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Theorem List for Intuitionistic Logic Explorer - 201-300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem3imtr3d 201 More general version of 3imtr3i 199. 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 202 More general version of 3imtr4i 200. 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 203 More general version of 3imtr3i 199. 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 204 More general version of 3imtr4i 200. 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 205 A chained inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ps  <->  ch )   &    |-  ( ch  <->  th )   =>    |-  ( ph  <->  th )
 
Theorem3bitrri 206 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  <->  ps )   &    |-  ( ps  <->  ch )   &    |-  ( ch  <->  th )   =>    |-  ( th  <->  ph )
 
Theorem3bitr2i 207 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ph  <->  th )
 
Theorem3bitr2ri 208 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( th  <->  ph )
 
Theorem3bitr3i 209 A chained inference from transitive law for logical equivalence. (Contributed by NM, 19-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ph  <->  ch )   &    |-  ( ps  <->  th )   =>    |-  ( ch  <->  th )
 
Theorem3bitr3ri 210 A chained inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ph  <->  ch )   &    |-  ( ps  <->  th )   =>    |-  ( th  <->  ch )
 
Theorem3bitr4i 211 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 212 A chained inference from transitive law for logical equivalence. (Contributed by NM, 2-Sep-1995.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  ph )   &    |-  ( th  <->  ps )   =>    |-  ( th  <->  ch )
 
Theorem3bitrd 213 Deduction from transitivity of biconditional. (Contributed by NM, 13-Aug-1999.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( ch  <->  th ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ps  <->  ta ) )
 
Theorem3bitrrd 214 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( ch  <->  th ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ta  <->  ps ) )
 
Theorem3bitr2d 215 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ps  <->  ta ) )
 
Theorem3bitr2rd 216 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ta  <->  ps ) )
 
Theorem3bitr3d 217 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 218 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( ps  <->  th ) )   &    |-  ( ph  ->  ( ch  <->  ta ) )   =>    |-  ( ph  ->  ( ta  <->  th ) )
 
Theorem3bitr4d 219 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 220 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ps ) )   &    |-  ( ph  ->  ( ta  <->  ch ) )   =>    |-  ( ph  ->  ( ta  <->  th ) )
 
Theorem3bitr3g 221 More general version of 3bitr3i 209. Useful for converting definitions in a formula. (Contributed by NM, 4-Jun-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ps 
 <-> 
 th )   &    |-  ( ch  <->  ta )   =>    |-  ( ph  ->  ( th 
 <->  ta ) )
 
Theorem3bitr4g 222 More general version of 3bitr4i 211. Useful for converting definitions in a formula. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( th 
 <->  ps )   &    |-  ( ta  <->  ch )   =>    |-  ( ph  ->  ( th 
 <->  ta ) )
 
Theorembi3ant 223 Construct a biconditional in antecedent position. (Contributed by Wolf Lammen, 14-May-2013.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ( ( th  ->  ta )  -> 
 ph )  ->  (
 ( ( ta  ->  th )  ->  ps )  ->  ( ( th  <->  ta )  ->  ch )
 ) )
 
Theorembisym 224 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 ) ) ) )
 
Theoremimbi2i 225 Introduce an antecedent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 6-Feb-2013.)
 |-  ( ph  <->  ps )   =>    |-  ( ( ch  ->  ph )  <->  ( ch  ->  ps ) )
 
Theorembibi2i 226 Inference adding a biconditional to the left in an equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 16-May-2013.)
 |-  ( ph  <->  ps )   =>    |-  ( ( ch  <->  ph )  <->  ( ch  <->  ps ) )
 
Theorembibi1i 227 Inference adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   =>    |-  ( ( ph  <->  ch )  <->  ( ps  <->  ch ) )
 
Theorembibi12i 228 The equivalence of two equivalences. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ( ph  <->  ch )  <->  ( ps  <->  th ) )
 
Theoremimbi2d 229 Deduction adding an antecedent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( th  ->  ps )  <->  ( th  ->  ch )
 ) )
 
Theoremimbi1d 230 Deduction adding a consequent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 17-Sep-2013.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( ps  ->  th )  <->  ( ch  ->  th )
 ) )
 
Theorembibi2d 231 Deduction adding a biconditional to the left in an equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( th  <->  ps )  <->  ( th  <->  ch ) ) )
 
Theorembibi1d 232 Deduction adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( ps  <->  th )  <->  ( ch  <->  th ) ) )
 
Theoremimbi12d 233 Deduction joining two equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ( ps  ->  th )  <->  ( ch  ->  ta )
 ) )
 
Theorembibi12d 234 Deduction joining two equivalences to form equivalence of biconditionals. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ( ps  <->  th )  <->  ( ch  <->  ta ) ) )
 
Theoremimbi1 235 Theorem *4.84 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  <->  ps )  ->  (
 ( ph  ->  ch )  <->  ( ps  ->  ch )
 ) )
 
Theoremimbi2 236 Theorem *4.85 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.)
 |-  ( ( ph  <->  ps )  ->  (
 ( ch  ->  ph )  <->  ( ch  ->  ps )
 ) )
 
Theoremimbi1i 237 Introduce a consequent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 17-Sep-2013.)
 |-  ( ph  <->  ps )   =>    |-  ( ( ph  ->  ch )  <->  ( ps  ->  ch ) )
 
Theoremimbi12i 238 Join two logical equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ( ph  ->  ch )  <->  ( ps  ->  th ) )
 
Theorembibi1 239 Theorem *4.86 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  <->  ps )  ->  (
 ( ph  <->  ch )  <->  ( ps  <->  ch ) ) )
 
Theorembiimt 240 A wff is equivalent to itself with true antecedent. (Contributed by NM, 28-Jan-1996.)
 |-  ( ph  ->  ( ps 
 <->  ( ph  ->  ps )
 ) )
 
Theorempm5.5 241 Theorem *5.5 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  ->  (
 ( ph  ->  ps )  <->  ps ) )
 
Theorema1bi 242 Inference introducing a theorem as an antecedent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)
 |-  ph   =>    |-  ( ps  <->  ( ph  ->  ps ) )
 
Theorempm5.501 243 Theorem *5.501 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 24-Jan-2013.)
 |-  ( ph  ->  ( ps 
 <->  ( ph  <->  ps ) ) )
 
Theoremibib 244 Implication in terms of implication and biconditional. (Contributed by NM, 31-Mar-1994.) (Proof shortened by Wolf Lammen, 24-Jan-2013.)
 |-  ( ( ph  ->  ps )  <->  ( ph  ->  (
 ph 
 <->  ps ) ) )
 
Theoremibibr 245 Implication in terms of implication and biconditional. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 21-Dec-2013.)
 |-  ( ( ph  ->  ps )  <->  ( ph  ->  ( ps  <->  ph ) ) )
 
Theoremtbt 246 A wff is equivalent to its equivalence with truth. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ph   =>    |-  ( ps  <->  ( ps  <->  ph ) )
 
Theorembi2.04 247 Logical equivalence of commuted antecedents. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 ) 
 <->  ( ps  ->  ( ph  ->  ch ) ) )
 
Theorempm5.4 248 Antecedent absorption implication. Theorem *5.4 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( ph  ->  (
 ph  ->  ps ) )  <->  ( ph  ->  ps ) )
 
Theoremimdi 249 Distributive law for implication. Compare Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 ) 
 <->  ( ( ph  ->  ps )  ->  ( ph  ->  ch ) ) )
 
Theorempm5.41 250 Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 12-Oct-2012.)
 |-  ( ( ( ph  ->  ps )  ->  ( ph  ->  ch ) )  <->  ( ph  ->  ( ps  ->  ch )
 ) )
 
Theoremimim21b 251 Simplify an implication between two implications when the antecedent of the first is a consequence of the antecedent of the second. The reverse form is useful in producing the successor step in induction proofs. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Wolf Lammen, 14-Sep-2013.)
 |-  ( ( ps  ->  ph )  ->  ( (
 ( ph  ->  ch )  ->  ( ps  ->  th )
 ) 
 <->  ( ps  ->  ( ch  ->  th ) ) ) )
 
Theoremimpd 252 Importation deduction. (Contributed by NM, 31-Mar-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   =>    |-  ( ph  ->  ( ( ps  /\  ch )  ->  th ) )
 
Theoremimpcomd 253 Importation deduction with commuted antecedents. (Contributed by Peter Mazsa, 24-Sep-2022.) (Proof shortened by Wolf Lammen, 22-Oct-2022.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   =>    |-  ( ph  ->  ( ( ch  /\  ps )  ->  th ) )
 
Theoremimp31 254 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   =>    |-  ( ( (
 ph  /\  ps )  /\  ch )  ->  th )
 
Theoremimp32 255 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   =>    |-  ( ( ph  /\  ( ps  /\  ch ) )  ->  th )
 
Theoremexpd 256 Exportation deduction. (Contributed by NM, 20-Aug-1993.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
 
Theoremexpdimp 257 A deduction version of exportation, followed by importation. (Contributed by NM, 6-Sep-2008.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ( ch  ->  th ) )
 
Theoremimpancom 258 Mixed importation/commutation inference. (Contributed by NM, 22-Jun-2013.)
 |-  ( ( ph  /\  ps )  ->  ( ch  ->  th ) )   =>    |-  ( ( ph  /\  ch )  ->  ( ps  ->  th ) )
 
Theorempm3.3 259 Theorem *3.3 (Exp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
 |-  ( ( ( ph  /\ 
 ps )  ->  ch )  ->  ( ph  ->  ( ps  ->  ch ) ) )
 
Theorempm3.31 260 Theorem *3.31 (Imp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 )  ->  ( ( ph  /\  ps )  ->  ch ) )
 
Theoremimpexp 261 Import-export theorem. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
 |-  ( ( ( ph  /\ 
 ps )  ->  ch )  <->  (
 ph  ->  ( ps  ->  ch ) ) )
 
Theorempm3.21 262 Join antecedents with conjunction. Theorem *3.21 of [WhiteheadRussell] p. 111. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps  ->  ( ps  /\  ph ) ) )
 
Theorempm3.22 263 Theorem *3.22 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Nov-2012.)
 |-  ( ( ph  /\  ps )  ->  ( ps  /\  ph ) )
 
Theoremancom 264 Commutative law for conjunction. Theorem *4.3 of [WhiteheadRussell] p. 118. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Wolf Lammen, 4-Nov-2012.)
 |-  ( ( ph  /\  ps ) 
 <->  ( ps  /\  ph )
 )
 
Theoremancomd 265 Commutation of conjuncts in consequent. (Contributed by Jeff Hankins, 14-Aug-2009.)
 |-  ( ph  ->  ( ps  /\  ch ) )   =>    |-  ( ph  ->  ( ch  /\ 
 ps ) )
 
Theoremancoms 266 Inference commuting conjunction in antecedent. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ps  /\  ph )  ->  ch )
 
Theoremancomsd 267 Deduction commuting conjunction in antecedent. (Contributed by NM, 12-Dec-2004.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   =>    |-  ( ph  ->  ( ( ch  /\  ps )  ->  th ) )
 
Theorembiancomi 268 Commuting conjunction in a biconditional. (Contributed by Peter Mazsa, 17-Jun-2018.)
 |-  ( ph  <->  ( ch  /\  ps ) )   =>    |-  ( ph  <->  ( ps  /\  ch ) )
 
Theorembiancomd 269 Commuting conjunction in a biconditional, deduction form. (Contributed by Peter Mazsa, 3-Oct-2018.)
 |-  ( ph  ->  ( ps 
 <->  ( th  /\  ch ) ) )   =>    |-  ( ph  ->  ( ps  <->  ( ch  /\  th ) ) )
 
Theorempm3.2i 270 Infer conjunction of premises. (Contributed by NM, 5-Aug-1993.)
 |-  ph   &    |- 
 ps   =>    |-  ( ph  /\  ps )
 
Theorempm3.43i 271 Nested conjunction of antecedents. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( ph  ->  ps )  ->  ( ( ph  ->  ch )  ->  ( ph  ->  ( ps  /\  ch ) ) ) )
 
Theoremsimplbi 272 Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
 |-  ( ph  <->  ( ps  /\  ch ) )   =>    |-  ( ph  ->  ps )
 
Theoremsimprbi 273 Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
 |-  ( ph  <->  ( ps  /\  ch ) )   =>    |-  ( ph  ->  ch )
 
Theoremadantr 274 Inference adding a conjunct to the right of an antecedent. (Contributed by NM, 30-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ph  /\  ch )  ->  ps )
 
Theoremadantl 275 Inference adding a conjunct to the left of an antecedent. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ch  /\  ph )  ->  ps )
 
Theoremadantld 276 Deduction adding a conjunct to the left of an antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2012.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ( th  /\  ps )  ->  ch ) )
 
Theoremadantrd 277 Deduction adding a conjunct to the right of an antecedent. (Contributed by NM, 4-May-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ( ps  /\  th )  ->  ch ) )
 
Theoremimpel 278 An inference for implication elimination. (Contributed by Giovanni Mascellani, 23-May-2019.) (Proof shortened by Wolf Lammen, 2-Sep-2020.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  ps )   =>    |-  ( ( ph  /\  th )  ->  ch )
 
Theoremmpan9 279 Modus ponens conjoining dissimilar antecedents. (Contributed by NM, 1-Feb-2008.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  ( ps  ->  th ) )   =>    |-  ( ( ph  /\  ch )  ->  th )
 
Theoremsyldan 280 A syllogism deduction with conjoined antecents. (Contributed by NM, 24-Feb-2005.) (Proof shortened by Wolf Lammen, 6-Apr-2013.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ps )  ->  th )
 
Theoremsylan 281 A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)
 |-  ( ph  ->  ps )   &    |-  (
 ( ps  /\  ch )  ->  th )   =>    |-  ( ( ph  /\  ch )  ->  th )
 
Theoremsylanb 282 A syllogism inference. (Contributed by NM, 18-May-1994.)
 |-  ( ph  <->  ps )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  (
 ( ph  /\  ch )  ->  th )
 
Theoremsylanbr 283 A syllogism inference. (Contributed by NM, 18-May-1994.)
 |-  ( ps  <->  ph )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  (
 ( ph  /\  ch )  ->  th )
 
Theoremsylan2 284 A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)
 |-  ( ph  ->  ch )   &    |-  (
 ( ps  /\  ch )  ->  th )   =>    |-  ( ( ps  /\  ph )  ->  th )
 
Theoremsylan2b 285 A syllogism inference. (Contributed by NM, 21-Apr-1994.)
 |-  ( ph  <->  ch )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  (
 ( ps  /\  ph )  ->  th )
 
Theoremsylan2br 286 A syllogism inference. (Contributed by NM, 21-Apr-1994.)
 |-  ( ch  <->  ph )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  (
 ( ps  /\  ph )  ->  th )
 
Theoremsyl2an 287 A double syllogism inference. (Contributed by NM, 31-Jan-1997.)
 |-  ( ph  ->  ps )   &    |-  ( ta  ->  ch )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  (
 ( ph  /\  ta )  ->  th )
 
Theoremsyl2anr 288 A double syllogism inference. (Contributed by NM, 17-Sep-2013.)
 |-  ( ph  ->  ps )   &    |-  ( ta  ->  ch )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  (
 ( ta  /\  ph )  ->  th )
 
Theoremsyl2anb 289 A double syllogism inference. (Contributed by NM, 29-Jul-1999.)
 |-  ( ph  <->  ps )   &    |-  ( ta  <->  ch )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  (
 ( ph  /\  ta )  ->  th )
 
Theoremsyl2anbr 290 A double syllogism inference. (Contributed by NM, 29-Jul-1999.)
 |-  ( ps  <->  ph )   &    |-  ( ch  <->  ta )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  (
 ( ph  /\  ta )  ->  th )
 
Theoremsyland 291 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ( ch 
 /\  th )  ->  ta )
 )   =>    |-  ( ph  ->  (
 ( ps  /\  th )  ->  ta ) )
 
Theoremsylan2d 292 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ( th  /\ 
 ch )  ->  ta )
 )   =>    |-  ( ph  ->  (
 ( th  /\  ps )  ->  ta ) )
 
Theoremsyl2and 293 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( th  ->  ta ) )   &    |-  ( ph  ->  ( ( ch  /\  ta )  ->  et ) )   =>    |-  ( ph  ->  ( ( ps  /\  th )  ->  et ) )
 
Theorembiimpa 294 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ch )
 
Theorembiimpar 295 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ( ph  /\ 
 ch )  ->  ps )
 
Theorembiimpac 296 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ( ps 
 /\  ph )  ->  ch )
 
Theorembiimparc 297 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ( ch 
 /\  ph )  ->  ps )
 
Theoremiba 298 Introduction of antecedent as conjunct. Theorem *4.73 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Mar-1994.) (Revised by NM, 24-Mar-2013.)
 |-  ( ph  ->  ( ps 
 <->  ( ps  /\  ph )
 ) )
 
Theoremibar 299 Introduction of antecedent as conjunct. (Contributed by NM, 5-Dec-1995.) (Revised by NM, 24-Mar-2013.)
 |-  ( ph  ->  ( ps 
 <->  ( ph  /\  ps ) ) )
 
Theorembiantru 300 A wff is equivalent to its conjunction with truth. (Contributed by NM, 5-Aug-1993.)
 |-  ph   =>    |-  ( ps  <->  ( ps  /\  ph ) )
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