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

Theoremmtoi 201 Modus tollens inference. (Contributed by NM, 5-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
¬ 𝜒    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜓)

Theoremmt2 202 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

Theoremmt3 203 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    &   𝜑𝜓)       𝜑

Theorempeirce 204 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 890. 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.)
(((𝜑𝜓) → 𝜑) → 𝜑)

Theoremlooinv 205 The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. Using dfor2 898, 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.)
(((𝜑𝜓) → 𝜓) → ((𝜓𝜑) → 𝜑))

Theorembijust0 206 A self-implication (see id 22) does not imply its own negation. The justification theorem bijust 207 is one of its instances. (Contributed by NM, 11-May-1999.) (Proof shortened by Josh Purinton, 29-Dec-2000.) Extract bijust0 206 from proof of bijust 207. (Revised by BJ, 19-Mar-2020.)
¬ ((𝜑𝜑) → ¬ (𝜑𝜑))

Theorembijust 207 Theorem used to justify the definition of the biconditional df-bi 209. Instance of bijust0 206. (Contributed by NM, 11-May-1999.)
¬ ((¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))) → ¬ (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))))

1.2.5  Logical equivalence

The definition df-bi 209 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 (𝜑𝜓) as an abbreviation for ¬ ((𝜑𝜓) → ¬ (𝜓𝜑)).

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 399 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.

Syntaxwb 208 Extend wff definition to include the biconditional connective.
wff (𝜑𝜓)

Definitiondf-bi 209 Define the biconditional (logical "iff" or "if and only if").

The definition df-bi 209 in this section is our first definition, which introduces and defines the biconditional connective . We define a wff of the form (𝜑𝜓) as an abbreviation for ¬ ((𝜑𝜓) → ¬ (𝜓𝜑)).

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 844 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 1085) 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 (𝜑𝜓) (the definiendum i.e. the thing being defined) with ¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) (the definiens i.e. the defining expression) in the definition, the definition becomes the previously proved theorem bijust 207. It is impossible to use df-bi 209 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 209 in the proof with the corresponding bijust 207 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 1540) and the constant false (df-fal 1550), we will be able to prove these truth table values: ((⊤ ↔ ⊤) ↔ ⊤) (trubitru 1566), ((⊤ ↔ ⊥) ↔ ⊥) (trubifal 1568), ((⊥ ↔ ⊤) ↔ ⊥) (falbitru 1567), and ((⊥ ↔ ⊥) ↔ ⊤) (falbifal 1569).

See dfbi1 215, dfbi2 477, and dfbi3 1044 for theorems suggesting typical textbook definitions of , showing that our definition has the properties we expect. Theorem dfbi1 215 is particularly useful if we want to eliminate from an expression to convert it to primitives. Theorem dfbi 478 shows this definition rewritten in an abbreviated form after conjunction is introduced, for easier understanding.

Contrast with (df-or 844), (wi 4), (df-nan 1482), and (df-xor 1502). In some sense returns true if two truth values are equal; = (df-cleq 2813) returns true if two classes are equal. (Contributed by NM, 27-Dec-1992.)

¬ (((𝜑𝜓) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))) → ¬ (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓)))

Theoremimpbi 210 Property of the biconditional connective. (Contributed by NM, 11-May-1999.)
((𝜑𝜓) → ((𝜓𝜑) → (𝜑𝜓)))

Theoremimpbii 211 Infer an equivalence from an implication and its converse. Inference associated with impbi 210. (Contributed by NM, 29-Dec-1992.)
(𝜑𝜓)    &   (𝜓𝜑)       (𝜑𝜓)

Theoremimpbidd 212 Deduce an equivalence from two implications. Double deduction associated with impbi 210 and impbii 211. Deduction associated with impbid 214. (Contributed by Rodolfo Medina, 12-Oct-2010.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓 → (𝜃𝜒)))       (𝜑 → (𝜓 → (𝜒𝜃)))

Theoremimpbid21d 213 Deduce an equivalence from two implications. (Contributed by Wolf Lammen, 12-May-2013.)
(𝜓 → (𝜒𝜃))    &   (𝜑 → (𝜃𝜒))       (𝜑 → (𝜓 → (𝜒𝜃)))

Theoremimpbid 214 Deduce an equivalence from two implications. Deduction associated with impbi 210 and impbii 211. (Contributed by NM, 24-Jan-1993.) Revised to prove it from impbid21d 213. (Revised by Wolf Lammen, 3-Nov-2012.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜓))       (𝜑 → (𝜓𝜒))

Theoremdfbi1 215 Relate the biconditional connective to primitive connectives. See dfbi1ALT 216 for an unusual version proved directly from axioms. (Contributed by NM, 29-Dec-1992.)
((𝜑𝜓) ↔ ¬ ((𝜑𝜓) → ¬ (𝜓𝜑)))

Theoremdfbi1ALT 216 Alternate proof of dfbi1 215. 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 209, compared to over 800 steps were the proof of dfbi1 215 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.)
((𝜑𝜓) ↔ ¬ ((𝜑𝜓) → ¬ (𝜓𝜑)))

Theorembiimp 217 Property of the biconditional connective. (Contributed by NM, 11-May-1999.)
((𝜑𝜓) → (𝜑𝜓))

Theorembiimpi 218 Infer an implication from a logical equivalence. Inference associated with biimp 217. (Contributed by NM, 29-Dec-1992.)
(𝜑𝜓)       (𝜑𝜓)

Theoremsylbi 219 A mixed syllogism inference from a biconditional and an implication. Useful for substituting an antecedent with a definition. (Contributed by NM, 3-Jan-1993.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑𝜒)

Theoremsylib 220 A mixed syllogism inference from an implication and a biconditional. (Contributed by NM, 3-Jan-1993.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑𝜒)

Theoremsylbb 221 A mixed syllogism inference from two biconditionals. (Contributed by BJ, 30-Mar-2019.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑𝜒)

Theorembiimpr 222 Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)
((𝜑𝜓) → (𝜓𝜑))

Theorembicom1 223 Commutative law for the biconditional. (Contributed by Wolf Lammen, 10-Nov-2012.)
((𝜑𝜓) → (𝜓𝜑))

Theorembicom 224 Commutative law for the biconditional. Theorem *4.21 of [WhiteheadRussell] p. 117. (Contributed by NM, 11-May-1993.)
((𝜑𝜓) ↔ (𝜓𝜑))

Theorembicomd 225 Commute two sides of a biconditional in a deduction. (Contributed by NM, 14-May-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜒𝜓))

Theorembicomi 226 Inference from commutative law for logical equivalence. (Contributed by NM, 3-Jan-1993.)
(𝜑𝜓)       (𝜓𝜑)

Theoremimpbid1 227 Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜓)       (𝜑 → (𝜓𝜒))

Theoremimpbid2 228 Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.) (Proof shortened by Wolf Lammen, 27-Sep-2013.)
(𝜓𝜒)    &   (𝜑 → (𝜒𝜓))       (𝜑 → (𝜓𝜒))

Theoremimpcon4bid 229 A variation on impbid 214 with contraposition. (Contributed by Jeff Hankins, 3-Jul-2009.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (¬ 𝜓 → ¬ 𝜒))       (𝜑 → (𝜓𝜒))

Theorembiimpri 230 Infer a converse implication from a logical equivalence. Inference associated with biimpr 222. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 16-Sep-2013.)
(𝜑𝜓)       (𝜓𝜑)

Theorembiimpd 231 Deduce an implication from a logical equivalence. Deduction associated with biimp 217 and biimpi 218. (Contributed by NM, 11-Jan-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓𝜒))

Theoremmpbi 232 An inference from a biconditional, related to modus ponens. (Contributed by NM, 11-May-1993.)
𝜑    &   (𝜑𝜓)       𝜓

Theoremmpbir 233 An inference from a biconditional, related to modus ponens. (Contributed by NM, 28-Dec-1992.)
𝜓    &   (𝜑𝜓)       𝜑

Theoremmpbid 234 A deduction from a biconditional, related to modus ponens. (Contributed by NM, 21-Jun-1993.)
(𝜑𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑𝜒)

Theoremmpbii 235 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 → (𝜓𝜒))       (𝜑𝜒)

Theoremsylibr 236 A mixed syllogism inference from an implication and a biconditional. Useful for substituting a consequent with a definition. (Contributed by NM, 3-Jan-1993.)
(𝜑𝜓)    &   (𝜒𝜓)       (𝜑𝜒)

Theoremsylbir 237 A mixed syllogism inference from a biconditional and an implication. (Contributed by NM, 3-Jan-1993.)
(𝜓𝜑)    &   (𝜓𝜒)       (𝜑𝜒)

Theoremsylbbr 238 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 219, sylib 220, sylbir 237, sylibr 236; four inferences inferring an implication from two biconditionals: sylbb 221, sylbbr 238, sylbb1 239, sylbb2 240; four inferences inferring a biconditional from two biconditionals: bitri 277, bitr2i 278, bitr3i 279, bitr4i 280 (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 234, bitrd 281, syl5bb 285, syl6bb 289 and variants. (Contributed by BJ, 21-Apr-2019.)

(𝜑𝜓)    &   (𝜓𝜒)       (𝜒𝜑)

Theoremsylbb1 239 A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.)
(𝜑𝜓)    &   (𝜑𝜒)       (𝜓𝜒)

Theoremsylbb2 240 A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.)
(𝜑𝜓)    &   (𝜒𝜓)       (𝜑𝜒)

Theoremsylibd 241 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))       (𝜑 → (𝜓𝜃))

Theoremsylbid 242 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))       (𝜑 → (𝜓𝜃))

Theoremmpbidi 243 A deduction from a biconditional, related to modus ponens. (Contributed by NM, 9-Aug-1994.)
(𝜃 → (𝜑𝜓))    &   (𝜑 → (𝜓𝜒))       (𝜃 → (𝜑𝜒))

Theoremsyl5bi 244 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.)
(𝜑𝜓)    &   (𝜒 → (𝜓𝜃))       (𝜒 → (𝜑𝜃))

Theoremsyl5bir 245 A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 21-Jun-1993.)
(𝜓𝜑)    &   (𝜒 → (𝜓𝜃))       (𝜒 → (𝜑𝜃))

Theoremsyl5ib 246 A mixed syllogism inference. (Contributed by NM, 12-Jan-1993.)
(𝜑𝜓)    &   (𝜒 → (𝜓𝜃))       (𝜒 → (𝜑𝜃))

Theoremsyl5ibcom 247 A mixed syllogism inference. (Contributed by NM, 19-Jun-2007.)
(𝜑𝜓)    &   (𝜒 → (𝜓𝜃))       (𝜑 → (𝜒𝜃))

Theoremsyl5ibr 248 A mixed syllogism inference. (Contributed by NM, 3-Apr-1994.)
(𝜑𝜃)    &   (𝜒 → (𝜓𝜃))       (𝜒 → (𝜑𝜓))

Theoremsyl5ibrcom 249 A mixed syllogism inference. (Contributed by NM, 20-Jun-2007.)
(𝜑𝜃)    &   (𝜒 → (𝜓𝜃))       (𝜑 → (𝜒𝜓))

Theorembiimprd 250 Deduce a converse implication from a logical equivalence. Deduction associated with biimpr 222 and biimpri 230. (Contributed by NM, 11-Jan-1993.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜒𝜓))

Theorembiimpcd 251 Deduce a commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)
(𝜑 → (𝜓𝜒))       (𝜓 → (𝜑𝜒))

Theorembiimprcd 252 Deduce a converse commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)
(𝜑 → (𝜓𝜒))       (𝜒 → (𝜑𝜓))

Theoremsyl6ib 253 A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 21-Jun-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜃)       (𝜑 → (𝜓𝜃))

Theoremsyl6ibr 254 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 → (𝜓𝜒))    &   (𝜃𝜒)       (𝜑 → (𝜓𝜃))

Theoremsyl6bi 255 A mixed syllogism inference. (Contributed by NM, 2-Jan-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜃)       (𝜑 → (𝜓𝜃))

Theoremsyl6bir 256 A mixed syllogism inference. (Contributed by NM, 18-May-1994.)
(𝜑 → (𝜒𝜓))    &   (𝜒𝜃)       (𝜑 → (𝜓𝜃))

Theoremsyl7bi 257 A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 14-May-1993.)
(𝜑𝜓)    &   (𝜒 → (𝜃 → (𝜓𝜏)))       (𝜒 → (𝜃 → (𝜑𝜏)))

Theoremsyl8ib 258 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 → (𝜒𝜃)))    &   (𝜃𝜏)       (𝜑 → (𝜓 → (𝜒𝜏)))

Theoremmpbird 259 A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜒)    &   (𝜑 → (𝜓𝜒))       (𝜑𝜓)

Theoremmpbiri 260 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 → (𝜓𝜒))       (𝜑𝜓)

Theoremsylibrd 261 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))       (𝜑 → (𝜓𝜃))

Theoremsylbird 262 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
(𝜑 → (𝜒𝜓))    &   (𝜑 → (𝜒𝜃))       (𝜑 → (𝜓𝜃))

Theorembiid 263 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 2820. (Contributed by NM, 2-Jun-1993.)
(𝜑𝜑)

Theorembiidd 264 Principle of identity with antecedent. (Contributed by NM, 25-Nov-1995.)
(𝜑 → (𝜓𝜓))

Theorempm5.1im 265 Two propositions are equivalent if they are both true. Closed form of 2th 266. Equivalent to a biimp 217-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 ↔ (𝜑𝜓))). (Contributed by Wolf Lammen, 12-May-2013.)
(𝜑 → (𝜓 → (𝜑𝜓)))

Theorem2th 266 Two truths are equivalent. (Contributed by NM, 18-Aug-1993.)
𝜑    &   𝜓       (𝜑𝜓)

Theorem2thd 267 Two truths are equivalent. Deduction form. (Contributed by NM, 3-Jun-2012.)
(𝜑𝜓)    &   (𝜑𝜒)       (𝜑 → (𝜓𝜒))

Theoremmonothetic 268 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 1539 is an instance. Relatedly, this would be the justification theorem if the definition of were dftru2 1542. (Contributed by BJ, 7-Sep-2022.)
((𝜑𝜑) ↔ (𝜓𝜓))

Theoremibi 269 Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 17-Oct-2003.)
(𝜑 → (𝜑𝜓))       (𝜑𝜓)

Theoremibir 270 Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 22-Jul-2004.)
(𝜑 → (𝜓𝜑))       (𝜑𝜓)

Theoremibd 271 Deduction that converts a biconditional implied by one of its arguments, into an implication. Deduction associated with ibi 269. (Contributed by NM, 26-Jun-2004.)
(𝜑 → (𝜓 → (𝜓𝜒)))       (𝜑 → (𝜓𝜒))

Theorempm5.74 272 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 → (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))

Theorempm5.74i 273 Distribution of implication over biconditional (inference form). (Contributed by NM, 1-Aug-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) ↔ (𝜑𝜒))

Theorempm5.74ri 274 Distribution of implication over biconditional (reverse inference form). (Contributed by NM, 1-Aug-1994.)
((𝜑𝜓) ↔ (𝜑𝜒))       (𝜑 → (𝜓𝜒))

Theorempm5.74d 275 Distribution of implication over biconditional (deduction form). (Contributed by NM, 21-Mar-1996.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))

Theorempm5.74rd 276 Distribution of implication over biconditional (deduction form). (Contributed by NM, 19-Mar-1997.)
(𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))       (𝜑 → (𝜓 → (𝜒𝜃)))

Theorembitri 277 An inference from transitive law for logical equivalence. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑𝜒)

Theorembitr2i 278 An inference from transitive law for logical equivalence. (Contributed by NM, 12-Mar-1993.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜒𝜑)

Theorembitr3i 279 An inference from transitive law for logical equivalence. (Contributed by NM, 2-Jun-1993.)
(𝜓𝜑)    &   (𝜓𝜒)       (𝜑𝜒)

Theorembitr4i 280 An inference from transitive law for logical equivalence. (Contributed by NM, 3-Jan-1993.)
(𝜑𝜓)    &   (𝜒𝜓)       (𝜑𝜒)

Theorembitrd 281 Deduction form of bitri 277. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 14-Apr-2013.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))       (𝜑 → (𝜓𝜃))

Theorembitr2d 282 Deduction form of bitr2i 278. (Contributed by NM, 9-Jun-2004.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))       (𝜑 → (𝜃𝜓))

Theorembitr3d 283 Deduction form of bitr3i 279. (Contributed by NM, 14-May-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))       (𝜑 → (𝜒𝜃))

Theorembitr4d 284 Deduction form of bitr4i 280. (Contributed by NM, 30-Jun-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))       (𝜑 → (𝜓𝜃))

Theoremsyl5bb 285 A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993.)
(𝜑𝜓)    &   (𝜒 → (𝜓𝜃))       (𝜒 → (𝜑𝜃))

Theoremsyl5rbb 286 A syllogism inference from two biconditionals. (Contributed by NM, 1-Aug-1993.)
(𝜑𝜓)    &   (𝜒 → (𝜓𝜃))       (𝜒 → (𝜃𝜑))

Theoremsyl5bbr 287 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
(𝜓𝜑)    &   (𝜒 → (𝜓𝜃))       (𝜒 → (𝜑𝜃))

Theorembitr3di 288 A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜓𝜃)       (𝜑 → (𝜒𝜃))

Theoremsyl6bb 289 A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜃)       (𝜑 → (𝜓𝜃))

Theoremsyl6rbb 290 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜃)       (𝜑 → (𝜃𝜓))

Theoremsyl6bbr 291 A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜃𝜒)       (𝜑 → (𝜓𝜃))

Theoremsyl6rbbr 292 A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜃𝜒)       (𝜑 → (𝜃𝜓))

Theorem3imtr3i 293 A mixed syllogism inference, useful for removing a definition from both sides of an implication. (Contributed by NM, 10-Aug-1994.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜓𝜃)       (𝜒𝜃)

Theorem3imtr4i 294 A mixed syllogism inference, useful for applying a definition to both sides of an implication. (Contributed by NM, 3-Jan-1993.)
(𝜑𝜓)    &   (𝜒𝜑)    &   (𝜃𝜓)       (𝜒𝜃)

Theorem3imtr3d 295 More general version of 3imtr3i 293. Useful for converting conditional definitions in a formula. (Contributed by NM, 8-Apr-1996.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜒𝜏))       (𝜑 → (𝜃𝜏))

Theorem3imtr4d 296 More general version of 3imtr4i 294. Useful for converting conditional definitions in a formula. (Contributed by NM, 26-Oct-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜓))    &   (𝜑 → (𝜏𝜒))       (𝜑 → (𝜃𝜏))

Theorem3imtr3g 297 More general version of 3imtr3i 293. Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)
(𝜑 → (𝜓𝜒))    &   (𝜓𝜃)    &   (𝜒𝜏)       (𝜑 → (𝜃𝜏))

Theorem3imtr4g 298 More general version of 3imtr4i 294. Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)
(𝜑 → (𝜓𝜒))    &   (𝜃𝜓)    &   (𝜏𝜒)       (𝜑 → (𝜃𝜏))

Theorem3bitri 299 A chained inference from transitive law for logical equivalence. (Contributed by NM, 3-Jan-1993.)
(𝜑𝜓)    &   (𝜓𝜒)    &   (𝜒𝜃)       (𝜑𝜃)

Theorem3bitrri 300 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
(𝜑𝜓)    &   (𝜓𝜒)    &   (𝜒𝜃)       (𝜃𝜑)

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