P. 3 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 201-300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mtod 201	Modus tollens deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
⊢ (𝜑 → ¬ 𝜒)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	mtoi 202	Modus tollens inference. (Contributed by NM, 5-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
⊢ ¬ 𝜒    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	mt2 203	A rule similar to modus tollens. Inference associated with con2i 141. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Sep-2013.)
⊢ 𝜓    &   ⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ ¬ 𝜑
Theorem	mt3 204	A rule similar to modus tollens. Inference associated with con1i 149. (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
⊢ ¬ 𝜓    &   ⊢ (¬ 𝜑 → 𝜓)    ⇒   ⊢ 𝜑
Theorem	peirce 205	Peirce's axiom. A non-intuitionistic implication-only statement. Added to intuitionistic (implicational) propositional calculus, it gives classical (implicational) propositional calculus. For another non-intuitionistic positive statement, see curryax 893. When ⊥ is substituted for 𝜓, then this becomes the Clavius law pm2.18 128. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 9-Oct-2012.)
⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)
Theorem	looinv 206	The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. Using dfor2 901, we can see that this essentially expresses "disjunction commutes". Theorem *2.69 of [WhiteheadRussell] p. 108. It is a special instance of the axiom "Roll", see peirceroll 85. (Contributed by NM, 12-Aug-2004.)
⊢ (((𝜑 → 𝜓) → 𝜓) → ((𝜓 → 𝜑) → 𝜑))
Theorem	bijust0 207	A self-implication (see id 22) does not imply its own negation. The justification theorem bijust 208 is one of its instances. (Contributed by NM, 11-May-1999.) (Proof shortened by Josh Purinton, 29-Dec-2000.) Extract bijust0 207 from proof of bijust 208. (Revised by BJ, 19-Mar-2020.)
⊢ ¬ ((𝜑 → 𝜑) → ¬ (𝜑 → 𝜑))
Theorem	bijust 208	Theorem used to justify the definition of the biconditional df-bi 210. Instance of bijust0 207. (Contributed by NM, 11-May-1999.)
⊢ ¬ ((¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑))) → ¬ (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑))))
1.2.5  Logical equivalence Definition df-bi 210 in this section is our first definition, which introduces and defines the biconditional connective ↔ used to denote logical equivalence. We define a wff of the form (𝜑 ↔ 𝜓) as an abbreviation for ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)). Unlike most traditional developments, we have chosen not to have a separate symbol such as "Df." to mean "is defined as". Instead, we will later use the biconditional connective for this purpose (df-an 400 is its first use), as it allows us to use logic to manipulate definitions directly. This greatly simplifies many proofs since it eliminates the need for a separate mechanism for introducing and eliminating definitions. A note on definitions: definitions are required to be eliminable (that is, a theorem stated in terms of the defined symbol can also be stated without it) and conservative (that is, a theorem whose statement does not contain the defined symbol can be proved without using that definition). This means that a definition does not increase the expressive power nor the deductive power, respectively, of a theory. On the other hand, definitions are often useful to write shorter proofs, so in (i)set.mm we will generally not try to avoid them. This is why, for instance, some theorems which do not contain disjunction in their statement are placed after the section on disjunction because a shorter proof using disjunction is possible.
Syntax	wb 209	Extend wff definition to include the biconditional connective.
wff (𝜑 ↔ 𝜓)
Definition	df-bi 210	Define the biconditional (logical "iff" or "if and only if"), also called biimplication. Definition df-bi 210 in this section is our first definition, which introduces and defines the biconditional connective ↔. We define a wff of the form (𝜑 ↔ 𝜓) as an abbreviation for ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)). Unlike most traditional developments, we have chosen not to have a separate symbol such as "Df." to mean "is defined as". Instead, we will later use the biconditional connective for this purpose (df-or 847 is its first use), as it allows us to use logic to manipulate definitions directly. This greatly simplifies many proofs since it eliminates the need for a separate mechanism for introducing and eliminating definitions. Of course, we cannot use this mechanism to define the biconditional itself, since it hasn't been introduced yet. Instead, we use a more general form of definition, described as follows. In its most general form, a definition is simply an assertion that introduces a new symbol (or a new combination of existing symbols, as in df-3an 1090) that is eliminable and does not strengthen the existing language. The latter requirement means that the set of provable statements not containing the new symbol (or new combination) should remain exactly the same after the definition is introduced. Our definition of the biconditional may look unusual compared to most definitions, but it strictly satisfies these requirements. The justification for our definition is that if we mechanically replace (𝜑 ↔ 𝜓) (the definiendum i.e. the thing being defined) with ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) (the definiens i.e. the defining expression) in the definition, the definition becomes the previously proved theorem bijust 208. It is impossible to use df-bi 210 to prove any statement expressed in the original language that can't be proved from the original axioms, because if we simply replace each instance of df-bi 210 in the proof with the corresponding bijust 208 instance, we will end up with a proof from the original axioms. Note that from Metamath's point of view, a definition is just another axiom - i.e. an assertion we claim to be true - but from our high level point of view, we are not strengthening the language. To indicate this fact, we prefix definition labels with "df-" instead of "ax-". (This prefixing is an informal convention that means nothing to the Metamath proof verifier; it is just a naming convention for human readability.) After we define the constant true ⊤ (df-tru 1545) and the constant false ⊥ (df-fal 1555), we will be able to prove these truth table values: ((⊤ ↔ ⊤) ↔ ⊤) (trubitru 1571), ((⊤ ↔ ⊥) ↔ ⊥) (trubifal 1573), ((⊥ ↔ ⊤) ↔ ⊥) (falbitru 1572), and ((⊥ ↔ ⊥) ↔ ⊤) (falbifal 1574). See dfbi1 216, dfbi2 478, and dfbi3 1049 for theorems suggesting typical textbook definitions of ↔, showing that our definition has the properties we expect. Theorem dfbi1 216 is particularly useful if we want to eliminate ↔ from an expression to convert it to primitives. Theorem dfbi 479 shows this definition rewritten in an abbreviated form after conjunction is introduced, for easier understanding. Contrast with ∨ (df-or 847), → (wi 4), ⊼ (df-nan 1487), and ⊻ (df-xor 1507). In some sense ↔ returns true if two truth values are equal; = (df-cleq 2730) returns true if two classes are equal. (Contributed by NM, 27-Dec-1992.)
⊢ ¬ (((𝜑 ↔ 𝜓) → ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑))) → ¬ (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓)))
Theorem	impbi 211	Property of the biconditional connective. (Contributed by NM, 11-May-1999.)
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → (𝜑 ↔ 𝜓)))
Theorem	impbii 212	Infer an equivalence from an implication and its converse. Inference associated with impbi 211. (Contributed by NM, 29-Dec-1992.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜓 → 𝜑)    ⇒   ⊢ (𝜑 ↔ 𝜓)
Theorem	impbidd 213	Deduce an equivalence from two implications. Double deduction associated with impbi 211 and impbii 212. Deduction associated with impbid 215. (Contributed by Rodolfo Medina, 12-Oct-2010.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))
Theorem	impbid21d 214	Deduce an equivalence from two implications. (Contributed by Wolf Lammen, 12-May-2013.)
⊢ (𝜓 → (𝜒 → 𝜃))    &   ⊢ (𝜑 → (𝜃 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))
Theorem	impbid 215	Deduce an equivalence from two implications. Deduction associated with impbi 211 and impbii 212. (Contributed by NM, 24-Jan-1993.) Revised to prove it from impbid21d 214. (Revised by Wolf Lammen, 3-Nov-2012.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜒 → 𝜓))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	dfbi1 216	Relate the biconditional connective to primitive connectives. See dfbi1ALT 217 for an unusual version proved directly from axioms. (Contributed by NM, 29-Dec-1992.)
⊢ ((𝜑 ↔ 𝜓) ↔ ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)))
Theorem	dfbi1ALT 217	Alternate proof of dfbi1 216. This proof, discovered by Gregory Bush on 8-Mar-2004, has several curious properties. First, it has only 17 steps directly from the axioms and df-bi 210, compared to over 800 steps were the proof of dfbi1 216 expanded into axioms. Second, step 2 demands only the property of "true"; any axiom (or theorem) could be used. It might be thought, therefore, that it is in some sense redundant, but in fact no proof is shorter than this (measured by number of steps). Third, it illustrates how intermediate steps can "blow up" in size even in short proofs. Fourth, the compressed proof is only 182 bytes (or 17 bytes in D-proof notation), but the generated web page is over 200kB with intermediate steps that are essentially incomprehensible to humans (other than Gregory Bush). If there were an obfuscated code contest for proofs, this would be a contender. This "blowing up" and incomprehensibility of the intermediate steps vividly demonstrate the advantages of using many layered intermediate theorems, since each theorem is easier to understand. (Contributed by Gregory Bush, 10-Mar-2004.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ((𝜑 ↔ 𝜓) ↔ ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)))
Theorem	biimp 218	Property of the biconditional connective. (Contributed by NM, 11-May-1999.)
⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
Theorem	biimpi 219	Infer an implication from a logical equivalence. Inference associated with biimp 218. (Contributed by NM, 29-Dec-1992.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	sylbi 220	A mixed syllogism inference from a biconditional and an implication. Useful for substituting an antecedent with a definition. (Contributed by NM, 3-Jan-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜓 → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	sylib 221	A mixed syllogism inference from an implication and a biconditional. (Contributed by NM, 3-Jan-1993.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	sylbb 222	A mixed syllogism inference from two biconditionals. (Contributed by BJ, 30-Mar-2019.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	biimpr 223	Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)
⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
Theorem	bicom1 224	Commutative law for the biconditional. (Contributed by Wolf Lammen, 10-Nov-2012.)
⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑))
Theorem	bicom 225	Commutative law for the biconditional. Theorem *4.21 of [WhiteheadRussell] p. 117. (Contributed by NM, 11-May-1993.)
⊢ ((𝜑 ↔ 𝜓) ↔ (𝜓 ↔ 𝜑))
Theorem	bicomd 226	Commute two sides of a biconditional in a deduction. (Contributed by NM, 14-May-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜒 ↔ 𝜓))
Theorem	bicomi 227	Inference from commutative law for logical equivalence. (Contributed by NM, 3-Jan-1993.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (𝜓 ↔ 𝜑)
Theorem	impbid1 228	Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜒 → 𝜓)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	impbid2 229	Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.) (Proof shortened by Wolf Lammen, 27-Sep-2013.)
⊢ (𝜓 → 𝜒)    &   ⊢ (𝜑 → (𝜒 → 𝜓))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	impcon4bid 230	A variation on impbid 215 with contraposition. (Contributed by Jeff Hankins, 3-Jul-2009.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (¬ 𝜓 → ¬ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	biimpri 231	Infer a converse implication from a logical equivalence. Inference associated with biimpr 223. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 16-Sep-2013.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (𝜓 → 𝜑)
Theorem	biimpd 232	Deduce an implication from a logical equivalence. Deduction associated with biimp 218 and biimpi 219. (Contributed by NM, 11-Jan-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜒))
Theorem	mpbi 233	An inference from a biconditional, related to modus ponens. (Contributed by NM, 11-May-1993.)
⊢ 𝜑    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ 𝜓
Theorem	mpbir 234	An inference from a biconditional, related to modus ponens. (Contributed by NM, 28-Dec-1992.)
⊢ 𝜓    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ 𝜑
Theorem	mpbid 235	A deduction from a biconditional, related to modus ponens. (Contributed by NM, 21-Jun-1993.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	mpbii 236	An inference from a nested biconditional, related to modus ponens. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
⊢ 𝜓    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	sylibr 237	A mixed syllogism inference from an implication and a biconditional. Useful for substituting a consequent with a definition. (Contributed by NM, 3-Jan-1993.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 ↔ 𝜓)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	sylbir 238	A mixed syllogism inference from a biconditional and an implication. (Contributed by NM, 3-Jan-1993.)
⊢ (𝜓 ↔ 𝜑)    &   ⊢ (𝜓 → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	sylbbr 239	A mixed syllogism inference from two biconditionals. Note on the various syllogism-like statements in set.mm. The hypothetical syllogism syl 17 infers an implication from two implications (and there are 3syl 18 and 4syl 19 for chaining more inferences). There are four inferences inferring an implication from one implication and one biconditional: sylbi 220, sylib 221, sylbir 238, sylibr 237; four inferences inferring an implication from two biconditionals: sylbb 222, sylbbr 239, sylbb1 240, sylbb2 241; four inferences inferring a biconditional from two biconditionals: bitri 278, bitr2i 279, bitr3i 280, bitr4i 281 (and more for chaining more biconditionals). There are also closed forms and deduction versions of these, like, among many others, syld 47, syl5 34, syl6 35, mpbid 235, bitrd 282, syl5bb 286, bitrdi 290 and variants. (Contributed by BJ, 21-Apr-2019.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (𝜒 → 𝜑)
Theorem	sylbb1 240	A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜑 ↔ 𝜒)    ⇒   ⊢ (𝜓 → 𝜒)
Theorem	sylbb2 241	A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜓)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	sylibd 242	A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜒 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	sylbid 243	A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜒 → 𝜃))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	mpbidi 244	A deduction from a biconditional, related to modus ponens. (Contributed by NM, 9-Aug-1994.)
⊢ (𝜃 → (𝜑 → 𝜓))    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜃 → (𝜑 → 𝜒))
Theorem	syl5bi 245	A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded antecedent with a definition. (Contributed by NM, 12-Jan-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 → (𝜓 → 𝜃))    ⇒   ⊢ (𝜒 → (𝜑 → 𝜃))
Theorem	syl5bir 246	A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 21-Jun-1993.)
⊢ (𝜓 ↔ 𝜑)    &   ⊢ (𝜒 → (𝜓 → 𝜃))    ⇒   ⊢ (𝜒 → (𝜑 → 𝜃))
Theorem	syl5ib 247	A mixed syllogism inference. (Contributed by NM, 12-Jan-1993.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜒 → (𝜑 → 𝜃))
Theorem	syl5ibcom 248	A mixed syllogism inference. (Contributed by NM, 19-Jun-2007.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (𝜒 → 𝜃))
Theorem	syl5ibr 249	A mixed syllogism inference. (Contributed by NM, 3-Apr-1994.)
⊢ (𝜑 → 𝜃)    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜒 → (𝜑 → 𝜓))
Theorem	syl5ibrcom 250	A mixed syllogism inference. (Contributed by NM, 20-Jun-2007.)
⊢ (𝜑 → 𝜃)    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (𝜒 → 𝜓))
Theorem	biimprd 251	Deduce a converse implication from a logical equivalence. Deduction associated with biimpr 223 and biimpri 231. (Contributed by NM, 11-Jan-1993.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜒 → 𝜓))
Theorem	biimpcd 252	Deduce a commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜓 → (𝜑 → 𝜒))
Theorem	biimprcd 253	Deduce a converse commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜒 → (𝜑 → 𝜓))
Theorem	syl6ib 254	A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 21-Jun-1993.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	syl6ibr 255	A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded consequent with a definition. (Contributed by NM, 10-Jan-1993.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜃 ↔ 𝜒)    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	syl6bi 256	A mixed syllogism inference. (Contributed by NM, 2-Jan-1994.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜒 → 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	syl6bir 257	A mixed syllogism inference. (Contributed by NM, 18-May-1994.)
⊢ (𝜑 → (𝜒 ↔ 𝜓))    &   ⊢ (𝜒 → 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	syl7bi 258	A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 14-May-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 → (𝜃 → (𝜓 → 𝜏)))    ⇒   ⊢ (𝜒 → (𝜃 → (𝜑 → 𝜏)))
Theorem	syl8ib 259	A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 1-Aug-1994.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜃 ↔ 𝜏)    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))
Theorem	mpbird 260	A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	mpbiri 261	An inference from a nested biconditional, related to modus ponens. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
⊢ 𝜒    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	sylibrd 262	A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	sylbird 263	A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
⊢ (𝜑 → (𝜒 ↔ 𝜓))    &   ⊢ (𝜑 → (𝜒 → 𝜃))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	biid 264	Principle of identity for logical equivalence. Theorem *4.2 of [WhiteheadRussell] p. 117. This is part of Frege's eighth axiom per Proposition 54 of [Frege1879] p. 50; see also eqid 2738. (Contributed by NM, 2-Jun-1993.)
⊢ (𝜑 ↔ 𝜑)
Theorem	biidd 265	Principle of identity with antecedent. (Contributed by NM, 25-Nov-1995.)
⊢ (𝜑 → (𝜓 ↔ 𝜓))
Theorem	pm5.1im 266	Two propositions are equivalent if they are both true. Closed form of 2th 267. Equivalent to a biimp 218-like version of the xor-connective. This theorem stays true, no matter how you permute its operands. This is evident from its sharper version (𝜑 ↔ (𝜓 ↔ (𝜑 ↔ 𝜓))). (Contributed by Wolf Lammen, 12-May-2013.)
⊢ (𝜑 → (𝜓 → (𝜑 ↔ 𝜓)))
Theorem	2th 267	Two truths are equivalent. (Contributed by NM, 18-Aug-1993.)
⊢ 𝜑    &   ⊢ 𝜓    ⇒   ⊢ (𝜑 ↔ 𝜓)
Theorem	2thd 268	Two truths are equivalent. Deduction form. (Contributed by NM, 3-Jun-2012.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	monothetic 269	Two self-implications (see id 22) are equivalent. This theorem, rather trivial in our axiomatization, is (the biconditional form of) a standard axiom for monothetic BCI logic. This is the most general theorem of which trujust 1544 is an instance. Relatedly, this would be the justification theorem if the definition of ⊤ were dftru2 1547. (Contributed by BJ, 7-Sep-2022.)
⊢ ((𝜑 → 𝜑) ↔ (𝜓 → 𝜓))
Theorem	ibi 270	Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 17-Oct-2003.)
⊢ (𝜑 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	ibir 271	Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 22-Jul-2004.)
⊢ (𝜑 → (𝜓 ↔ 𝜑))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	ibd 272	Deduction that converts a biconditional implied by one of its arguments, into an implication. Deduction associated with ibi 270. (Contributed by NM, 26-Jun-2004.)
⊢ (𝜑 → (𝜓 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜒))
Theorem	pm5.74 273	Distribution of implication over biconditional. Theorem *5.74 of [WhiteheadRussell] p. 126. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 11-Apr-2013.)
⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)))
Theorem	pm5.74i 274	Distribution of implication over biconditional (inference form). (Contributed by NM, 1-Aug-1994.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))
Theorem	pm5.74ri 275	Distribution of implication over biconditional (reverse inference form). (Contributed by NM, 1-Aug-1994.)
⊢ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	pm5.74d 276	Distribution of implication over biconditional (deduction form). (Contributed by NM, 21-Mar-1996.)
⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))    ⇒   ⊢ (𝜑 → ((𝜓 → 𝜒) ↔ (𝜓 → 𝜃)))
Theorem	pm5.74rd 277	Distribution of implication over biconditional (deduction form). (Contributed by NM, 19-Mar-1997.)
⊢ (𝜑 → ((𝜓 → 𝜒) ↔ (𝜓 → 𝜃)))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))
Theorem	bitri 278	An inference from transitive law for logical equivalence. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (𝜑 ↔ 𝜒)
Theorem	bitr2i 279	An inference from transitive law for logical equivalence. (Contributed by NM, 12-Mar-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (𝜒 ↔ 𝜑)
Theorem	bitr3i 280	An inference from transitive law for logical equivalence. (Contributed by NM, 2-Jun-1993.)
⊢ (𝜓 ↔ 𝜑)    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (𝜑 ↔ 𝜒)
Theorem	bitr4i 281	An inference from transitive law for logical equivalence. (Contributed by NM, 3-Jan-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜓)    ⇒   ⊢ (𝜑 ↔ 𝜒)
Theorem	bitrd 282	Deduction form of bitri 278. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 14-Apr-2013.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜒 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜃))
Theorem	bitr2d 283	Deduction form of bitr2i 279. (Contributed by NM, 9-Jun-2004.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜒 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (𝜃 ↔ 𝜓))
Theorem	bitr3d 284	Deduction form of bitr3i 280. (Contributed by NM, 14-May-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (𝜒 ↔ 𝜃))
Theorem	bitr4d 285	Deduction form of bitr4i 281. (Contributed by NM, 30-Jun-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜃))
Theorem	syl5bb 286	A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜒 → (𝜑 ↔ 𝜃))
Theorem	syl5rbb 287	A syllogism inference from two biconditionals. (Contributed by NM, 1-Aug-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜒 → (𝜃 ↔ 𝜑))
Theorem	bitr3id 288	A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜓 ↔ 𝜑)    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜒 → (𝜑 ↔ 𝜃))
Theorem	bitr3di 289	A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜓 ↔ 𝜃)    ⇒   ⊢ (𝜑 → (𝜒 ↔ 𝜃))
Theorem	bitrdi 290	A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜃))
Theorem	bitr2di 291	A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ (𝜑 → (𝜃 ↔ 𝜓))
Theorem	bitr4di 292	A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜃 ↔ 𝜒)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜃))
Theorem	bitr4id 293	A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)
⊢ (𝜓 ↔ 𝜒)    &   ⊢ (𝜑 → (𝜃 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜃))
Theorem	3imtr3i 294	A mixed syllogism inference, useful for removing a definition from both sides of an implication. (Contributed by NM, 10-Aug-1994.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 ↔ 𝜒)    &   ⊢ (𝜓 ↔ 𝜃)    ⇒   ⊢ (𝜒 → 𝜃)
Theorem	3imtr4i 295	A mixed syllogism inference, useful for applying a definition to both sides of an implication. (Contributed by NM, 3-Jan-1993.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 ↔ 𝜑)    &   ⊢ (𝜃 ↔ 𝜓)    ⇒   ⊢ (𝜒 → 𝜃)
Theorem	3imtr3d 296	More general version of 3imtr3i 294. Useful for converting conditional definitions in a formula. (Contributed by NM, 8-Apr-1996.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜓 ↔ 𝜃))    &   ⊢ (𝜑 → (𝜒 ↔ 𝜏))    ⇒   ⊢ (𝜑 → (𝜃 → 𝜏))
Theorem	3imtr4d 297	More general version of 3imtr4i 295. Useful for converting conditional definitions in a formula. (Contributed by NM, 26-Oct-1995.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜓))    &   ⊢ (𝜑 → (𝜏 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜃 → 𝜏))
Theorem	3imtr3g 298	More general version of 3imtr3i 294. Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜓 ↔ 𝜃)    &   ⊢ (𝜒 ↔ 𝜏)    ⇒   ⊢ (𝜑 → (𝜃 → 𝜏))
Theorem	3imtr4g 299	More general version of 3imtr4i 295. Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜃 ↔ 𝜓)    &   ⊢ (𝜏 ↔ 𝜒)    ⇒   ⊢ (𝜑 → (𝜃 → 𝜏))
Theorem	3bitri 300	A chained inference from transitive law for logical equivalence. (Contributed by NM, 3-Jan-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜓 ↔ 𝜒)    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ (𝜑 ↔ 𝜃)