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Theorem List for Intuitionistic Logic Explorer - 201-300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem3bitr2i 201 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
(𝜑𝜓)    &   (𝜒𝜓)    &   (𝜒𝜃)       (𝜑𝜃)
 
Theorem3bitr2ri 202 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
(𝜑𝜓)    &   (𝜒𝜓)    &   (𝜒𝜃)       (𝜃𝜑)
 
Theorem3bitr3i 203 A chained inference from transitive law for logical equivalence. (Contributed by NM, 19-Aug-1993.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜓𝜃)       (𝜒𝜃)
 
Theorem3bitr3ri 204 A chained inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜓𝜃)       (𝜃𝜒)
 
Theorem3bitr4i 205 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 206 A chained inference from transitive law for logical equivalence. (Contributed by NM, 2-Sep-1995.)
(𝜑𝜓)    &   (𝜒𝜑)    &   (𝜃𝜓)       (𝜃𝜒)
 
Theorem3bitrd 207 Deduction from transitivity of biconditional. (Contributed by NM, 13-Aug-1999.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))    &   (𝜑 → (𝜃𝜏))       (𝜑 → (𝜓𝜏))
 
Theorem3bitrrd 208 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))    &   (𝜑 → (𝜃𝜏))       (𝜑 → (𝜏𝜓))
 
Theorem3bitr2d 209 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → (𝜓𝜏))
 
Theorem3bitr2rd 210 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → (𝜏𝜓))
 
Theorem3bitr3d 211 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 212 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜒𝜏))       (𝜑 → (𝜏𝜃))
 
Theorem3bitr4d 213 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 214 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜓))    &   (𝜑 → (𝜏𝜒))       (𝜑 → (𝜏𝜃))
 
Theorem3bitr3g 215 More general version of 3bitr3i 203. Useful for converting definitions in a formula. (Contributed by NM, 4-Jun-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜓𝜃)    &   (𝜒𝜏)       (𝜑 → (𝜃𝜏))
 
Theorem3bitr4g 216 More general version of 3bitr4i 205. Useful for converting definitions in a formula. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜃𝜓)    &   (𝜏𝜒)       (𝜑 → (𝜃𝜏))
 
Theorembi3ant 217 Construct a biconditional in antecedent position. (Contributed by Wolf Lammen, 14-May-2013.)
(𝜑 → (𝜓𝜒))       (((𝜃𝜏) → 𝜑) → (((𝜏𝜃) → 𝜓) → ((𝜃𝜏) → 𝜒)))
 
Theorembisym 218 Express symmetries of theorems in terms of biconditionals. (Contributed by Wolf Lammen, 14-May-2013.)
(((𝜑𝜓) → (𝜒𝜃)) → (((𝜓𝜑) → (𝜃𝜒)) → ((𝜑𝜓) → (𝜒𝜃))))
 
Theoremimbi2i 219 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 220 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 221 Inference adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)       ((𝜑𝜒) ↔ (𝜓𝜒))
 
Theorembibi12i 222 The equivalence of two equivalences. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))
 
Theoremimbi2d 223 Deduction adding an antecedent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) ↔ (𝜃𝜒)))
 
Theoremimbi1d 224 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 225 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 226 Deduction adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))
 
Theoremimbi12d 227 Deduction joining two equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
 
Theorembibi12d 228 Deduction joining two equivalences to form equivalence of biconditionals. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
 
Theoremimbi1 229 Theorem *4.84 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))
 
Theoremimbi2 230 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 231 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 232 Join two logical equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))
 
Theorembibi1 233 Theorem *4.86 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))
 
Theorembiimt 234 A wff is equivalent to itself with true antecedent. (Contributed by NM, 28-Jan-1996.)
(𝜑 → (𝜓 ↔ (𝜑𝜓)))
 
Theorempm5.5 235 Theorem *5.5 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
(𝜑 → ((𝜑𝜓) ↔ 𝜓))
 
Theorema1bi 236 Inference rule introducing a theorem as an antecedent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)
𝜑       (𝜓 ↔ (𝜑𝜓))
 
Theorempm5.501 237 Theorem *5.501 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 24-Jan-2013.)
(𝜑 → (𝜓 ↔ (𝜑𝜓)))
 
Theoremibib 238 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 239 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 240 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))
 
Theorembi2.04 241 Logical equivalence of commuted antecedents. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 5-Aug-1993.)
((𝜑 → (𝜓𝜒)) ↔ (𝜓 → (𝜑𝜒)))
 
Theorempm5.4 242 Antecedent absorption implication. Theorem *5.4 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
((𝜑 → (𝜑𝜓)) ↔ (𝜑𝜓))
 
Theoremimdi 243 Distributive law for implication. Compare Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒)))
 
Theorempm5.41 244 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 245 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 246 Importation deduction. (Contributed by NM, 31-Mar-1994.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → ((𝜓𝜒) → 𝜃))
 
Theoremimp31 247 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (((𝜑𝜓) ∧ 𝜒) → 𝜃)
 
Theoremimp32 248 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒𝜃)))       ((𝜑 ∧ (𝜓𝜒)) → 𝜃)
 
Theoremexpd 249 Exportation deduction. (Contributed by NM, 20-Aug-1993.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑 → (𝜓 → (𝜒𝜃)))
 
Theoremexpdimp 250 A deduction version of exportation, followed by importation. (Contributed by NM, 6-Sep-2008.)
(𝜑 → ((𝜓𝜒) → 𝜃))       ((𝜑𝜓) → (𝜒𝜃))
 
Theoremimpancom 251 Mixed importation/commutation inference. (Contributed by NM, 22-Jun-2013.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑𝜒) → (𝜓𝜃))
 
Theorempm3.3 252 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)))
 
Theorempm3.31 253 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 254 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)))
 
Theorempm3.21 255 Join antecedents with conjunction. Theorem *3.21 of [WhiteheadRussell] p. 111. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓 → (𝜓𝜑)))
 
Theorempm3.22 256 Theorem *3.22 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Nov-2012.)
((𝜑𝜓) → (𝜓𝜑))
 
Theoremancom 257 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 258 Commutation of conjuncts in consequent. (Contributed by Jeff Hankins, 14-Aug-2009.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜒𝜓))
 
Theoremancoms 259 Inference commuting conjunction in antecedent. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓) → 𝜒)       ((𝜓𝜑) → 𝜒)
 
Theoremancomsd 260 Deduction commuting conjunction in antecedent. (Contributed by NM, 12-Dec-2004.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑 → ((𝜒𝜓) → 𝜃))
 
Theorempm3.2i 261 Infer conjunction of premises. (Contributed by NM, 5-Aug-1993.)
𝜑    &   𝜓       (𝜑𝜓)
 
Theorempm3.43i 262 Nested conjunction of antecedents. (Contributed by NM, 5-Aug-1993.)
((𝜑𝜓) → ((𝜑𝜒) → (𝜑 → (𝜓𝜒))))
 
Theoremsimplbi 263 Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
(𝜑 ↔ (𝜓𝜒))       (𝜑𝜓)
 
Theoremsimprbi 264 Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
(𝜑 ↔ (𝜓𝜒))       (𝜑𝜒)
 
Theoremadantr 265 Inference adding a conjunct to the right of an antecedent. (Contributed by NM, 30-Aug-1993.)
(𝜑𝜓)       ((𝜑𝜒) → 𝜓)
 
Theoremadantl 266 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 267 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 268 Deduction adding a conjunct to the right of an antecedent. (Contributed by NM, 4-May-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) → 𝜒))
 
Theoremmpan9 269 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 270 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 271 A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)
(𝜑𝜓)    &   ((𝜓𝜒) → 𝜃)       ((𝜑𝜒) → 𝜃)
 
Theoremsylanb 272 A syllogism inference. (Contributed by NM, 18-May-1994.)
(𝜑𝜓)    &   ((𝜓𝜒) → 𝜃)       ((𝜑𝜒) → 𝜃)
 
Theoremsylanbr 273 A syllogism inference. (Contributed by NM, 18-May-1994.)
(𝜓𝜑)    &   ((𝜓𝜒) → 𝜃)       ((𝜑𝜒) → 𝜃)
 
Theoremsylan2 274 A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)
(𝜑𝜒)    &   ((𝜓𝜒) → 𝜃)       ((𝜓𝜑) → 𝜃)
 
Theoremsylan2b 275 A syllogism inference. (Contributed by NM, 21-Apr-1994.)
(𝜑𝜒)    &   ((𝜓𝜒) → 𝜃)       ((𝜓𝜑) → 𝜃)
 
Theoremsylan2br 276 A syllogism inference. (Contributed by NM, 21-Apr-1994.)
(𝜒𝜑)    &   ((𝜓𝜒) → 𝜃)       ((𝜓𝜑) → 𝜃)
 
Theoremsyl2an 277 A double syllogism inference. (Contributed by NM, 31-Jan-1997.)
(𝜑𝜓)    &   (𝜏𝜒)    &   ((𝜓𝜒) → 𝜃)       ((𝜑𝜏) → 𝜃)
 
Theoremsyl2anr 278 A double syllogism inference. (Contributed by NM, 17-Sep-2013.)
(𝜑𝜓)    &   (𝜏𝜒)    &   ((𝜓𝜒) → 𝜃)       ((𝜏𝜑) → 𝜃)
 
Theoremsyl2anb 279 A double syllogism inference. (Contributed by NM, 29-Jul-1999.)
(𝜑𝜓)    &   (𝜏𝜒)    &   ((𝜓𝜒) → 𝜃)       ((𝜑𝜏) → 𝜃)
 
Theoremsyl2anbr 280 A double syllogism inference. (Contributed by NM, 29-Jul-1999.)
(𝜓𝜑)    &   (𝜒𝜏)    &   ((𝜓𝜒) → 𝜃)       ((𝜑𝜏) → 𝜃)
 
Theoremsyland 281 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → ((𝜒𝜃) → 𝜏))       (𝜑 → ((𝜓𝜃) → 𝜏))
 
Theoremsylan2d 282 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → ((𝜃𝜒) → 𝜏))       (𝜑 → ((𝜃𝜓) → 𝜏))
 
Theoremsyl2and 283 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))    &   (𝜑 → ((𝜒𝜏) → 𝜂))       (𝜑 → ((𝜓𝜃) → 𝜂))
 
Theorembiimpa 284 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) → 𝜒)
 
Theorembiimpar 285 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜒) → 𝜓)
 
Theorembiimpac 286 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))       ((𝜓𝜑) → 𝜒)
 
Theorembiimparc 287 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))       ((𝜒𝜑) → 𝜓)
 
Theoremiba 288 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 289 Introduction of antecedent as conjunct. (Contributed by NM, 5-Dec-1995.) (Revised by NM, 24-Mar-2013.)
(𝜑 → (𝜓 ↔ (𝜑𝜓)))
 
Theorembiantru 290 A wff is equivalent to its conjunction with truth. (Contributed by NM, 5-Aug-1993.)
𝜑       (𝜓 ↔ (𝜓𝜑))
 
Theorembiantrur 291 A wff is equivalent to its conjunction with truth. (Contributed by NM, 3-Aug-1994.)
𝜑       (𝜓 ↔ (𝜑𝜓))
 
Theorembiantrud 292 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 293 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)))
 
Theoremjca 294 Deduce conjunction of the consequents of two implications ("join consequents with 'and'"). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
(𝜑𝜓)    &   (𝜑𝜒)       (𝜑 → (𝜓𝜒))
 
Theoremjcad 295 Deduction conjoining the consequents of two implications. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 23-Jul-2013.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))       (𝜑 → (𝜓 → (𝜒𝜃)))
 
Theoremjca31 296 Join three consequents. (Contributed by Jeff Hankins, 1-Aug-2009.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)       (𝜑 → ((𝜓𝜒) ∧ 𝜃))
 
Theoremjca32 297 Join three consequents. (Contributed by FL, 1-Aug-2009.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)       (𝜑 → (𝜓 ∧ (𝜒𝜃)))
 
Theoremjcai 298 Deduction replacing implication with conjunction. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝜓𝜒))
 
Theoremjctil 299 Inference conjoining a theorem to left of consequent in an implication. (Contributed by NM, 31-Dec-1993.)
(𝜑𝜓)    &   𝜒       (𝜑 → (𝜒𝜓))
 
Theoremjctir 300 Inference conjoining a theorem to right of consequent in an implication. (Contributed by NM, 31-Dec-1993.)
(𝜑𝜓)    &   𝜒       (𝜑 → (𝜓𝜒))
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