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Theorem List for Metamath Proof Explorer - 1201-1300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsimp312 1201 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏)) → 𝜓)
 
Theoremsimp313 1202 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏)) → 𝜒)
 
Theoremsimp321 1203 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏)) → 𝜑)
 
Theoremsimp322 1204 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏)) → 𝜓)
 
Theoremsimp323 1205 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏)) → 𝜒)
 
Theoremsimp331 1206 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒))) → 𝜑)
 
Theoremsimp332 1207 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒))) → 𝜓)
 
Theoremsimp333 1208 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒))) → 𝜒)
 
Theorem3adantl1 1209 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜏𝜑𝜓) ∧ 𝜒) → 𝜃)
 
Theorem3adantl2 1210 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜑𝜏𝜓) ∧ 𝜒) → 𝜃)
 
Theorem3adantl3 1211 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜑𝜓𝜏) ∧ 𝜒) → 𝜃)
 
Theorem3adantr1 1212 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑 ∧ (𝜏𝜓𝜒)) → 𝜃)
 
Theorem3adantr2 1213 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑 ∧ (𝜓𝜏𝜒)) → 𝜃)
 
Theorem3adantr3 1214 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑 ∧ (𝜓𝜒𝜏)) → 𝜃)
 
Theorem3ad2antl1 1215 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
((𝜑𝜒) → 𝜃)       (((𝜑𝜓𝜏) ∧ 𝜒) → 𝜃)
 
Theorem3ad2antl2 1216 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
((𝜑𝜒) → 𝜃)       (((𝜓𝜑𝜏) ∧ 𝜒) → 𝜃)
 
Theorem3ad2antl3 1217 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
((𝜑𝜒) → 𝜃)       (((𝜓𝜏𝜑) ∧ 𝜒) → 𝜃)
 
Theorem3ad2antr1 1218 Deduction adding conjuncts to antecedent. (Contributed by NM, 25-Dec-2007.)
((𝜑𝜒) → 𝜃)       ((𝜑 ∧ (𝜒𝜓𝜏)) → 𝜃)
 
Theorem3ad2antr2 1219 Deduction adding conjuncts to antecedent. (Contributed by NM, 27-Dec-2007.)
((𝜑𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜒𝜏)) → 𝜃)
 
Theorem3ad2antr3 1220 Deduction adding conjuncts to antecedent. (Contributed by NM, 30-Dec-2007.)
((𝜑𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜏𝜒)) → 𝜃)
 
Theorem3anibar 1221 Remove a hypothesis from the second member of a biimplication. (Contributed by FL, 22-Jul-2008.)
((𝜑𝜓𝜒) → (𝜃 ↔ (𝜒𝜏)))       ((𝜑𝜓𝜒) → (𝜃𝜏))
 
Theorem3mix1 1222 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
(𝜑 → (𝜑𝜓𝜒))
 
Theorem3mix2 1223 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
(𝜑 → (𝜓𝜑𝜒))
 
Theorem3mix3 1224 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
(𝜑 → (𝜓𝜒𝜑))
 
Theorem3mix1i 1225 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
𝜑       (𝜑𝜓𝜒)
 
Theorem3mix2i 1226 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
𝜑       (𝜓𝜑𝜒)
 
Theorem3mix3i 1227 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
𝜑       (𝜓𝜒𝜑)
 
Theorem3mix1d 1228 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
(𝜑𝜓)       (𝜑 → (𝜓𝜒𝜃))
 
Theorem3mix2d 1229 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
(𝜑𝜓)       (𝜑 → (𝜒𝜓𝜃))
 
Theorem3mix3d 1230 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
(𝜑𝜓)       (𝜑 → (𝜒𝜃𝜓))
 
Theorem3pm3.2i 1231 Infer conjunction of premises. (Contributed by NM, 10-Feb-1995.)
𝜑    &   𝜓    &   𝜒       (𝜑𝜓𝜒)
 
Theorempm3.2an3 1232 Version of pm3.2 461 for a triple conjunction. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by Kyle Wyonch, 24-Apr-2021.)
(𝜑 → (𝜓 → (𝜒 → (𝜑𝜓𝜒))))
 
Theorempm3.2an3OLD 1233 Obsolete proof of pm3.2an3 1232 as of 24-Apr-2021. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒 → (𝜑𝜓𝜒))))
 
Theorem3jca 1234 Join consequents with conjunction. (Contributed by NM, 9-Apr-1994.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)       (𝜑 → (𝜓𝜒𝜃))
 
Theorem3jcad 1235 Deduction conjoining the consequents of three implications. (Contributed by NM, 25-Sep-2005.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜓𝜏))       (𝜑 → (𝜓 → (𝜒𝜃𝜏)))
 
Theoremmpbir3an 1236 Detach a conjunction of truths in a biconditional. (Contributed by NM, 16-Sep-2011.)
𝜓    &   𝜒    &   𝜃    &   (𝜑 ↔ (𝜓𝜒𝜃))       𝜑
 
Theoremmpbir3and 1237 Detach a conjunction of truths in a biconditional. (Contributed by Mario Carneiro, 11-May-2014.) (Revised by Mario Carneiro, 9-Jan-2015.)
(𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑 → (𝜓 ↔ (𝜒𝜃𝜏)))       (𝜑𝜓)
 
Theoremsyl3anbrc 1238 Syllogism inference. (Contributed by Mario Carneiro, 11-May-2014.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜏 ↔ (𝜓𝜒𝜃))       (𝜑𝜏)
 
Theorem3anim123i 1239 Join antecedents and consequents with conjunction. (Contributed by NM, 8-Apr-1994.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)       ((𝜑𝜒𝜏) → (𝜓𝜃𝜂))
 
Theorem3anim1i 1240 Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 16-Aug-2009.)
(𝜑𝜓)       ((𝜑𝜒𝜃) → (𝜓𝜒𝜃))
 
Theorem3anim2i 1241 Add two conjuncts to antecedent and consequent. (Contributed by AV, 21-Nov-2019.)
(𝜑𝜓)       ((𝜒𝜑𝜃) → (𝜒𝜓𝜃))
 
Theorem3anim3i 1242 Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 19-Aug-2009.)
(𝜑𝜓)       ((𝜒𝜃𝜑) → (𝜒𝜃𝜓))
 
Theorem3anbi123i 1243 Join 3 biconditionals with conjunction. (Contributed by NM, 21-Apr-1994.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)       ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂))
 
Theorem3orbi123i 1244 Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)       ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂))
 
Theorem3anbi1i 1245 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(𝜑𝜓)       ((𝜑𝜒𝜃) ↔ (𝜓𝜒𝜃))
 
Theorem3anbi2i 1246 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(𝜑𝜓)       ((𝜒𝜑𝜃) ↔ (𝜒𝜓𝜃))
 
Theorem3anbi3i 1247 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(𝜑𝜓)       ((𝜒𝜃𝜑) ↔ (𝜒𝜃𝜓))
 
Theorem3imp 1248 Importation inference. (Contributed by NM, 8-Apr-1994.)
(𝜑 → (𝜓 → (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)
 
Theorem3imp31 1249 The importation inference 3imp 1248 with commutation of the first and third conjuncts of the assertion relative to the hypothesis. (Contributed by Alan Sare, 11-Sep-2016.)
(𝜑 → (𝜓 → (𝜒𝜃)))       ((𝜒𝜓𝜑) → 𝜃)
 
Theorem3impa 1250 Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theorem3impb 1251 Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theorem3impia 1252 Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑𝜓𝜒) → 𝜃)
 
Theorem3impib 1253 Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.)
(𝜑 → ((𝜓𝜒) → 𝜃))       ((𝜑𝜓𝜒) → 𝜃)
 
Theoremex3 1254 Apply ex 448 to a hypothesis with a 3-right-nested conjunction antecedent, with the antecedent of the assertion being a triple conjunction rather than a 2-right-nested conjunction. (Contributed by Alan Sare, 22-Apr-2018.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       ((𝜑𝜓𝜒) → (𝜃𝜏))
 
Theorem3exp 1255 Exportation inference. (Contributed by NM, 30-May-1994.)
((𝜑𝜓𝜒) → 𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))
 
Theorem3expa 1256 Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.)
((𝜑𝜓𝜒) → 𝜃)       (((𝜑𝜓) ∧ 𝜒) → 𝜃)
 
Theorem3expb 1257 Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜒)) → 𝜃)
 
Theorem3expia 1258 Exportation from triple conjunction. (Contributed by NM, 19-May-2007.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜓) → (𝜒𝜃))
 
Theorem3expib 1259 Exportation from triple conjunction. (Contributed by NM, 19-May-2007.)
((𝜑𝜓𝜒) → 𝜃)       (𝜑 → ((𝜓𝜒) → 𝜃))
 
Theorem3com12 1260 Commutation in antecedent. Swap 1st and 2nd. (Contributed by NM, 28-Jan-1996.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜓𝜑𝜒) → 𝜃)
 
Theorem3com13 1261 Commutation in antecedent. Swap 1st and 3rd. (Contributed by NM, 28-Jan-1996.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜒𝜓𝜑) → 𝜃)
 
Theorem3com23 1262 Commutation in antecedent. Swap 2nd and 3rd. (Contributed by NM, 28-Jan-1996.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜒𝜓) → 𝜃)
 
Theorem3coml 1263 Commutation in antecedent. Rotate left. (Contributed by NM, 28-Jan-1996.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜓𝜒𝜑) → 𝜃)
 
Theorem3comr 1264 Commutation in antecedent. Rotate right. (Contributed by NM, 28-Jan-1996.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜒𝜑𝜓) → 𝜃)
 
Theorem3adant3r1 1265 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Feb-2008.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜏𝜓𝜒)) → 𝜃)
 
Theorem3adant3r2 1266 Deduction adding a conjunct to antecedent. (Contributed by NM, 17-Feb-2008.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜏𝜒)) → 𝜃)
 
Theorem3adant3r3 1267 Deduction adding a conjunct to antecedent. (Contributed by NM, 18-Feb-2008.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜒𝜏)) → 𝜃)
 
Theorem3imp21 1268 The importation inference 3imp 1248 with commutation of the first and second conjuncts of the assertion relative to the hypothesis. (Contributed by Alan Sare, 11-Sep-2016.)
(𝜑 → (𝜓 → (𝜒𝜃)))       ((𝜓𝜑𝜒) → 𝜃)
 
Theorem3imp3i2an 1269 An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓𝜒) → 𝜃)    &   ((𝜑𝜒) → 𝜏)    &   ((𝜃𝜏) → 𝜂)       ((𝜑𝜓𝜒) → 𝜂)
 
Theorem3an1rs 1270 Swap conjuncts. (Contributed by NM, 16-Dec-2007.)
(((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜑𝜓𝜃) ∧ 𝜒) → 𝜏)
 
Theorem3imp1 1271 Importation to left triple conjunction. (Contributed by NM, 24-Feb-2005.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)
 
Theorem3impd 1272 Importation deduction for triple conjunction. (Contributed by NM, 26-Oct-2006.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → ((𝜓𝜒𝜃) → 𝜏))
 
Theorem3imp2 1273 Importation to right triple conjunction. (Contributed by NM, 26-Oct-2006.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)
 
Theorem3exp1 1274 Exportation from left triple conjunction. (Contributed by NM, 24-Feb-2005.)
(((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theorem3expd 1275 Exportation deduction for triple conjunction. (Contributed by NM, 26-Oct-2006.)
(𝜑 → ((𝜓𝜒𝜃) → 𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theorem3exp2 1276 Exportation from right triple conjunction. (Contributed by NM, 26-Oct-2006.)
((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp5o 1277 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
((𝜑𝜓𝜒) → ((𝜃𝜏) → 𝜂))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp516 1278 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
(((𝜑 ∧ (𝜓𝜒𝜃)) ∧ 𝜏) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp520 1279 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
(((𝜑𝜓𝜒) ∧ (𝜃𝜏)) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theorem3impexp 1280 Version of impexp 460 for a triple conjunction. (Contributed by Alan Sare, 31-Dec-2011.)
(((𝜑𝜓𝜒) → 𝜃) ↔ (𝜑 → (𝜓 → (𝜒𝜃))))
 
Theorem3anassrs 1281 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by Mario Carneiro, 4-Jan-2017.)
((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)       ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
 
Theorem3an4anass 1282 Associative law for four conjunctions with a triple conjunction. (Contributed by Alexander van der Vekens, 24-Jun-2018.)
(((𝜑𝜓𝜒) ∧ 𝜃) ↔ ((𝜑𝜓) ∧ (𝜒𝜃)))
 
Theoremad4ant13 1283 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓) → 𝜒)       ((((𝜑𝜃) ∧ 𝜓) ∧ 𝜏) → 𝜒)
 
Theoremad4ant14 1284 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓) → 𝜒)       ((((𝜑𝜃) ∧ 𝜏) ∧ 𝜓) → 𝜒)
 
Theoremad4ant123 1285 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓𝜒) → 𝜃)       ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜃)
 
Theoremad4ant124 1286 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓𝜒) → 𝜃)       ((((𝜑𝜓) ∧ 𝜏) ∧ 𝜒) → 𝜃)
 
Theoremad4ant134 1287 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓𝜒) → 𝜃)       ((((𝜑𝜏) ∧ 𝜓) ∧ 𝜒) → 𝜃)
 
Theoremad4ant23 1288 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓) → 𝜒)       ((((𝜃𝜑) ∧ 𝜓) ∧ 𝜏) → 𝜒)
 
Theoremad4ant24 1289 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓) → 𝜒)       ((((𝜃𝜑) ∧ 𝜏) ∧ 𝜓) → 𝜒)
 
Theoremad4ant234 1290 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓𝜒) → 𝜃)       ((((𝜏𝜑) ∧ 𝜓) ∧ 𝜒) → 𝜃)
 
Theoremad5ant12 1291 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓) → 𝜒)       (((((𝜑𝜓) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜒)
 
Theoremad5ant13 1292 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓) → 𝜒)       (((((𝜑𝜃) ∧ 𝜓) ∧ 𝜏) ∧ 𝜂) → 𝜒)
 
Theoremad5ant14 1293 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓) → 𝜒)       (((((𝜑𝜃) ∧ 𝜏) ∧ 𝜓) ∧ 𝜂) → 𝜒)
 
Theoremad5ant15 1294 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓) → 𝜒)       (((((𝜑𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜓) → 𝜒)
 
Theoremad5ant23 1295 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓) → 𝜒)       (((((𝜃𝜑) ∧ 𝜓) ∧ 𝜏) ∧ 𝜂) → 𝜒)
 
Theoremad5ant24 1296 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓) → 𝜒)       (((((𝜃𝜑) ∧ 𝜏) ∧ 𝜓) ∧ 𝜂) → 𝜒)
 
Theoremad5ant25 1297 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓) → 𝜒)       (((((𝜃𝜑) ∧ 𝜏) ∧ 𝜂) ∧ 𝜓) → 𝜒)
 
Theoremad5ant245 1298 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓𝜒) → 𝜃)       (((((𝜏𝜑) ∧ 𝜂) ∧ 𝜓) ∧ 𝜒) → 𝜃)
 
Theoremad5ant234 1299 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓𝜒) → 𝜃)       (((((𝜏𝜑) ∧ 𝜓) ∧ 𝜒) ∧ 𝜂) → 𝜃)
 
Theoremad5ant235 1300 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓𝜒) → 𝜃)       (((((𝜏𝜑) ∧ 𝜓) ∧ 𝜂) ∧ 𝜒) → 𝜃)
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