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

Theoremimpbid 201 Deduce an equivalence from two implications. Deduction associated with impbi 197 and impbii 198. (Contributed by NM, 24-Jan-1993.) Revised to prove it from impbid21d 200. (Revised by Wolf Lammen, 3-Nov-2012.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜓))       (𝜑 → (𝜓𝜒))

Theoremdfbi1 202 Relate the biconditional connective to primitive connectives. See dfbi1ALT 203 for an unusual version proved directly from axioms. (Contributed by NM, 29-Dec-1992.)
((𝜑𝜓) ↔ ¬ ((𝜑𝜓) → ¬ (𝜓𝜑)))

Theoremdfbi1ALT 203 Alternate proof of dfbi1 202. 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 196, compared to over 800 steps were the proof of dfbi1 202 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 204 Property of the biconditional connective. (Contributed by NM, 11-May-1999.)
((𝜑𝜓) → (𝜑𝜓))

Theorembiimpi 205 Infer an implication from a logical equivalence. Inference associated with biimp 204. (Contributed by NM, 29-Dec-1992.)
(𝜑𝜓)       (𝜑𝜓)

Theoremsylbi 206 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 207 A mixed syllogism inference from an implication and a biconditional. (Contributed by NM, 3-Jan-1993.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑𝜒)

Theoremsylbb 208 A mixed syllogism inference from two biconditionals. (Contributed by BJ, 30-Mar-2019.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑𝜒)

Theorembiimpr 209 Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)
((𝜑𝜓) → (𝜓𝜑))

Theorembicom1 210 Commutative law for the biconditional. (Contributed by Wolf Lammen, 10-Nov-2012.)
((𝜑𝜓) → (𝜓𝜑))

Theorembicom 211 Commutative law for the biconditional. Theorem *4.21 of [WhiteheadRussell] p. 117. (Contributed by NM, 11-May-1993.)
((𝜑𝜓) ↔ (𝜓𝜑))

Theorembicomd 212 Commute two sides of a biconditional in a deduction. (Contributed by NM, 14-May-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜒𝜓))

Theorembicomi 213 Inference from commutative law for logical equivalence. (Contributed by NM, 3-Jan-1993.)
(𝜑𝜓)       (𝜓𝜑)

Theoremimpbid1 214 Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜓)       (𝜑 → (𝜓𝜒))

Theoremimpbid2 215 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 216 A variation on impbid 201 with contraposition. (Contributed by Jeff Hankins, 3-Jul-2009.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (¬ 𝜓 → ¬ 𝜒))       (𝜑 → (𝜓𝜒))

Theorembiimpri 217 Infer a converse implication from a logical equivalence. Inference associated with biimpr 209. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 16-Sep-2013.)
(𝜑𝜓)       (𝜓𝜑)

Theorembiimpd 218 Deduce an implication from a logical equivalence. Deduction associated with biimp 204 and biimpi 205. (Contributed by NM, 11-Jan-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓𝜒))

Theoremmpbi 219 An inference from a biconditional, related to modus ponens. (Contributed by NM, 11-May-1993.)
𝜑    &   (𝜑𝜓)       𝜓

Theoremmpbir 220 An inference from a biconditional, related to modus ponens. (Contributed by NM, 28-Dec-1992.)
𝜓    &   (𝜑𝜓)       𝜑

Theoremmpbid 221 A deduction from a biconditional, related to modus ponens. (Contributed by NM, 21-Jun-1993.)
(𝜑𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑𝜒)

Theoremmpbii 222 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 223 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 224 A mixed syllogism inference from a biconditional and an implication. (Contributed by NM, 3-Jan-1993.)
(𝜓𝜑)    &   (𝜓𝜒)       (𝜑𝜒)

Theoremsylbbr 225 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 206, sylib 207, sylbir 224, sylibr 223; four inferences inferring an implication from two biconditionals: sylbb 208, sylbbr 225, sylbb1 226, sylbb2 227; four inferences inferring a biconditional from two biconditionals: bitri 263, bitr2i 264, bitr3i 265, bitr4i 266 (and more for chaining more biconditionals). There are also closed forms and deduction versions of these, like, among many others, syld 46, syl5 33, syl6 34, mpbid 221, bitrd 267, syl5bb 271, syl6bb 275 and variants. (Contributed by BJ, 21-Apr-2019.)

(𝜑𝜓)    &   (𝜓𝜒)       (𝜒𝜑)

Theoremsylbb1 226 A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.)
(𝜑𝜓)    &   (𝜑𝜒)       (𝜓𝜒)

Theoremsylbb2 227 A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.)
(𝜑𝜓)    &   (𝜒𝜓)       (𝜑𝜒)

Theoremsylibd 228 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))       (𝜑 → (𝜓𝜃))

Theoremsylbid 229 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))       (𝜑 → (𝜓𝜃))

Theoremmpbidi 230 A deduction from a biconditional, related to modus ponens. (Contributed by NM, 9-Aug-1994.)
(𝜃 → (𝜑𝜓))    &   (𝜑 → (𝜓𝜒))       (𝜃 → (𝜑𝜒))

Theoremsyl5bi 231 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 232 A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 21-Jun-1993.)
(𝜓𝜑)    &   (𝜒 → (𝜓𝜃))       (𝜒 → (𝜑𝜃))

Theoremsyl5ib 233 A mixed syllogism inference. (Contributed by NM, 12-Jan-1993.)
(𝜑𝜓)    &   (𝜒 → (𝜓𝜃))       (𝜒 → (𝜑𝜃))

Theoremsyl5ibcom 234 A mixed syllogism inference. (Contributed by NM, 19-Jun-2007.)
(𝜑𝜓)    &   (𝜒 → (𝜓𝜃))       (𝜑 → (𝜒𝜃))

Theoremsyl5ibr 235 A mixed syllogism inference. (Contributed by NM, 3-Apr-1994.)
(𝜑𝜃)    &   (𝜒 → (𝜓𝜃))       (𝜒 → (𝜑𝜓))

Theoremsyl5ibrcom 236 A mixed syllogism inference. (Contributed by NM, 20-Jun-2007.)
(𝜑𝜃)    &   (𝜒 → (𝜓𝜃))       (𝜑 → (𝜒𝜓))

Theorembiimprd 237 Deduce a converse implication from a logical equivalence. Deduction associated with biimpr 209 and biimpri 217. (Contributed by NM, 11-Jan-1993.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜒𝜓))

Theorembiimpcd 238 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 239 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 240 A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 21-Jun-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜃)       (𝜑 → (𝜓𝜃))

Theoremsyl6ibr 241 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 242 A mixed syllogism inference. (Contributed by NM, 2-Jan-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜃)       (𝜑 → (𝜓𝜃))

Theoremsyl6bir 243 A mixed syllogism inference. (Contributed by NM, 18-May-1994.)
(𝜑 → (𝜒𝜓))    &   (𝜒𝜃)       (𝜑 → (𝜓𝜃))

Theoremsyl7bi 244 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 245 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 246 A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜒)    &   (𝜑 → (𝜓𝜒))       (𝜑𝜓)

Theoremmpbiri 247 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 248 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))       (𝜑 → (𝜓𝜃))

Theoremsylbird 249 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
(𝜑 → (𝜒𝜓))    &   (𝜑 → (𝜒𝜃))       (𝜑 → (𝜓𝜃))

Theorembiid 250 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 2610. (Contributed by NM, 2-Jun-1993.)
(𝜑𝜑)

Theorembiidd 251 Principle of identity with antecedent. (Contributed by NM, 25-Nov-1995.)
(𝜑 → (𝜓𝜓))

Theorempm5.1im 252 Two propositions are equivalent if they are both true. Closed form of 2th 253. Equivalent to a biimp 204-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 253 Two truths are equivalent. (Contributed by NM, 18-Aug-1993.)
𝜑    &   𝜓       (𝜑𝜓)

Theorem2thd 254 Two truths are equivalent (deduction rule). (Contributed by NM, 3-Jun-2012.)
(𝜑𝜓)    &   (𝜑𝜒)       (𝜑 → (𝜓𝜒))

Theoremibi 255 Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 17-Oct-2003.)
(𝜑 → (𝜑𝜓))       (𝜑𝜓)

Theoremibir 256 Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 22-Jul-2004.)
(𝜑 → (𝜓𝜑))       (𝜑𝜓)

Theoremibd 257 Deduction that converts a biconditional implied by one of its arguments, into an implication. Deduction associated with ibi 255. (Contributed by NM, 26-Jun-2004.)
(𝜑 → (𝜓 → (𝜓𝜒)))       (𝜑 → (𝜓𝜒))

Theorempm5.74 258 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 259 Distribution of implication over biconditional (inference rule). (Contributed by NM, 1-Aug-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) ↔ (𝜑𝜒))

Theorempm5.74ri 260 Distribution of implication over biconditional (reverse inference rule). (Contributed by NM, 1-Aug-1994.)
((𝜑𝜓) ↔ (𝜑𝜒))       (𝜑 → (𝜓𝜒))

Theorempm5.74d 261 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 21-Mar-1996.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))

Theorempm5.74rd 262 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 19-Mar-1997.)
(𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))       (𝜑 → (𝜓 → (𝜒𝜃)))

Theorembitri 263 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 264 An inference from transitive law for logical equivalence. (Contributed by NM, 12-Mar-1993.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜒𝜑)

Theorembitr3i 265 An inference from transitive law for logical equivalence. (Contributed by NM, 2-Jun-1993.)
(𝜓𝜑)    &   (𝜓𝜒)       (𝜑𝜒)

Theorembitr4i 266 An inference from transitive law for logical equivalence. (Contributed by NM, 3-Jan-1993.)
(𝜑𝜓)    &   (𝜒𝜓)       (𝜑𝜒)

Theorembitrd 267 Deduction form of bitri 263. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 14-Apr-2013.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))       (𝜑 → (𝜓𝜃))

Theorembitr2d 268 Deduction form of bitr2i 264. (Contributed by NM, 9-Jun-2004.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))       (𝜑 → (𝜃𝜓))

Theorembitr3d 269 Deduction form of bitr3i 265. (Contributed by NM, 14-May-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))       (𝜑 → (𝜒𝜃))

Theorembitr4d 270 Deduction form of bitr4i 266. (Contributed by NM, 30-Jun-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))       (𝜑 → (𝜓𝜃))

Theoremsyl5bb 271 A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993.)
(𝜑𝜓)    &   (𝜒 → (𝜓𝜃))       (𝜒 → (𝜑𝜃))

Theoremsyl5rbb 272 A syllogism inference from two biconditionals. (Contributed by NM, 1-Aug-1993.)
(𝜑𝜓)    &   (𝜒 → (𝜓𝜃))       (𝜒 → (𝜃𝜑))

Theoremsyl5bbr 273 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
(𝜓𝜑)    &   (𝜒 → (𝜓𝜃))       (𝜒 → (𝜑𝜃))

Theoremsyl5rbbr 274 A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)
(𝜓𝜑)    &   (𝜒 → (𝜓𝜃))       (𝜒 → (𝜃𝜑))

Theoremsyl6bb 275 A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜃)       (𝜑 → (𝜓𝜃))

Theoremsyl6rbb 276 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜃)       (𝜑 → (𝜃𝜓))

Theoremsyl6bbr 277 A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜃𝜒)       (𝜑 → (𝜓𝜃))

Theoremsyl6rbbr 278 A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜃𝜒)       (𝜑 → (𝜃𝜓))

Theorem3imtr3i 279 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 280 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 281 More general version of 3imtr3i 279. Useful for converting conditional definitions in a formula. (Contributed by NM, 8-Apr-1996.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜒𝜏))       (𝜑 → (𝜃𝜏))

Theorem3imtr4d 282 More general version of 3imtr4i 280. Useful for converting conditional definitions in a formula. (Contributed by NM, 26-Oct-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜓))    &   (𝜑 → (𝜏𝜒))       (𝜑 → (𝜃𝜏))

Theorem3imtr3g 283 More general version of 3imtr3i 279. 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 284 More general version of 3imtr4i 280. 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 285 A chained inference from transitive law for logical equivalence. (Contributed by NM, 3-Jan-1993.)
(𝜑𝜓)    &   (𝜓𝜒)    &   (𝜒𝜃)       (𝜑𝜃)

Theorem3bitrri 286 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
(𝜑𝜓)    &   (𝜓𝜒)    &   (𝜒𝜃)       (𝜃𝜑)

Theorem3bitr2i 287 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
(𝜑𝜓)    &   (𝜒𝜓)    &   (𝜒𝜃)       (𝜑𝜃)

Theorem3bitr2ri 288 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
(𝜑𝜓)    &   (𝜒𝜓)    &   (𝜒𝜃)       (𝜃𝜑)

Theorem3bitr3i 289 A chained inference from transitive law for logical equivalence. (Contributed by NM, 19-Aug-1993.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜓𝜃)       (𝜒𝜃)

Theorem3bitr3ri 290 A chained inference from transitive law for logical equivalence. (Contributed by NM, 21-Jun-1993.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜓𝜃)       (𝜃𝜒)

Theorem3bitr4i 291 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.)
(𝜑𝜓)    &   (𝜒𝜑)    &   (𝜃𝜓)       (𝜒𝜃)

Theorem3bitr4ri 292 A chained inference from transitive law for logical equivalence. (Contributed by NM, 2-Sep-1995.)
(𝜑𝜓)    &   (𝜒𝜑)    &   (𝜃𝜓)       (𝜃𝜒)

Theorem3bitrd 293 Deduction from transitivity of biconditional. (Contributed by NM, 13-Aug-1999.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))    &   (𝜑 → (𝜃𝜏))       (𝜑 → (𝜓𝜏))

Theorem3bitrrd 294 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))    &   (𝜑 → (𝜃𝜏))       (𝜑 → (𝜏𝜓))

Theorem3bitr2d 295 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → (𝜓𝜏))

Theorem3bitr2rd 296 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → (𝜏𝜓))

Theorem3bitr3d 297 Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 24-Apr-1996.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜒𝜏))       (𝜑 → (𝜃𝜏))

Theorem3bitr3rd 298 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜒𝜏))       (𝜑 → (𝜏𝜃))

Theorem3bitr4d 299 Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 18-Oct-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜓))    &   (𝜑 → (𝜏𝜒))       (𝜑 → (𝜃𝜏))

Theorem3bitr4rd 300 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜓))    &   (𝜑 → (𝜏𝜒))       (𝜑 → (𝜏𝜃))

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