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Theorem List for Metamath Proof Explorer - 201-300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdfbi1 201 Relate the biconditional connective to primitive connectives. See dfbi1ALT 202 for an unusual version proved directly from axioms. (Contributed by NM, 29-Dec-1992.)
((𝜑𝜓) ↔ ¬ ((𝜑𝜓) → ¬ (𝜓𝜑)))
 
Theoremdfbi1ALT 202 Alternate proof of dfbi1 201. 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 195, compared to over 800 steps were the proof of dfbi1 201 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 203 Property of the biconditional connective. (Contributed by NM, 11-May-1999.)
((𝜑𝜓) → (𝜑𝜓))
 
Theorembiimpi 204 Infer an implication from a logical equivalence. Inference associated with biimp 203. (Contributed by NM, 29-Dec-1992.)
(𝜑𝜓)       (𝜑𝜓)
 
Theoremsylbi 205 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 206 A mixed syllogism inference from an implication and a biconditional. (Contributed by NM, 3-Jan-1993.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑𝜒)
 
Theoremsylbb 207 A mixed syllogism inference from two biconditionals. (Contributed by BJ, 30-Mar-2019.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑𝜒)
 
Theorembiimpr 208 Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)
((𝜑𝜓) → (𝜓𝜑))
 
Theorembicom1 209 Commutative law for the biconditional. (Contributed by Wolf Lammen, 10-Nov-2012.)
((𝜑𝜓) → (𝜓𝜑))
 
Theorembicom 210 Commutative law for the biconditional. Theorem *4.21 of [WhiteheadRussell] p. 117. (Contributed by NM, 11-May-1993.)
((𝜑𝜓) ↔ (𝜓𝜑))
 
Theorembicomd 211 Commute two sides of a biconditional in a deduction. (Contributed by NM, 14-May-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜒𝜓))
 
Theorembicomi 212 Inference from commutative law for logical equivalence. (Contributed by NM, 3-Jan-1993.)
(𝜑𝜓)       (𝜓𝜑)
 
Theoremimpbid1 213 Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜓)       (𝜑 → (𝜓𝜒))
 
Theoremimpbid2 214 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 215 A variation on impbid 200 with contraposition. (Contributed by Jeff Hankins, 3-Jul-2009.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (¬ 𝜓 → ¬ 𝜒))       (𝜑 → (𝜓𝜒))
 
Theorembiimpri 216 Infer a converse implication from a logical equivalence. Inference associated with biimpr 208. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 16-Sep-2013.)
(𝜑𝜓)       (𝜓𝜑)
 
Theorembiimpd 217 Deduce an implication from a logical equivalence. Deduction associated with biimp 203 and biimpi 204. (Contributed by NM, 11-Jan-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓𝜒))
 
Theoremmpbi 218 An inference from a biconditional, related to modus ponens. (Contributed by NM, 11-May-1993.)
𝜑    &   (𝜑𝜓)       𝜓
 
Theoremmpbir 219 An inference from a biconditional, related to modus ponens. (Contributed by NM, 28-Dec-1992.)
𝜓    &   (𝜑𝜓)       𝜑
 
Theoremmpbid 220 A deduction from a biconditional, related to modus ponens. (Contributed by NM, 21-Jun-1993.)
(𝜑𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑𝜒)
 
Theoremmpbii 221 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 222 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 223 A mixed syllogism inference from a biconditional and an implication. (Contributed by NM, 3-Jan-1993.)
(𝜓𝜑)    &   (𝜓𝜒)       (𝜑𝜒)
 
Theoremsylbbr 224 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 205, sylib 206, sylbir 223, sylibr 222; four inferences inferring an implication from two biconditionals: sylbb 207, sylbbr 224, sylbb1 225, sylbb2 226; four inferences inferring a biconditional from two biconditionals: bitri 262, bitr2i 263, bitr3i 264, bitr4i 265 (and more for chaining more biconditionals). There are also closed forms and deduction versions of these, like, among many others, syld 45, syl5 33, syl6 34, mpbid 220, bitrd 266, syl5bb 270, syl6bb 274 and variants. (Contributed by BJ, 21-Apr-2019.)

(𝜑𝜓)    &   (𝜓𝜒)       (𝜒𝜑)
 
Theoremsylbb1 225 A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.)
(𝜑𝜓)    &   (𝜑𝜒)       (𝜓𝜒)
 
Theoremsylbb2 226 A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.)
(𝜑𝜓)    &   (𝜒𝜓)       (𝜑𝜒)
 
Theoremsylibd 227 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))       (𝜑 → (𝜓𝜃))
 
Theoremsylbid 228 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))       (𝜑 → (𝜓𝜃))
 
Theoremmpbidi 229 A deduction from a biconditional, related to modus ponens. (Contributed by NM, 9-Aug-1994.)
(𝜃 → (𝜑𝜓))    &   (𝜑 → (𝜓𝜒))       (𝜃 → (𝜑𝜒))
 
Theoremsyl5bi 230 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 231 A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 21-Jun-1993.)
(𝜓𝜑)    &   (𝜒 → (𝜓𝜃))       (𝜒 → (𝜑𝜃))
 
Theoremsyl5ib 232 A mixed syllogism inference. (Contributed by NM, 12-Jan-1993.)
(𝜑𝜓)    &   (𝜒 → (𝜓𝜃))       (𝜒 → (𝜑𝜃))
 
Theoremsyl5ibcom 233 A mixed syllogism inference. (Contributed by NM, 19-Jun-2007.)
(𝜑𝜓)    &   (𝜒 → (𝜓𝜃))       (𝜑 → (𝜒𝜃))
 
Theoremsyl5ibr 234 A mixed syllogism inference. (Contributed by NM, 3-Apr-1994.)
(𝜑𝜃)    &   (𝜒 → (𝜓𝜃))       (𝜒 → (𝜑𝜓))
 
Theoremsyl5ibrcom 235 A mixed syllogism inference. (Contributed by NM, 20-Jun-2007.)
(𝜑𝜃)    &   (𝜒 → (𝜓𝜃))       (𝜑 → (𝜒𝜓))
 
Theorembiimprd 236 Deduce a converse implication from a logical equivalence. Deduction associated with biimpr 208 and biimpri 216. (Contributed by NM, 11-Jan-1993.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜒𝜓))
 
Theorembiimpcd 237 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 238 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 239 A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 21-Jun-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜃)       (𝜑 → (𝜓𝜃))
 
Theoremsyl6ibr 240 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 241 A mixed syllogism inference. (Contributed by NM, 2-Jan-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜃)       (𝜑 → (𝜓𝜃))
 
Theoremsyl6bir 242 A mixed syllogism inference. (Contributed by NM, 18-May-1994.)
(𝜑 → (𝜒𝜓))    &   (𝜒𝜃)       (𝜑 → (𝜓𝜃))
 
Theoremsyl7bi 243 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 244 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 245 A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜒)    &   (𝜑 → (𝜓𝜒))       (𝜑𝜓)
 
Theoremmpbiri 246 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 247 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))       (𝜑 → (𝜓𝜃))
 
Theoremsylbird 248 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
(𝜑 → (𝜒𝜓))    &   (𝜑 → (𝜒𝜃))       (𝜑 → (𝜓𝜃))
 
Theorembiid 249 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 2514. (Contributed by NM, 2-Jun-1993.)
(𝜑𝜑)
 
Theorembiidd 250 Principle of identity with antecedent. (Contributed by NM, 25-Nov-1995.)
(𝜑 → (𝜓𝜓))
 
Theorempm5.1im 251 Two propositions are equivalent if they are both true. Closed form of 2th 252. Equivalent to a biimp 203-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 252 Two truths are equivalent. (Contributed by NM, 18-Aug-1993.)
𝜑    &   𝜓       (𝜑𝜓)
 
Theorem2thd 253 Two truths are equivalent (deduction rule). (Contributed by NM, 3-Jun-2012.)
(𝜑𝜓)    &   (𝜑𝜒)       (𝜑 → (𝜓𝜒))
 
Theoremibi 254 Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 17-Oct-2003.)
(𝜑 → (𝜑𝜓))       (𝜑𝜓)
 
Theoremibir 255 Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 22-Jul-2004.)
(𝜑 → (𝜓𝜑))       (𝜑𝜓)
 
Theoremibd 256 Deduction that converts a biconditional implied by one of its arguments, into an implication. Deduction associated with ibi 254. (Contributed by NM, 26-Jun-2004.)
(𝜑 → (𝜓 → (𝜓𝜒)))       (𝜑 → (𝜓𝜒))
 
Theorempm5.74 257 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 258 Distribution of implication over biconditional (inference rule). (Contributed by NM, 1-Aug-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) ↔ (𝜑𝜒))
 
Theorempm5.74ri 259 Distribution of implication over biconditional (reverse inference rule). (Contributed by NM, 1-Aug-1994.)
((𝜑𝜓) ↔ (𝜑𝜒))       (𝜑 → (𝜓𝜒))
 
Theorempm5.74d 260 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 21-Mar-1996.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))
 
Theorempm5.74rd 261 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 19-Mar-1997.)
(𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))       (𝜑 → (𝜓 → (𝜒𝜃)))
 
Theorembitri 262 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 263 An inference from transitive law for logical equivalence. (Contributed by NM, 12-Mar-1993.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜒𝜑)
 
Theorembitr3i 264 An inference from transitive law for logical equivalence. (Contributed by NM, 2-Jun-1993.)
(𝜓𝜑)    &   (𝜓𝜒)       (𝜑𝜒)
 
Theorembitr4i 265 An inference from transitive law for logical equivalence. (Contributed by NM, 3-Jan-1993.)
(𝜑𝜓)    &   (𝜒𝜓)       (𝜑𝜒)
 
Theorembitrd 266 Deduction form of bitri 262. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 14-Apr-2013.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))       (𝜑 → (𝜓𝜃))
 
Theorembitr2d 267 Deduction form of bitr2i 263. (Contributed by NM, 9-Jun-2004.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))       (𝜑 → (𝜃𝜓))
 
Theorembitr3d 268 Deduction form of bitr3i 264. (Contributed by NM, 14-May-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))       (𝜑 → (𝜒𝜃))
 
Theorembitr4d 269 Deduction form of bitr4i 265. (Contributed by NM, 30-Jun-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))       (𝜑 → (𝜓𝜃))
 
Theoremsyl5bb 270 A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993.)
(𝜑𝜓)    &   (𝜒 → (𝜓𝜃))       (𝜒 → (𝜑𝜃))
 
Theoremsyl5rbb 271 A syllogism inference from two biconditionals. (Contributed by NM, 1-Aug-1993.)
(𝜑𝜓)    &   (𝜒 → (𝜓𝜃))       (𝜒 → (𝜃𝜑))
 
Theoremsyl5bbr 272 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
(𝜓𝜑)    &   (𝜒 → (𝜓𝜃))       (𝜒 → (𝜑𝜃))
 
Theoremsyl5rbbr 273 A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)
(𝜓𝜑)    &   (𝜒 → (𝜓𝜃))       (𝜒 → (𝜃𝜑))
 
Theoremsyl6bb 274 A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜃)       (𝜑 → (𝜓𝜃))
 
Theoremsyl6rbb 275 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜃)       (𝜑 → (𝜃𝜓))
 
Theoremsyl6bbr 276 A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜃𝜒)       (𝜑 → (𝜓𝜃))
 
Theoremsyl6rbbr 277 A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜃𝜒)       (𝜑 → (𝜃𝜓))
 
Theorem3imtr3i 278 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 279 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 280 More general version of 3imtr3i 278. Useful for converting conditional definitions in a formula. (Contributed by NM, 8-Apr-1996.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜒𝜏))       (𝜑 → (𝜃𝜏))
 
Theorem3imtr4d 281 More general version of 3imtr4i 279. Useful for converting conditional definitions in a formula. (Contributed by NM, 26-Oct-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜓))    &   (𝜑 → (𝜏𝜒))       (𝜑 → (𝜃𝜏))
 
Theorem3imtr3g 282 More general version of 3imtr3i 278. 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 283 More general version of 3imtr4i 279. 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 284 A chained inference from transitive law for logical equivalence. (Contributed by NM, 3-Jan-1993.)
(𝜑𝜓)    &   (𝜓𝜒)    &   (𝜒𝜃)       (𝜑𝜃)
 
Theorem3bitrri 285 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
(𝜑𝜓)    &   (𝜓𝜒)    &   (𝜒𝜃)       (𝜃𝜑)
 
Theorem3bitr2i 286 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
(𝜑𝜓)    &   (𝜒𝜓)    &   (𝜒𝜃)       (𝜑𝜃)
 
Theorem3bitr2ri 287 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
(𝜑𝜓)    &   (𝜒𝜓)    &   (𝜒𝜃)       (𝜃𝜑)
 
Theorem3bitr3i 288 A chained inference from transitive law for logical equivalence. (Contributed by NM, 19-Aug-1993.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜓𝜃)       (𝜒𝜃)
 
Theorem3bitr3ri 289 A chained inference from transitive law for logical equivalence. (Contributed by NM, 21-Jun-1993.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜓𝜃)       (𝜃𝜒)
 
Theorem3bitr4i 290 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 291 A chained inference from transitive law for logical equivalence. (Contributed by NM, 2-Sep-1995.)
(𝜑𝜓)    &   (𝜒𝜑)    &   (𝜃𝜓)       (𝜃𝜒)
 
Theorem3bitrd 292 Deduction from transitivity of biconditional. (Contributed by NM, 13-Aug-1999.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))    &   (𝜑 → (𝜃𝜏))       (𝜑 → (𝜓𝜏))
 
Theorem3bitrrd 293 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))    &   (𝜑 → (𝜃𝜏))       (𝜑 → (𝜏𝜓))
 
Theorem3bitr2d 294 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → (𝜓𝜏))
 
Theorem3bitr2rd 295 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → (𝜏𝜓))
 
Theorem3bitr3d 296 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 297 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜒𝜏))       (𝜑 → (𝜏𝜃))
 
Theorem3bitr4d 298 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 299 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜓))    &   (𝜑 → (𝜏𝜒))       (𝜑 → (𝜏𝜃))
 
Theorem3bitr3g 300 More general version of 3bitr3i 288. Useful for converting definitions in a formula. (Contributed by NM, 4-Jun-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜓𝜃)    &   (𝜒𝜏)       (𝜑 → (𝜃𝜏))
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