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Theorem List for Metamath Proof Explorer - 201-300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmpbi 201 An inference from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)
 |-  ph   &    |-  ( ph  <->  ps )   =>    |- 
 ps
 
Theoremmpbir 202 An inference from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)
 |- 
 ps   &    |-  ( ph  <->  ps )   =>    |-  ph
 
Theoremmpbid 203 A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ch )
 
Theoremmpbii 204 An inference from a nested biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
 |- 
 ps   &    |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ch )
 
Theoremsylibr 205 A mixed syllogism inference from an implication and a biconditional. Useful for substituting a consequent with a definition. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   &    |-  ( ch 
 <->  ps )   =>    |-  ( ph  ->  ch )
 
Theoremsylbir 206 A mixed syllogism inference from a biconditional and an implication. (Contributed by NM, 5-Aug-1993.)
 |-  ( ps  <->  ph )   &    |-  ( ps  ->  ch )   =>    |-  ( ph  ->  ch )
 
Theoremsylibd 207 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ch  <->  th ) )   =>    |-  ( ph  ->  ( ps  ->  th )
 )
 
Theoremsylbid 208 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( ch  ->  th ) )   =>    |-  ( ph  ->  ( ps  ->  th ) )
 
Theoremmpbidi 209 A deduction from a biconditional, related to modus ponens. (Contributed by NM, 9-Aug-1994.)
 |-  ( th  ->  ( ph  ->  ps ) )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( th  ->  (
 ph  ->  ch ) )
 
Theoremsyl5bi 210 A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded antecedent with a definition. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ch  ->  ( ps  ->  th )
 )   =>    |-  ( ch  ->  ( ph  ->  th ) )
 
Theoremsyl5bir 211 A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.)
 |-  ( ps  <->  ph )   &    |-  ( ch  ->  ( ps  ->  th )
 )   =>    |-  ( ch  ->  ( ph  ->  th ) )
 
Theoremsyl5ib 212 A mixed syllogism inference. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  ( ps  <->  th ) )   =>    |-  ( ch  ->  (
 ph  ->  th ) )
 
Theoremsyl5ibcom 213 A mixed syllogism inference. (Contributed by NM, 19-Jun-2007.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  ( ps  <->  th ) )   =>    |-  ( ph  ->  ( ch  ->  th )
 )
 
Theoremsyl5ibr 214 A mixed syllogism inference. (Contributed by NM, 3-Apr-1994.)
 |-  ( ph  ->  th )   &    |-  ( ch  ->  ( ps  <->  th ) )   =>    |-  ( ch  ->  (
 ph  ->  ps ) )
 
Theoremsyl5ibrcom 215 A mixed syllogism inference. (Contributed by NM, 20-Jun-2007.)
 |-  ( ph  ->  th )   &    |-  ( ch  ->  ( ps  <->  th ) )   =>    |-  ( ph  ->  ( ch  ->  ps )
 )
 
Theorembiimprd 216 Deduce a converse implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ch  ->  ps )
 )
 
Theorembiimpcd 217 Deduce a commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ps  ->  (
 ph  ->  ch ) )
 
Theorembiimprcd 218 Deduce a converse commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ch  ->  (
 ph  ->  ps ) )
 
Theoremsyl6ib 219 A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ch 
 <-> 
 th )   =>    |-  ( ph  ->  ( ps  ->  th ) )
 
Theoremsyl6ibr 220 A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded consequent with a definition. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th 
 <->  ch )   =>    |-  ( ph  ->  ( ps  ->  th ) )
 
Theoremsyl6bi 221 A mixed syllogism inference. (Contributed by NM, 2-Jan-1994.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ch  ->  th )   =>    |-  ( ph  ->  ( ps  ->  th ) )
 
Theoremsyl6bir 222 A mixed syllogism inference. (Contributed by NM, 18-May-1994.)
 |-  ( ph  ->  ( ch 
 <->  ps ) )   &    |-  ( ch  ->  th )   =>    |-  ( ph  ->  ( ps  ->  th ) )
 
Theoremsyl7bi 223 A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ch  ->  ( th  ->  ( ps  ->  ta ) ) )   =>    |-  ( ch  ->  ( th  ->  ( ph  ->  ta )
 ) )
 
Theoremsyl8ib 224 A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 1-Aug-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   &    |-  ( th 
 <->  ta )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )
 
Theoremmpbird 225 A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ch )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ps )
 
Theoremmpbiri 226 An inference from a nested biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
 |- 
 ch   &    |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ps )
 
Theoremsylibrd 227 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( th  <->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  th )
 )
 
Theoremsylbird 228 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
 |-  ( ph  ->  ( ch 
 <->  ps ) )   &    |-  ( ph  ->  ( ch  ->  th ) )   =>    |-  ( ph  ->  ( ps  ->  th ) )
 
Theorembiid 229 Principle of identity for logical equivalence. Theorem *4.2 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ph )
 
Theorembiidd 230 Principle of identity with antecedent. (Contributed by NM, 25-Nov-1995.)
 |-  ( ph  ->  ( ps 
 <->  ps ) )
 
Theorempm5.1im 231 Two propositions are equivalent if they are both true. Closed form of 2th 232. Equivalent to a bi1 180-like version of the xor-connective. This theorem stays true, no matter how you permute its operands. This is evident from its sharper version  ( ph  <->  ( ps  <->  (
ph 
<->  ps ) ) ). (Contributed by Wolf Lammen, 12-May-2013.)
 |-  ( ph  ->  ( ps  ->  ( ph  <->  ps ) ) )
 
Theorem2th 232 Two truths are equivalent. (Contributed by NM, 18-Aug-1993.)
 |-  ph   &    |- 
 ps   =>    |-  ( ph  <->  ps )
 
Theorem2thd 233 Two truths are equivalent (deduction rule). (Contributed by NM, 3-Jun-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   =>    |-  ( ph  ->  ( ps 
 <->  ch ) )
 
Theoremibi 234 Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 17-Oct-2003.)
 |-  ( ph  ->  ( ph 
 <->  ps ) )   =>    |-  ( ph  ->  ps )
 
Theoremibir 235 Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 22-Jul-2004.)
 |-  ( ph  ->  ( ps 
 <-> 
 ph ) )   =>    |-  ( ph  ->  ps )
 
Theoremibd 236 Deduction that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 26-Jun-2004.)
 |-  ( ph  ->  ( ps  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  ( ps  ->  ch ) )
 
Theorempm5.74 237 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.)
 |-  ( ( ph  ->  ( ps  <->  ch ) )  <->  ( ( ph  ->  ps )  <->  ( ph  ->  ch ) ) )
 
Theorempm5.74i 238 Distribution of implication over biconditional (inference rule). (Contributed by NM, 1-Aug-1994.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ( ph  ->  ps )  <->  ( ph  ->  ch ) )
 
Theorempm5.74ri 239 Distribution of implication over biconditional (reverse inference rule). (Contributed by NM, 1-Aug-1994.)
 |-  ( ( ph  ->  ps )  <->  ( ph  ->  ch ) )   =>    |-  ( ph  ->  ( ps 
 <->  ch ) )
 
Theorempm5.74d 240 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 21-Mar-1996.)
 |-  ( ph  ->  ( ps  ->  ( ch  <->  th ) ) )   =>    |-  ( ph  ->  ( ( ps  ->  ch )  <->  ( ps  ->  th ) ) )
 
Theorempm5.74rd 241 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 19-Mar-1997.)
 |-  ( ph  ->  (
 ( ps  ->  ch )  <->  ( ps  ->  th )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  <->  th ) ) )
 
Theorembitri 242 An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.)
 |-  ( ph  <->  ps )   &    |-  ( ps  <->  ch )   =>    |-  ( ph  <->  ch )
 
Theorembitr2i 243 An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ps  <->  ch )   =>    |-  ( ch  <->  ph )
 
Theorembitr3i 244 An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ps  <->  ph )   &    |-  ( ps  <->  ch )   =>    |-  ( ph  <->  ch )
 
Theorembitr4i 245 An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  ps )   =>    |-  ( ph  <->  ch )
 
Theorembitrd 246 Deduction form of bitri 242. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Apr-2013.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( ch  <->  th ) )   =>    |-  ( ph  ->  ( ps  <->  th ) )
 
Theorembitr2d 247 Deduction form of bitr2i 243. (Contributed by NM, 9-Jun-2004.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( ch  <->  th ) )   =>    |-  ( ph  ->  ( th  <->  ps ) )
 
Theorembitr3d 248 Deduction form of bitr3i 244. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( ps  <->  th ) )   =>    |-  ( ph  ->  ( ch  <->  th ) )
 
Theorembitr4d 249 Deduction form of bitr4i 245. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ch ) )   =>    |-  ( ph  ->  ( ps  <->  th ) )
 
Theoremsyl5bb 250 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ch  ->  ( ps  <->  th ) )   =>    |-  ( ch  ->  (
 ph 
 <-> 
 th ) )
 
Theoremsyl5rbb 251 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ch  ->  ( ps  <->  th ) )   =>    |-  ( ch  ->  ( th  <->  ph ) )
 
Theoremsyl5bbr 252 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
 |-  ( ps  <->  ph )   &    |-  ( ch  ->  ( ps  <->  th ) )   =>    |-  ( ch  ->  (
 ph 
 <-> 
 th ) )
 
Theoremsyl5rbbr 253 A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)
 |-  ( ps  <->  ph )   &    |-  ( ch  ->  ( ps  <->  th ) )   =>    |-  ( ch  ->  ( th  <->  ph ) )
 
Theoremsyl6bb 254 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ch 
 <-> 
 th )   =>    |-  ( ph  ->  ( ps 
 <-> 
 th ) )
 
Theoremsyl6rbb 255 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ch 
 <-> 
 th )   =>    |-  ( ph  ->  ( th 
 <->  ps ) )
 
Theoremsyl6bbr 256 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( th 
 <->  ch )   =>    |-  ( ph  ->  ( ps 
 <-> 
 th ) )
 
Theoremsyl6rbbr 257 A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( th 
 <->  ch )   =>    |-  ( ph  ->  ( th 
 <->  ps ) )
 
Theorem3imtr3i 258 A mixed syllogism inference, useful for removing a definition from both sides of an implication. (Contributed by NM, 10-Aug-1994.)
 |-  ( ph  ->  ps )   &    |-  ( ph 
 <->  ch )   &    |-  ( ps  <->  th )   =>    |-  ( ch  ->  th )
 
Theorem3imtr4i 259 A mixed syllogism inference, useful for applying a definition to both sides of an implication. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   &    |-  ( ch 
 <-> 
 ph )   &    |-  ( th  <->  ps )   =>    |-  ( ch  ->  th )
 
Theorem3imtr3d 260 More general version of 3imtr3i 258. Useful for converting conditional definitions in a formula. (Contributed by NM, 8-Apr-1996.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  <->  th ) )   &    |-  ( ph  ->  ( ch  <->  ta ) )   =>    |-  ( ph  ->  ( th  ->  ta )
 )
 
Theorem3imtr4d 261 More general version of 3imtr4i 259. Useful for converting conditional definitions in a formula. (Contributed by NM, 26-Oct-1995.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( th  <->  ps ) )   &    |-  ( ph  ->  ( ta  <->  ch ) )   =>    |-  ( ph  ->  ( th  ->  ta )
 )
 
Theorem3imtr3g 262 More general version of 3imtr3i 258. Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ps 
 <-> 
 th )   &    |-  ( ch  <->  ta )   =>    |-  ( ph  ->  ( th  ->  ta ) )
 
Theorem3imtr4g 263 More general version of 3imtr4i 259. Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th 
 <->  ps )   &    |-  ( ta  <->  ch )   =>    |-  ( ph  ->  ( th  ->  ta ) )
 
Theorem3bitri 264 A chained inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ps  <->  ch )   &    |-  ( ch  <->  th )   =>    |-  ( ph  <->  th )
 
Theorem3bitrri 265 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  <->  ps )   &    |-  ( ps  <->  ch )   &    |-  ( ch  <->  th )   =>    |-  ( th  <->  ph )
 
Theorem3bitr2i 266 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ph  <->  th )
 
Theorem3bitr2ri 267 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( th  <->  ph )
 
Theorem3bitr3i 268 A chained inference from transitive law for logical equivalence. (Contributed by NM, 19-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ph  <->  ch )   &    |-  ( ps  <->  th )   =>    |-  ( ch  <->  th )
 
Theorem3bitr3ri 269 A chained inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ph  <->  ch )   &    |-  ( ps  <->  th )   =>    |-  ( th  <->  ch )
 
Theorem3bitr4i 270 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, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  ph )   &    |-  ( th  <->  ps )   =>    |-  ( ch  <->  th )
 
Theorem3bitr4ri 271 A chained inference from transitive law for logical equivalence. (Contributed by NM, 2-Sep-1995.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  ph )   &    |-  ( th  <->  ps )   =>    |-  ( th  <->  ch )
 
Theorem3bitrd 272 Deduction from transitivity of biconditional. (Contributed by NM, 13-Aug-1999.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( ch  <->  th ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ps  <->  ta ) )
 
Theorem3bitrrd 273 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( ch  <->  th ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ta  <->  ps ) )
 
Theorem3bitr2d 274 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ps  <->  ta ) )
 
Theorem3bitr2rd 275 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ta  <->  ps ) )
 
Theorem3bitr3d 276 Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 24-Apr-1996.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( ps  <->  th ) )   &    |-  ( ph  ->  ( ch  <->  ta ) )   =>    |-  ( ph  ->  ( th  <->  ta ) )
 
Theorem3bitr3rd 277 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( ps  <->  th ) )   &    |-  ( ph  ->  ( ch  <->  ta ) )   =>    |-  ( ph  ->  ( ta  <->  th ) )
 
Theorem3bitr4d 278 Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 18-Oct-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ps ) )   &    |-  ( ph  ->  ( ta  <->  ch ) )   =>    |-  ( ph  ->  ( th  <->  ta ) )
 
Theorem3bitr4rd 279 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ps ) )   &    |-  ( ph  ->  ( ta  <->  ch ) )   =>    |-  ( ph  ->  ( ta  <->  th ) )
 
Theorem3bitr3g 280 More general version of 3bitr3i 268. Useful for converting definitions in a formula. (Contributed by NM, 4-Jun-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ps 
 <-> 
 th )   &    |-  ( ch  <->  ta )   =>    |-  ( ph  ->  ( th 
 <->  ta ) )
 
Theorem3bitr4g 281 More general version of 3bitr4i 270. Useful for converting definitions in a formula. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( th 
 <->  ps )   &    |-  ( ta  <->  ch )   =>    |-  ( ph  ->  ( th 
 <->  ta ) )
 
Theorembi3ant 282 Construct a bi-conditional in antecedent position. (Contributed by Wolf Lammen, 14-May-2013.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ( ( th  ->  ta )  -> 
 ph )  ->  (
 ( ( ta  ->  th )  ->  ps )  ->  ( ( th  <->  ta )  ->  ch )
 ) )
 
Theorembisym 283 Express symmetries of theorems in terms of biconditionals. (Contributed by Wolf Lammen, 14-May-2013.)
 |-  ( ( ( ph  ->  ps )  ->  ( ch  ->  th ) )  ->  ( ( ( ps 
 ->  ph )  ->  ( th  ->  ch ) )  ->  ( ( ph  <->  ps )  ->  ( ch 
 <-> 
 th ) ) ) )
 
Theoremnotnot 284 Double negation. Theorem *4.13 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  -.  -.  ph )
 
Theoremcon34b 285 Contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( ph  ->  ps )  <->  ( -.  ps  ->  -.  ph ) )
 
Theoremcon4bid 286 A contraposition deduction. (Contributed by NM, 21-May-1994.)
 |-  ( ph  ->  ( -.  ps  <->  -.  ch ) )   =>    |-  ( ph  ->  ( ps  <->  ch ) )
 
Theoremnotbid 287 Deduction negating both sides of a logical equivalence. (Contributed by NM, 21-May-1994.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( -.  ps  <->  -.  ch ) )
 
Theoremnotbi 288 Contraposition. Theorem *4.11 of [WhiteheadRussell] p. 117. (Contributed by NM, 21-May-1994.) (Proof shortened by Wolf Lammen, 12-Jun-2013.)
 |-  ( ( ph  <->  ps )  <->  ( -.  ph  <->  -.  ps ) )
 
Theoremnotbii 289 Negate both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.)
 |-  ( ph  <->  ps )   =>    |-  ( -.  ph  <->  -.  ps )
 
Theoremcon4bii 290 A contraposition inference. (Contributed by NM, 21-May-1994.)
 |-  ( -.  ph  <->  -.  ps )   =>    |-  ( ph  <->  ps )
 
Theoremmtbi 291 An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
 |- 
 -.  ph   &    |-  ( ph  <->  ps )   =>    |- 
 -.  ps
 
Theoremmtbir 292 An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 14-Oct-2012.)
 |- 
 -.  ps   &    |-  ( ph  <->  ps )   =>    |- 
 -.  ph
 
Theoremmtbid 293 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.)
 |-  ( ph  ->  -.  ps )   &    |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  -. 
 ch )
 
Theoremmtbird 294 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  -. 
 ps )
 
Theoremmtbii 295 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.)
 |- 
 -.  ps   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  -. 
 ch )
 
Theoremmtbiri 296 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.)
 |- 
 -.  ch   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  -. 
 ps )
 
Theoremsylnib 297 A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.)
 |-  ( ph  ->  -.  ps )   &    |-  ( ps  <->  ch )   =>    |-  ( ph  ->  -.  ch )
 
Theoremsylnibr 298 A mixed syllogism inference from an implication and a biconditional. Useful for substituting an consequent with a definition. (Contributed by Wolf Lammen, 16-Dec-2013.)
 |-  ( ph  ->  -.  ps )   &    |-  ( ch  <->  ps )   =>    |-  ( ph  ->  -.  ch )
 
Theoremsylnbi 299 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.)
 |-  ( ph  <->  ps )   &    |-  ( -.  ps  ->  ch )   =>    |-  ( -.  ph  ->  ch )
 
Theoremsylnbir 300 A mixed syllogism inference from a biconditional and an implication. (Contributed by Wolf Lammen, 16-Dec-2013.)
 |-  ( ps  <->  ph )   &    |-  ( -.  ps  ->  ch )   =>    |-  ( -.  ph  ->  ch )
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