mmtheorems13 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 1201-1300 - Page 13 of 191

Type	Label	Description
Statement
Theorem	simpl31 1201	Simplification of conjunction.
|- (((th /\ ta /\ (ph /\ ps /\ ch)) /\ et) -> ph)
Theorem	simpl32 1202	Simplification of conjunction.
|- (((th /\ ta /\ (ph /\ ps /\ ch)) /\ et) -> ps)
Theorem	simpl33 1203	Simplification of conjunction.
|- (((th /\ ta /\ (ph /\ ps /\ ch)) /\ et) -> ch)
Theorem	simpr11 1204	Simplification of conjunction.
|- ((et /\ ((ph /\ ps /\ ch) /\ th /\ ta)) -> ph)
Theorem	simpr12 1205	Simplification of conjunction.
|- ((et /\ ((ph /\ ps /\ ch) /\ th /\ ta)) -> ps)
Theorem	simpr13 1206	Simplification of conjunction.
|- ((et /\ ((ph /\ ps /\ ch) /\ th /\ ta)) -> ch)
Theorem	simpr21 1207	Simplification of conjunction.
|- ((et /\ (th /\ (ph /\ ps /\ ch) /\ ta)) -> ph)
Theorem	simpr22 1208	Simplification of conjunction.
|- ((et /\ (th /\ (ph /\ ps /\ ch) /\ ta)) -> ps)
Theorem	simpr23 1209	Simplification of conjunction.
|- ((et /\ (th /\ (ph /\ ps /\ ch) /\ ta)) -> ch)
Theorem	simpr31 1210	Simplification of conjunction.
|- ((et /\ (th /\ ta /\ (ph /\ ps /\ ch))) -> ph)
Theorem	simpr32 1211	Simplification of conjunction.
|- ((et /\ (th /\ ta /\ (ph /\ ps /\ ch))) -> ps)
Theorem	simpr33 1212	Simplification of conjunction.
|- ((et /\ (th /\ ta /\ (ph /\ ps /\ ch))) -> ch)
Theorem	simp1l1 1213	Simplification of conjunction.
|- ((((ph /\ ps /\ ch) /\ th) /\ ta /\ et) -> ph)
Theorem	simp1l2 1214	Simplification of conjunction.
|- ((((ph /\ ps /\ ch) /\ th) /\ ta /\ et) -> ps)
Theorem	simp1l3 1215	Simplification of conjunction.
|- ((((ph /\ ps /\ ch) /\ th) /\ ta /\ et) -> ch)
Theorem	simp1r1 1216	Simplification of conjunction.
|- (((th /\ (ph /\ ps /\ ch)) /\ ta /\ et) -> ph)
Theorem	simp1r2 1217	Simplification of conjunction.
|- (((th /\ (ph /\ ps /\ ch)) /\ ta /\ et) -> ps)
Theorem	simp1r3 1218	Simplification of conjunction.
|- (((th /\ (ph /\ ps /\ ch)) /\ ta /\ et) -> ch)
Theorem	simp2l1 1219	Simplification of conjunction.
|- ((ta /\ ((ph /\ ps /\ ch) /\ th) /\ et) -> ph)
Theorem	simp2l2 1220	Simplification of conjunction.
|- ((ta /\ ((ph /\ ps /\ ch) /\ th) /\ et) -> ps)
Theorem	simp2l3 1221	Simplification of conjunction.
|- ((ta /\ ((ph /\ ps /\ ch) /\ th) /\ et) -> ch)
Theorem	simp2r1 1222	Simplification of conjunction.
|- ((ta /\ (th /\ (ph /\ ps /\ ch)) /\ et) -> ph)
Theorem	simp2r2 1223	Simplification of conjunction.
|- ((ta /\ (th /\ (ph /\ ps /\ ch)) /\ et) -> ps)
Theorem	simp2r3 1224	Simplification of conjunction.
|- ((ta /\ (th /\ (ph /\ ps /\ ch)) /\ et) -> ch)
Theorem	simp3l1 1225	Simplification of conjunction.
|- ((ta /\ et /\ ((ph /\ ps /\ ch) /\ th)) -> ph)
Theorem	simp3l2 1226	Simplification of conjunction.
|- ((ta /\ et /\ ((ph /\ ps /\ ch) /\ th)) -> ps)
Theorem	simp3l3 1227	Simplification of conjunction.
|- ((ta /\ et /\ ((ph /\ ps /\ ch) /\ th)) -> ch)
Theorem	simp3r1 1228	Simplification of conjunction.
|- ((ta /\ et /\ (th /\ (ph /\ ps /\ ch))) -> ph)
Theorem	simp3r2 1229	Simplification of conjunction.
|- ((ta /\ et /\ (th /\ (ph /\ ps /\ ch))) -> ps)
Theorem	simp3r3 1230	Simplification of conjunction.
|- ((ta /\ et /\ (th /\ (ph /\ ps /\ ch))) -> ch)
Theorem	simp11l 1231	Simplification of conjunction.
|- ((((ph /\ ps) /\ ch /\ th) /\ ta /\ et) -> ph)
Theorem	simp11r 1232	Simplification of conjunction.
|- ((((ph /\ ps) /\ ch /\ th) /\ ta /\ et) -> ps)
Theorem	simp12l 1233	Simplification of conjunction.
|- (((ch /\ (ph /\ ps) /\ th) /\ ta /\ et) -> ph)
Theorem	simp12r 1234	Simplification of conjunction.
|- (((ch /\ (ph /\ ps) /\ th) /\ ta /\ et) -> ps)
Theorem	simp13l 1235	Simplification of conjunction.
|- (((ch /\ th /\ (ph /\ ps)) /\ ta /\ et) -> ph)
Theorem	simp13r 1236	Simplification of conjunction.
|- (((ch /\ th /\ (ph /\ ps)) /\ ta /\ et) -> ps)
Theorem	simp21l 1237	Simplification of conjunction.
|- ((ta /\ ((ph /\ ps) /\ ch /\ th) /\ et) -> ph)
Theorem	simp21r 1238	Simplification of conjunction.
|- ((ta /\ ((ph /\ ps) /\ ch /\ th) /\ et) -> ps)
Theorem	simp22l 1239	Simplification of conjunction.
|- ((ta /\ (ch /\ (ph /\ ps) /\ th) /\ et) -> ph)
Theorem	simp22r 1240	Simplification of conjunction.
|- ((ta /\ (ch /\ (ph /\ ps) /\ th) /\ et) -> ps)
Theorem	simp23l 1241	Simplification of conjunction.
|- ((ta /\ (ch /\ th /\ (ph /\ ps)) /\ et) -> ph)
Theorem	simp23r 1242	Simplification of conjunction.
|- ((ta /\ (ch /\ th /\ (ph /\ ps)) /\ et) -> ps)
Theorem	simp31l 1243	Simplification of conjunction.
|- ((ta /\ et /\ ((ph /\ ps) /\ ch /\ th)) -> ph)
Theorem	simp31r 1244	Simplification of conjunction.
|- ((ta /\ et /\ ((ph /\ ps) /\ ch /\ th)) -> ps)
Theorem	simp32l 1245	Simplification of conjunction.
|- ((ta /\ et /\ (ch /\ (ph /\ ps) /\ th)) -> ph)
Theorem	simp32r 1246	Simplification of conjunction.
|- ((ta /\ et /\ (ch /\ (ph /\ ps) /\ th)) -> ps)
Theorem	simp33l 1247	Simplification of conjunction.
|- ((ta /\ et /\ (ch /\ th /\ (ph /\ ps))) -> ph)
Theorem	simp33r 1248	Simplification of conjunction.
|- ((ta /\ et /\ (ch /\ th /\ (ph /\ ps))) -> ps)
Theorem	simp111 1249	Simplification of conjunction.
|- ((((ph /\ ps /\ ch) /\ th /\ ta) /\ et /\ ze) -> ph)
Theorem	simp112 1250	Simplification of conjunction.
|- ((((ph /\ ps /\ ch) /\ th /\ ta) /\ et /\ ze) -> ps)
Theorem	simp113 1251	Simplification of conjunction.
|- ((((ph /\ ps /\ ch) /\ th /\ ta) /\ et /\ ze) -> ch)
Theorem	simp121 1252	Simplification of conjunction.
|- (((th /\ (ph /\ ps /\ ch) /\ ta) /\ et /\ ze) -> ph)
Theorem	simp122 1253	Simplification of conjunction.
|- (((th /\ (ph /\ ps /\ ch) /\ ta) /\ et /\ ze) -> ps)
Theorem	simp123 1254	Simplification of conjunction.
|- (((th /\ (ph /\ ps /\ ch) /\ ta) /\ et /\ ze) -> ch)
Theorem	simp131 1255	Simplification of conjunction.
|- (((th /\ ta /\ (ph /\ ps /\ ch)) /\ et /\ ze) -> ph)
Theorem	simp132 1256	Simplification of conjunction.
|- (((th /\ ta /\ (ph /\ ps /\ ch)) /\ et /\ ze) -> ps)
Theorem	simp133 1257	Simplification of conjunction.
|- (((th /\ ta /\ (ph /\ ps /\ ch)) /\ et /\ ze) -> ch)
Theorem	simp211 1258	Simplification of conjunction.
|- ((et /\ ((ph /\ ps /\ ch) /\ th /\ ta) /\ ze) -> ph)
Theorem	simp212 1259	Simplification of conjunction.
|- ((et /\ ((ph /\ ps /\ ch) /\ th /\ ta) /\ ze) -> ps)
Theorem	simp213 1260	Simplification of conjunction.
|- ((et /\ ((ph /\ ps /\ ch) /\ th /\ ta) /\ ze) -> ch)
Theorem	simp221 1261	Simplification of conjunction.
|- ((et /\ (th /\ (ph /\ ps /\ ch) /\ ta) /\ ze) -> ph)
Theorem	simp222 1262	Simplification of conjunction.
|- ((et /\ (th /\ (ph /\ ps /\ ch) /\ ta) /\ ze) -> ps)
Theorem	simp223 1263	Simplification of conjunction.
|- ((et /\ (th /\ (ph /\ ps /\ ch) /\ ta) /\ ze) -> ch)
Theorem	simp231 1264	Simplification of conjunction.
|- ((et /\ (th /\ ta /\ (ph /\ ps /\ ch)) /\ ze) -> ph)
Theorem	simp232 1265	Simplification of conjunction.
|- ((et /\ (th /\ ta /\ (ph /\ ps /\ ch)) /\ ze) -> ps)
Theorem	simp233 1266	Simplification of conjunction.
|- ((et /\ (th /\ ta /\ (ph /\ ps /\ ch)) /\ ze) -> ch)
Theorem	simp311 1267	Simplification of conjunction.
|- ((et /\ ze /\ ((ph /\ ps /\ ch) /\ th /\ ta)) -> ph)
Theorem	simp312 1268	Simplification of conjunction.
|- ((et /\ ze /\ ((ph /\ ps /\ ch) /\ th /\ ta)) -> ps)
Theorem	simp313 1269	Simplification of conjunction.
|- ((et /\ ze /\ ((ph /\ ps /\ ch) /\ th /\ ta)) -> ch)
Theorem	simp321 1270	Simplification of conjunction.
|- ((et /\ ze /\ (th /\ (ph /\ ps /\ ch) /\ ta)) -> ph)
Theorem	simp322 1271	Simplification of conjunction.
|- ((et /\ ze /\ (th /\ (ph /\ ps /\ ch) /\ ta)) -> ps)
Theorem	simp323 1272	Simplification of conjunction.
|- ((et /\ ze /\ (th /\ (ph /\ ps /\ ch) /\ ta)) -> ch)
Theorem	simp331 1273	Simplification of conjunction.
|- ((et /\ ze /\ (th /\ ta /\ (ph /\ ps /\ ch))) -> ph)
Theorem	simp332 1274	Simplification of conjunction.
|- ((et /\ ze /\ (th /\ ta /\ (ph /\ ps /\ ch))) -> ps)
Theorem	simp333 1275	Simplification of conjunction.
|- ((et /\ ze /\ (th /\ ta /\ (ph /\ ps /\ ch))) -> ch)
Theorem	3adantl1 1276	Deduction adding a conjunct to antecedent. (The proof was shortened by NM, 4-Dec-2012.)
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (((ta /\ ph /\ ps) /\ ch) -> th)
Theorem	3adantl1OLD 1277	Obsolete proof of 3adantl1 1276.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (((ta /\ ph /\ ps) /\ ch) -> th)
Theorem	3adantl2 1278	Deduction adding a conjunct to antecedent. (The proof was shortened by NM, 5-Dec-2012.)
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (((ph /\ ta /\ ps) /\ ch) -> th)
Theorem	3adantl2OLD 1279	Obsolete proof of 3adantl2 1278.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (((ph /\ ta /\ ps) /\ ch) -> th)
Theorem	3adantl3 1280	Deduction adding a conjunct to antecedent. (The proof was shortened by NM, 5-Dec-2012.)
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (((ph /\ ps /\ ta) /\ ch) -> th)
Theorem	3adantl3OLD 1281	Obsolete proof of 3adantl3 1280.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (((ph /\ ps /\ ta) /\ ch) -> th)
Theorem	3adantr1 1282	Deduction adding a conjunct to antecedent. (The proof was shortened by NM, 7-Dec-2012.)
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ta /\ ps /\ ch)) -> th)
Theorem	3adantr1OLD 1283	Obsolete proof of 3adantr1 1282.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ta /\ ps /\ ch)) -> th)
Theorem	3adantr2 1284	Deduction adding a conjunct to antecedent. (The proof was shortened by NM, 7-Dec-2012.)
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ps /\ ta /\ ch)) -> th)
Theorem	3adantr2OLD 1285	Obsolete proof of 3adantr1 1282.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ps /\ ta /\ ch)) -> th)
Theorem	3adantr3 1286	Deduction adding a conjunct to antecedent. (The proof was shortened by NM, 7-Dec-2012.)
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ps /\ ch /\ ta)) -> th)
Theorem	3adantr3OLD 1287	Obsolete proof of 3adantr1 1282.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ps /\ ch /\ ta)) -> th)
Theorem	3ad2antl1 1288	Deduction adding conjuncts to antecedent.
|- ((ph /\ ch) -> th)   =>   |- (((ph /\ ps /\ ta) /\ ch) -> th)
Theorem	3ad2antl2 1289	Deduction adding conjuncts to antecedent.
|- ((ph /\ ch) -> th)   =>   |- (((ps /\ ph /\ ta) /\ ch) -> th)
Theorem	3ad2antl3 1290	Deduction adding conjuncts to antecedent.
|- ((ph /\ ch) -> th)   =>   |- (((ps /\ ta /\ ph) /\ ch) -> th)
Theorem	3ad2antr1 1291	Deduction adding a conjuncts to antecedent.
|- ((ph /\ ch) -> th)   =>   |- ((ph /\ (ch /\ ps /\ ta)) -> th)
Theorem	3ad2antr2 1292	Deduction adding a conjuncts to antecedent.
|- ((ph /\ ch) -> th)   =>   |- ((ph /\ (ps /\ ch /\ ta)) -> th)
Theorem	3ad2antr3 1293	Deduction adding a conjuncts to antecedent.
|- ((ph /\ ch) -> th)   =>   |- ((ph /\ (ps /\ ta /\ ch)) -> th)
Theorem	3anibar 1294	Remove a hypothesis from the second member of a biimplication. (Contributed by FL, 22-Jul-2008.)
|- ((ph /\ ps /\ ch) -> (th <-> (ch /\ ta)))   =>   |- ((ph /\ ps /\ ch) -> (th <-> ta))
Theorem	3mix1 1295	Introduction in triple disjunction.
|- (ph -> (ph \/ ps \/ ch))
Theorem	3mix2 1296	Introduction in triple disjunction.
|- (ph -> (ps \/ ph \/ ch))
Theorem	3mix3 1297	Introduction in triple disjunction.
|- (ph -> (ps \/ ch \/ ph))
Theorem	3pm3.2i 1298	Infer conjunction of premises.
|- ph   &   |- ps   &   |- ch   =>   |- (ph /\ ps /\ ch)
Theorem	pm3.2an3 1299	pm3.2 410 for a triple conjunction. (Contributed by Alan Sare, 24-Oct-2011.)
|- (ph -> (ps -> (ch -> (ph /\ ps /\ ch))))
Theorem	3jca 1300	Join consequents with conjunction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   =>   |- (ph -> (ps /\ ch /\ th))