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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.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜒𝜏))       (𝜑 → (𝜃𝜏))

Theorem3imtr4d 202 More general version of 3imtr4i 200. Useful for converting conditional definitions in a formula. (Contributed by NM, 26-Oct-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜓))    &   (𝜑 → (𝜏𝜒))       (𝜑 → (𝜃𝜏))

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.)
(𝜑 → (𝜓𝜒))    &   (𝜓𝜃)    &   (𝜒𝜏)       (𝜑 → (𝜃𝜏))

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.)
(𝜑 → (𝜓𝜒))    &   (𝜃𝜓)    &   (𝜏𝜒)       (𝜑 → (𝜃𝜏))

Theorem3bitri 205 A chained inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)    &   (𝜓𝜒)    &   (𝜒𝜃)       (𝜑𝜃)

Theorem3bitrri 206 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
(𝜑𝜓)    &   (𝜓𝜒)    &   (𝜒𝜃)       (𝜃𝜑)

Theorem3bitr2i 207 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
(𝜑𝜓)    &   (𝜒𝜓)    &   (𝜒𝜃)       (𝜑𝜃)

Theorem3bitr2ri 208 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
(𝜑𝜓)    &   (𝜒𝜓)    &   (𝜒𝜃)       (𝜃𝜑)

Theorem3bitr3i 209 A chained inference from transitive law for logical equivalence. (Contributed by NM, 19-Aug-1993.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜓𝜃)       (𝜒𝜃)

Theorem3bitr3ri 210 A chained inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜓𝜃)       (𝜃𝜒)

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

Theorem3bitr4ri 212 A chained inference from transitive law for logical equivalence. (Contributed by NM, 2-Sep-1995.)
(𝜑𝜓)    &   (𝜒𝜑)    &   (𝜃𝜓)       (𝜃𝜒)

Theorem3bitrd 213 Deduction from transitivity of biconditional. (Contributed by NM, 13-Aug-1999.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))    &   (𝜑 → (𝜃𝜏))       (𝜑 → (𝜓𝜏))

Theorem3bitrrd 214 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))    &   (𝜑 → (𝜃𝜏))       (𝜑 → (𝜏𝜓))

Theorem3bitr2d 215 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → (𝜓𝜏))

Theorem3bitr2rd 216 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → (𝜏𝜓))

Theorem3bitr3d 217 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 218 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜒𝜏))       (𝜑 → (𝜏𝜃))

Theorem3bitr4d 219 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 220 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜓))    &   (𝜑 → (𝜏𝜒))       (𝜑 → (𝜏𝜃))

Theorem3bitr3g 221 More general version of 3bitr3i 209. Useful for converting definitions in a formula. (Contributed by NM, 4-Jun-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜓𝜃)    &   (𝜒𝜏)       (𝜑 → (𝜃𝜏))

Theorem3bitr4g 222 More general version of 3bitr4i 211. Useful for converting definitions in a formula. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜃𝜓)    &   (𝜏𝜒)       (𝜑 → (𝜃𝜏))

Theorembi3ant 223 Construct a biconditional in antecedent position. (Contributed by Wolf Lammen, 14-May-2013.)
(𝜑 → (𝜓𝜒))       (((𝜃𝜏) → 𝜑) → (((𝜏𝜃) → 𝜓) → ((𝜃𝜏) → 𝜒)))

Theorembisym 224 Express symmetries of theorems in terms of biconditionals. (Contributed by Wolf Lammen, 14-May-2013.)
(((𝜑𝜓) → (𝜒𝜃)) → (((𝜓𝜑) → (𝜃𝜒)) → ((𝜑𝜓) → (𝜒𝜃))))

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.)
(𝜑𝜓)       ((𝜒𝜑) ↔ (𝜒𝜓))

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.)
(𝜑𝜓)       ((𝜒𝜑) ↔ (𝜒𝜓))

Theorembibi1i 227 Inference adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)       ((𝜑𝜒) ↔ (𝜓𝜒))

Theorembibi12i 228 The equivalence of two equivalences. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))

Theoremimbi2d 229 Deduction adding an antecedent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) ↔ (𝜃𝜒)))

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.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))

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.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) ↔ (𝜃𝜒)))

Theorembibi1d 232 Deduction adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))

Theoremimbi12d 233 Deduction joining two equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))

Theorembibi12d 234 Deduction joining two equivalences to form equivalence of biconditionals. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))

Theoremimbi1 235 Theorem *4.84 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))

Theoremimbi2 236 Theorem *4.85 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.)
((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))

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.)
(𝜑𝜓)       ((𝜑𝜒) ↔ (𝜓𝜒))

Theoremimbi12i 238 Join two logical equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))

Theorembibi1 239 Theorem *4.86 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))

Theorembiimt 240 A wff is equivalent to itself with true antecedent. (Contributed by NM, 28-Jan-1996.)
(𝜑 → (𝜓 ↔ (𝜑𝜓)))

Theorempm5.5 241 Theorem *5.5 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
(𝜑 → ((𝜑𝜓) ↔ 𝜓))

Theorema1bi 242 Inference introducing a theorem as an antecedent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)
𝜑       (𝜓 ↔ (𝜑𝜓))

Theorempm5.501 243 Theorem *5.501 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 24-Jan-2013.)
(𝜑 → (𝜓 ↔ (𝜑𝜓)))

Theoremibib 244 Implication in terms of implication and biconditional. (Contributed by NM, 31-Mar-1994.) (Proof shortened by Wolf Lammen, 24-Jan-2013.)
((𝜑𝜓) ↔ (𝜑 → (𝜑𝜓)))

Theoremibibr 245 Implication in terms of implication and biconditional. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 21-Dec-2013.)
((𝜑𝜓) ↔ (𝜑 → (𝜓𝜑)))

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.)
𝜑       (𝜓 ↔ (𝜓𝜑))

Theorembi2.04 247 Logical equivalence of commuted antecedents. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 5-Aug-1993.)
((𝜑 → (𝜓𝜒)) ↔ (𝜓 → (𝜑𝜒)))

Theorempm5.4 248 Antecedent absorption implication. Theorem *5.4 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
((𝜑 → (𝜑𝜓)) ↔ (𝜑𝜓))

Theoremimdi 249 Distributive law for implication. Compare Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒)))

Theorempm5.41 250 Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 12-Oct-2012.)
(((𝜑𝜓) → (𝜑𝜒)) ↔ (𝜑 → (𝜓𝜒)))

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.)
((𝜓𝜑) → (((𝜑𝜒) → (𝜓𝜃)) ↔ (𝜓 → (𝜒𝜃))))

Theoremimpd 252 Importation deduction. (Contributed by NM, 31-Mar-1994.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → ((𝜓𝜒) → 𝜃))

Theoremimp31 253 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (((𝜑𝜓) ∧ 𝜒) → 𝜃)

Theoremimp32 254 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒𝜃)))       ((𝜑 ∧ (𝜓𝜒)) → 𝜃)

Theoremexpd 255 Exportation deduction. (Contributed by NM, 20-Aug-1993.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑 → (𝜓 → (𝜒𝜃)))

Theoremexpdimp 256 A deduction version of exportation, followed by importation. (Contributed by NM, 6-Sep-2008.)
(𝜑 → ((𝜓𝜒) → 𝜃))       ((𝜑𝜓) → (𝜒𝜃))

Theoremimpancom 257 Mixed importation/commutation inference. (Contributed by NM, 22-Jun-2013.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑𝜒) → (𝜓𝜃))

Theorempm3.3 258 Theorem *3.3 (Exp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
(((𝜑𝜓) → 𝜒) → (𝜑 → (𝜓𝜒)))

Theorempm3.31 259 Theorem *3.31 (Imp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → 𝜒))

Theoremimpexp 260 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.)
(((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒)))

Theorempm3.21 261 Join antecedents with conjunction. Theorem *3.21 of [WhiteheadRussell] p. 111. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓 → (𝜓𝜑)))

Theorempm3.22 262 Theorem *3.22 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Nov-2012.)
((𝜑𝜓) → (𝜓𝜑))

Theoremancom 263 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.)
((𝜑𝜓) ↔ (𝜓𝜑))

Theoremancomd 264 Commutation of conjuncts in consequent. (Contributed by Jeff Hankins, 14-Aug-2009.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜒𝜓))

Theoremancoms 265 Inference commuting conjunction in antecedent. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓) → 𝜒)       ((𝜓𝜑) → 𝜒)

Theoremancomsd 266 Deduction commuting conjunction in antecedent. (Contributed by NM, 12-Dec-2004.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑 → ((𝜒𝜓) → 𝜃))

Theorempm3.2i 267 Infer conjunction of premises. (Contributed by NM, 5-Aug-1993.)
𝜑    &   𝜓       (𝜑𝜓)

Theorempm3.43i 268 Nested conjunction of antecedents. (Contributed by NM, 5-Aug-1993.)
((𝜑𝜓) → ((𝜑𝜒) → (𝜑 → (𝜓𝜒))))

Theoremsimplbi 269 Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
(𝜑 ↔ (𝜓𝜒))       (𝜑𝜓)

Theoremsimprbi 270 Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
(𝜑 ↔ (𝜓𝜒))       (𝜑𝜒)

Theoremadantr 271 Inference adding a conjunct to the right of an antecedent. (Contributed by NM, 30-Aug-1993.)
(𝜑𝜓)       ((𝜑𝜒) → 𝜓)

Theoremadantl 272 Inference adding a conjunct to the left of an antecedent. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)
(𝜑𝜓)       ((𝜒𝜑) → 𝜓)

Theoremadantld 273 Deduction adding a conjunct to the left of an antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2012.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) → 𝜒))

Theoremadantrd 274 Deduction adding a conjunct to the right of an antecedent. (Contributed by NM, 4-May-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) → 𝜒))

Theoremimpel 275 An inference for implication elimination. (Contributed by Giovanni Mascellani, 23-May-2019.) (Proof shortened by Wolf Lammen, 2-Sep-2020.)
(𝜑 → (𝜓𝜒))    &   (𝜃𝜓)       ((𝜑𝜃) → 𝜒)

Theoremmpan9 276 Modus ponens conjoining dissimilar antecedents. (Contributed by NM, 1-Feb-2008.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(𝜑𝜓)    &   (𝜒 → (𝜓𝜃))       ((𝜑𝜒) → 𝜃)

Theoremsyldan 277 A syllogism deduction with conjoined antecents. (Contributed by NM, 24-Feb-2005.) (Proof shortened by Wolf Lammen, 6-Apr-2013.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)

Theoremsylan 278 A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)
(𝜑𝜓)    &   ((𝜓𝜒) → 𝜃)       ((𝜑𝜒) → 𝜃)

Theoremsylanb 279 A syllogism inference. (Contributed by NM, 18-May-1994.)
(𝜑𝜓)    &   ((𝜓𝜒) → 𝜃)       ((𝜑𝜒) → 𝜃)

Theoremsylanbr 280 A syllogism inference. (Contributed by NM, 18-May-1994.)
(𝜓𝜑)    &   ((𝜓𝜒) → 𝜃)       ((𝜑𝜒) → 𝜃)

Theoremsylan2 281 A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)
(𝜑𝜒)    &   ((𝜓𝜒) → 𝜃)       ((𝜓𝜑) → 𝜃)

Theoremsylan2b 282 A syllogism inference. (Contributed by NM, 21-Apr-1994.)
(𝜑𝜒)    &   ((𝜓𝜒) → 𝜃)       ((𝜓𝜑) → 𝜃)

Theoremsylan2br 283 A syllogism inference. (Contributed by NM, 21-Apr-1994.)
(𝜒𝜑)    &   ((𝜓𝜒) → 𝜃)       ((𝜓𝜑) → 𝜃)

Theoremsyl2an 284 A double syllogism inference. (Contributed by NM, 31-Jan-1997.)
(𝜑𝜓)    &   (𝜏𝜒)    &   ((𝜓𝜒) → 𝜃)       ((𝜑𝜏) → 𝜃)

Theoremsyl2anr 285 A double syllogism inference. (Contributed by NM, 17-Sep-2013.)
(𝜑𝜓)    &   (𝜏𝜒)    &   ((𝜓𝜒) → 𝜃)       ((𝜏𝜑) → 𝜃)

Theoremsyl2anb 286 A double syllogism inference. (Contributed by NM, 29-Jul-1999.)
(𝜑𝜓)    &   (𝜏𝜒)    &   ((𝜓𝜒) → 𝜃)       ((𝜑𝜏) → 𝜃)

Theoremsyl2anbr 287 A double syllogism inference. (Contributed by NM, 29-Jul-1999.)
(𝜓𝜑)    &   (𝜒𝜏)    &   ((𝜓𝜒) → 𝜃)       ((𝜑𝜏) → 𝜃)

Theoremsyland 288 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → ((𝜒𝜃) → 𝜏))       (𝜑 → ((𝜓𝜃) → 𝜏))

Theoremsylan2d 289 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → ((𝜃𝜒) → 𝜏))       (𝜑 → ((𝜃𝜓) → 𝜏))

Theoremsyl2and 290 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))    &   (𝜑 → ((𝜒𝜏) → 𝜂))       (𝜑 → ((𝜓𝜃) → 𝜂))

Theorembiimpa 291 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) → 𝜒)

Theorembiimpar 292 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜒) → 𝜓)

Theorembiimpac 293 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))       ((𝜓𝜑) → 𝜒)

Theorembiimparc 294 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))       ((𝜒𝜑) → 𝜓)

Theoremiba 295 Introduction of antecedent as conjunct. Theorem *4.73 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Mar-1994.) (Revised by NM, 24-Mar-2013.)
(𝜑 → (𝜓 ↔ (𝜓𝜑)))

Theoremibar 296 Introduction of antecedent as conjunct. (Contributed by NM, 5-Dec-1995.) (Revised by NM, 24-Mar-2013.)
(𝜑 → (𝜓 ↔ (𝜑𝜓)))

Theorembiantru 297 A wff is equivalent to its conjunction with truth. (Contributed by NM, 5-Aug-1993.)
𝜑       (𝜓 ↔ (𝜓𝜑))

Theorembiantrur 298 A wff is equivalent to its conjunction with truth. (Contributed by NM, 3-Aug-1994.)
𝜑       (𝜓 ↔ (𝜑𝜓))

Theorembiantrud 299 A wff is equivalent to its conjunction with truth. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Wolf Lammen, 23-Oct-2013.)
(𝜑𝜓)       (𝜑 → (𝜒 ↔ (𝜒𝜓)))

Theorembiantrurd 300 A wff is equivalent to its conjunction with truth. (Contributed by NM, 1-May-1995.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(𝜑𝜓)       (𝜑 → (𝜒 ↔ (𝜓𝜒)))

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