 3bitr3i 301. 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 314	More general version of 3bitr4i 303. 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 315	Double negation. Theorem *4.13 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-1993.)
⊢ (𝜑 ↔ ¬ ¬ 𝜑)
Theorem	con34b 316	A biconditional form of contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116. (Contributed by NM, 11-May-1993.)
⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜓 → ¬ 𝜑))
Theorem	con4bid 317	A contraposition deduction. (Contributed by NM, 21-May-1994.)
⊢ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	notbid 318	Deduction negating both sides of a logical equivalence. (Contributed by NM, 21-May-1994.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))
Theorem	notbi 319	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 320	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 321	A contraposition inference. (Contributed by NM, 21-May-1994.)
⊢ (¬ 𝜑 ↔ ¬ 𝜓)    ⇒   ⊢ (𝜑 ↔ 𝜓)
Theorem	mtbi 322	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 323	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 324	A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.)
⊢ (𝜑 → ¬ 𝜓)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜒)
Theorem	mtbird 325	A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.)
⊢ (𝜑 → ¬ 𝜒)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	mtbii 326	An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.)
⊢ ¬ 𝜓    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜒)
Theorem	mtbiri 327	An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.)
⊢ ¬ 𝜒    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	sylnib 328	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 329	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 330	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 331	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 332	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
⊢ (¬ 𝜑 ↔ 𝜓)    &   ⊢ (𝜑 ↔ 𝜒)    ⇒   ⊢ (¬ 𝜒 ↔ 𝜓)
Theorem	xchnxbir 333	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
⊢ (¬ 𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜑)    ⇒   ⊢ (¬ 𝜒 ↔ 𝜓)
Theorem	xchbinx 334	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
⊢ (𝜑 ↔ ¬ 𝜓)    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (𝜑 ↔ ¬ 𝜒)
Theorem	xchbinxr 335	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
⊢ (𝜑 ↔ ¬ 𝜓)    &   ⊢ (𝜒 ↔ 𝜓)    ⇒   ⊢ (𝜑 ↔ ¬ 𝜒)
Theorem	imbi2i 336	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	jcndOLD 337	Obsolete version of jcnd 163 as of 10-Apr-2024. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → ¬ 𝜒)    ⇒   ⊢ (𝜑 → ¬ (𝜓 → 𝜒))
Theorem	bibi2i 338	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 339	Inference adding a biconditional to the right in an equivalence. (Contributed by NM, 26-May-1993.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒))
Theorem	bibi12i 340	The equivalence of two equivalences. (Contributed by NM, 26-May-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜃))
Theorem	imbi2d 341	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 342	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 343	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 344	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 345	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 346	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 347	Closed form of imbi12i 351. Was automatically derived from its "Virtual Deduction" version and the Metamath program "MM-PA> MINIMIZE_WITH *" command. (Contributed by Alan Sare, 18-Mar-2012.)
⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))))
Theorem	imbi1 348	Theorem *4.84 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ↔ 𝜓) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜒)))
Theorem	imbi2 349	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 350	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 351	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 352	Theorem *4.86 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒)))
Theorem	bitr3 353	Closed nested implication form of bitr3i 276. Derived automatically from bitr3VD 42476. (Contributed by Alan Sare, 31-Dec-2011.)
⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ 𝜒) → (𝜓 ↔ 𝜒)))
Theorem	con2bi 354	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 355	A contraposition deduction. (Contributed by NM, 15-Apr-1995.)
⊢ (𝜑 → (𝜓 ↔ ¬ 𝜒))    ⇒   ⊢ (𝜑 → (𝜒 ↔ ¬ 𝜓))
Theorem	con1bid 356	A contraposition deduction. (Contributed by NM, 9-Oct-1999.)
⊢ (𝜑 → (¬ 𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (¬ 𝜒 ↔ 𝜓))
Theorem	con1bii 357	A contraposition inference. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.)
⊢ (¬ 𝜑 ↔ 𝜓)    ⇒   ⊢ (¬ 𝜓 ↔ 𝜑)
Theorem	con2bii 358	A contraposition inference. (Contributed by NM, 12-Mar-1993.)
⊢ (𝜑 ↔ ¬ 𝜓)    ⇒   ⊢ (𝜓 ↔ ¬ 𝜑)
Theorem	con1b 359	Contraposition. Bidirectional version of con1 146. (Contributed by NM, 3-Jan-1993.)
⊢ ((¬ 𝜑 → 𝜓) ↔ (¬ 𝜓 → 𝜑))
Theorem	con2b 360	Contraposition. Bidirectional version of con2 135. (Contributed by NM, 12-Mar-1993.)
⊢ ((𝜑 → ¬ 𝜓) ↔ (𝜓 → ¬ 𝜑))
Theorem	biimt 361	A wff is equivalent to itself with true antecedent. (Contributed by NM, 28-Jan-1996.)
⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓)))
Theorem	pm5.5 362	Theorem *5.5 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ (𝜑 → ((𝜑 → 𝜓) ↔ 𝜓))
Theorem	a1bi 363	Inference 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 364	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 365	Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Proof shortened by Wolf Lammen, 12-Nov-2012.)
⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (𝜓 → 𝜑)))
Theorem	imnot 366	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), pm2.21 123 (false antecedent), pm5.5 362 (true antecedent). (Contributed by Mario Carneiro, 26-Apr-2019.) (Proof shortened by Wolf Lammen, 26-May-2019.)
⊢ (¬ 𝜓 → ((𝜑 → 𝜓) ↔ ¬ 𝜑))
Theorem	pm5.501 367	Theorem *5.501 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ (𝜑 → (𝜓 ↔ (𝜑 ↔ 𝜓)))
Theorem	ibib 368	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 369	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 370	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 371	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 372	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 373	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 374	Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.)
⊢ 𝜑    ⇒   ⊢ (¬ 𝜓 ↔ (𝜓 ↔ ¬ 𝜑))
Theorem	pm5.21im 375	Two propositions are equivalent if they are both false. Closed form of 2false 376. Equivalent to a biimpr 219-like version of the xor-connective. (Contributed by Wolf Lammen, 13-May-2013.)
⊢ (¬ 𝜑 → (¬ 𝜓 → (𝜑 ↔ 𝜓)))
Theorem	2false 376	Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.)
⊢ ¬ 𝜑    &   ⊢ ¬ 𝜓    ⇒   ⊢ (𝜑 ↔ 𝜓)
Theorem	2falsed 377	Two falsehoods are equivalent (deduction form). (Contributed by NM, 11-Oct-2013.) (Proof shortened by Wolf Lammen, 11-Apr-2024.)
⊢ (𝜑 → ¬ 𝜓)    &   ⊢ (𝜑 → ¬ 𝜒)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	2falsedOLD 378	Obsolete version of 2falsed 377 as of 11-Apr-2024. (Contributed by NM, 11-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → ¬ 𝜓)    &   ⊢ (𝜑 → ¬ 𝜒)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	pm5.21ni 379	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 380	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 381	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 382	Combine antecedents into a single biconditional. This inference, reminiscent of ja 186, is reversible: The hypotheses can be deduced from the conclusion alone (see pm5.1im 262 and pm5.21im 375). (Contributed by Wolf Lammen, 13-May-2013.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (¬ 𝜑 → (¬ 𝜓 → 𝜒))    ⇒   ⊢ ((𝜑 ↔ 𝜓) → 𝜒)
Theorem	pm5.18 383	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 384	Two ways to express "exclusive or". (Contributed by NM, 1-Jan-2006.)
⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
Theorem	nbbn 385	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 386	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	biluk 387	Lukasiewicz's shortest axiom for equivalential calculus. Storrs McCall, ed., Polish Logic 1920-1939 (Oxford, 1967), p. 96. (Contributed by NM, 10-Jan-2005.)
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜒 ↔ 𝜓) ↔ (𝜑 ↔ 𝜒)))
Theorem	pm5.19 388	Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.)
⊢ ¬ (𝜑 ↔ ¬ 𝜑)
Theorem	bi2.04 389	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	pm5.4 390	Antecedent absorption implication. Theorem *5.4 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
⊢ ((𝜑 → (𝜑 → 𝜓)) ↔ (𝜑 → 𝜓))
Theorem	imdi 391	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	pm5.41 392	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	pm4.8 393	Theorem *4.8 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 → ¬ 𝜑) ↔ ¬ 𝜑)
Theorem	pm4.81 394	A formula is equivalent to its negation implying it. Theorem *4.81 of [WhiteheadRussell] p. 122. Note that the second step, using pm2.24 124, could also use ax-1 6. (Contributed by NM, 3-Jan-2005.)
⊢ ((¬ 𝜑 → 𝜑) ↔ 𝜑)
Theorem	imim21b 395	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 conjunction This section defines conjunction of two formulas, denoted by infix "∧ " and read "and". It is defined in terms of implication and negation, which is possible in classical logic (but not in intuitionistic logic: see iset.mm). After the definition, we briefly introduce conversion of simple expressions to and from conjunction. Two simple operations called importation (imp 407) and exportation (ex 413) follow. In the propositions-as-types interpretation, they correspond to uncurrying and currying respectively. They are foundational for this section. Most of the theorems proved here trace back to them, mostly indirectly, in a layered fashion, where more complex expressions are built from simpler ones. Here are some of these successive layers: importation and exportation, commutativity and associativity laws, adding antecedents and simplifying, conjunction of consequents, syllogisms, etc. As indicated in the "note on definitions" in the section comment for logical equivalence, some theorems containing only implication, negation and conjunction are placed in the section after disjunction since theirs proofs use disjunction (although this is not required since definitions are conservative, see said section comment).
Syntax	wa 396	Extend wff definition to include conjunction ("and").
wff (𝜑 ∧ 𝜓)
Definition	df-an 397	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 1542) and the constant false ⊥ (df-fal 1552), we will be able to prove these truth table values: ((⊤ ∧ ⊤) ↔ ⊤) (truantru 1572), ((⊤ ∧ ⊥) ↔ ⊥) (truanfal 1573), ((⊥ ∧ ⊤) ↔ ⊥) (falantru 1574), and ((⊥ ∧ ⊥) ↔ ⊥) (falanfal 1575). 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 260. This is the justification for the definition, along with the fact that it introduces a new symbol ∧. Contrast with ∨ (df-or 845), → (wi 4), ⊼ (df-nan 1487), and ⊻ (df-xor 1507). (Contributed by NM, 5-Jan-1993.)
⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))
Theorem	pm4.63 398	Theorem *4.63 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ (𝜑 → ¬ 𝜓) ↔ (𝜑 ∧ 𝜓))
Theorem	pm4.67 399	Theorem *4.67 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ (¬ 𝜑 → ¬ 𝜓) ↔ (¬ 𝜑 ∧ 𝜓))
Theorem	imnan 400	Express an implication in terms of a negated conjunction. (Contributed by NM, 9-Apr-1994.)
⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))