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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | 3adant3r3 1201 | Deduction adding a conjunct to antecedent. (Contributed by NM, 18-Feb-2008.) |
| ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜏)) → 𝜃) | ||
| Theorem | 3ad2antl1 1202 | Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.) |
| ⊢ ((𝜑 ∧ 𝜒) → 𝜃) ⇒ ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜏) ∧ 𝜒) → 𝜃) | ||
| Theorem | 3ad2antl2 1203 | Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.) |
| ⊢ ((𝜑 ∧ 𝜒) → 𝜃) ⇒ ⊢ (((𝜓 ∧ 𝜑 ∧ 𝜏) ∧ 𝜒) → 𝜃) | ||
| Theorem | 3ad2antl3 1204 | Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.) |
| ⊢ ((𝜑 ∧ 𝜒) → 𝜃) ⇒ ⊢ (((𝜓 ∧ 𝜏 ∧ 𝜑) ∧ 𝜒) → 𝜃) | ||
| Theorem | 3ad2antr1 1205 | Deduction adding conjuncts to antecedent. (Contributed by NM, 25-Dec-2007.) |
| ⊢ ((𝜑 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ (𝜒 ∧ 𝜓 ∧ 𝜏)) → 𝜃) | ||
| Theorem | 3ad2antr2 1206 | Deduction adding conjuncts to antecedent. (Contributed by NM, 27-Dec-2007.) |
| ⊢ ((𝜑 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜏)) → 𝜃) | ||
| Theorem | 3ad2antr3 1207 | Deduction adding conjuncts to antecedent. (Contributed by NM, 30-Dec-2007.) |
| ⊢ ((𝜑 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜏 ∧ 𝜒)) → 𝜃) | ||
| Theorem | simpl1 1208 | Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
| ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜑) | ||
| Theorem | simpl2 1209 | Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
| ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜓) | ||
| Theorem | simpl3 1210 | Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
| ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜒) | ||
| Theorem | simpr1 1211 | Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
| ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜓) | ||
| Theorem | simpr2 1212 | Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
| ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜒) | ||
| Theorem | simpr3 1213 | Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
| ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜃) | ||
| Theorem | simp1l 1214 | Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) |
| ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃) → 𝜑) | ||
| Theorem | simp1r 1215 | Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) |
| ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃) → 𝜓) | ||
| Theorem | simp2l 1216 | Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) |
| ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜓) | ||
| Theorem | simp2r 1217 | Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) |
| ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜒) | ||
| Theorem | simp3l 1218 | Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) |
| ⊢ ((𝜑 ∧ 𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜒) | ||
| Theorem | simp3r 1219 | Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) |
| ⊢ ((𝜑 ∧ 𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜃) | ||
| Theorem | simp11 1220 | Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.) |
| ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃 ∧ 𝜏) → 𝜑) | ||
| Theorem | simp12 1221 | Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.) |
| ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃 ∧ 𝜏) → 𝜓) | ||
| Theorem | simp13 1222 | Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.) |
| ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃 ∧ 𝜏) → 𝜒) | ||
| Theorem | simp21 1223 | Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.) |
| ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜓) | ||
| Theorem | simp22 1224 | Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.) |
| ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜒) | ||
| Theorem | simp23 1225 | Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.) |
| ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜃) | ||
| Theorem | simp31 1226 | Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.) |
| ⊢ ((𝜑 ∧ 𝜓 ∧ (𝜒 ∧ 𝜃 ∧ 𝜏)) → 𝜒) | ||
| Theorem | simp32 1227 | Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.) |
| ⊢ ((𝜑 ∧ 𝜓 ∧ (𝜒 ∧ 𝜃 ∧ 𝜏)) → 𝜃) | ||
| Theorem | simp33 1228 | Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.) |
| ⊢ ((𝜑 ∧ 𝜓 ∧ (𝜒 ∧ 𝜃 ∧ 𝜏)) → 𝜏) | ||
| Theorem | simpll1 1229 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
| ⊢ ((((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜑) | ||
| Theorem | simpll2 1230 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
| ⊢ ((((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜓) | ||
| Theorem | simpll3 1231 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
| ⊢ ((((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜒) | ||
| Theorem | simplr1 1232 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
| ⊢ (((𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒)) ∧ 𝜏) → 𝜑) | ||
| Theorem | simplr2 1233 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
| ⊢ (((𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒)) ∧ 𝜏) → 𝜓) | ||
| Theorem | simplr3 1234 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
| ⊢ (((𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒)) ∧ 𝜏) → 𝜒) | ||
| Theorem | simprl1 1235 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
| ⊢ ((𝜏 ∧ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃)) → 𝜑) | ||
| Theorem | simprl2 1236 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
| ⊢ ((𝜏 ∧ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃)) → 𝜓) | ||
| Theorem | simprl3 1237 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
| ⊢ ((𝜏 ∧ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃)) → 𝜒) | ||
| Theorem | simprr1 1238 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
| ⊢ ((𝜏 ∧ (𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒))) → 𝜑) | ||
| Theorem | simprr2 1239 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
| ⊢ ((𝜏 ∧ (𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒))) → 𝜓) | ||
| Theorem | simprr3 1240 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
| ⊢ ((𝜏 ∧ (𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒))) → 𝜒) | ||
| Theorem | simpl1l 1241 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
| ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜑) | ||
| Theorem | simpl1r 1242 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
| ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜓) | ||
| Theorem | simpl2l 1243 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
| ⊢ (((𝜒 ∧ (𝜑 ∧ 𝜓) ∧ 𝜃) ∧ 𝜏) → 𝜑) | ||
| Theorem | simpl2r 1244 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
| ⊢ (((𝜒 ∧ (𝜑 ∧ 𝜓) ∧ 𝜃) ∧ 𝜏) → 𝜓) | ||
| Theorem | simpl3l 1245 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
| ⊢ (((𝜒 ∧ 𝜃 ∧ (𝜑 ∧ 𝜓)) ∧ 𝜏) → 𝜑) | ||
| Theorem | simpl3r 1246 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
| ⊢ (((𝜒 ∧ 𝜃 ∧ (𝜑 ∧ 𝜓)) ∧ 𝜏) → 𝜓) | ||
| Theorem | simpr1l 1247 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.) |
| ⊢ ((𝜏 ∧ ((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃)) → 𝜑) | ||
| Theorem | simpr1r 1248 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.) |
| ⊢ ((𝜏 ∧ ((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃)) → 𝜓) | ||
| Theorem | simpr2l 1249 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.) |
| ⊢ ((𝜏 ∧ (𝜒 ∧ (𝜑 ∧ 𝜓) ∧ 𝜃)) → 𝜑) | ||
| Theorem | simpr2r 1250 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.) |
| ⊢ ((𝜏 ∧ (𝜒 ∧ (𝜑 ∧ 𝜓) ∧ 𝜃)) → 𝜓) | ||
| Theorem | simpr3l 1251 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.) |
| ⊢ ((𝜏 ∧ (𝜒 ∧ 𝜃 ∧ (𝜑 ∧ 𝜓))) → 𝜑) | ||
| Theorem | simpr3r 1252 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.) |
| ⊢ ((𝜏 ∧ (𝜒 ∧ 𝜃 ∧ (𝜑 ∧ 𝜓))) → 𝜓) | ||
| Theorem | simp1ll 1253 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃 ∧ 𝜏) → 𝜑) | ||
| Theorem | simp1lr 1254 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃 ∧ 𝜏) → 𝜓) | ||
| Theorem | simp1rl 1255 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ (((𝜒 ∧ (𝜑 ∧ 𝜓)) ∧ 𝜃 ∧ 𝜏) → 𝜑) | ||
| Theorem | simp1rr 1256 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ (((𝜒 ∧ (𝜑 ∧ 𝜓)) ∧ 𝜃 ∧ 𝜏) → 𝜓) | ||
| Theorem | simp2ll 1257 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ ((𝜃 ∧ ((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜑) | ||
| Theorem | simp2lr 1258 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ ((𝜃 ∧ ((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜓) | ||
| Theorem | simp2rl 1259 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ ((𝜃 ∧ (𝜒 ∧ (𝜑 ∧ 𝜓)) ∧ 𝜏) → 𝜑) | ||
| Theorem | simp2rr 1260 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ ((𝜃 ∧ (𝜒 ∧ (𝜑 ∧ 𝜓)) ∧ 𝜏) → 𝜓) | ||
| Theorem | simp3ll 1261 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ ((𝜃 ∧ 𝜏 ∧ ((𝜑 ∧ 𝜓) ∧ 𝜒)) → 𝜑) | ||
| Theorem | simp3lr 1262 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ ((𝜃 ∧ 𝜏 ∧ ((𝜑 ∧ 𝜓) ∧ 𝜒)) → 𝜓) | ||
| Theorem | simp3rl 1263 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ ((𝜃 ∧ 𝜏 ∧ (𝜒 ∧ (𝜑 ∧ 𝜓))) → 𝜑) | ||
| Theorem | simp3rr 1264 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ ((𝜃 ∧ 𝜏 ∧ (𝜒 ∧ (𝜑 ∧ 𝜓))) → 𝜓) | ||
| Theorem | simpl11 1265 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.) |
| ⊢ ((((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃 ∧ 𝜏) ∧ 𝜂) → 𝜑) | ||
| Theorem | simpl12 1266 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.) |
| ⊢ ((((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃 ∧ 𝜏) ∧ 𝜂) → 𝜓) | ||
| Theorem | simpl13 1267 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.) |
| ⊢ ((((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃 ∧ 𝜏) ∧ 𝜂) → 𝜒) | ||
| Theorem | simpl21 1268 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.) |
| ⊢ (((𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜏) ∧ 𝜂) → 𝜑) | ||
| Theorem | simpl22 1269 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.) |
| ⊢ (((𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜏) ∧ 𝜂) → 𝜓) | ||
| Theorem | simpl23 1270 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.) |
| ⊢ (((𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜏) ∧ 𝜂) → 𝜒) | ||
| Theorem | simpl31 1271 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.) |
| ⊢ (((𝜃 ∧ 𝜏 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒)) ∧ 𝜂) → 𝜑) | ||
| Theorem | simpl32 1272 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.) |
| ⊢ (((𝜃 ∧ 𝜏 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒)) ∧ 𝜂) → 𝜓) | ||
| Theorem | simpl33 1273 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.) |
| ⊢ (((𝜃 ∧ 𝜏 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒)) ∧ 𝜂) → 𝜒) | ||
| Theorem | simpr11 1274 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.) |
| ⊢ ((𝜂 ∧ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃 ∧ 𝜏)) → 𝜑) | ||
| Theorem | simpr12 1275 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.) |
| ⊢ ((𝜂 ∧ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃 ∧ 𝜏)) → 𝜓) | ||
| Theorem | simpr13 1276 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.) |
| ⊢ ((𝜂 ∧ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃 ∧ 𝜏)) → 𝜒) | ||
| Theorem | simpr21 1277 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.) |
| ⊢ ((𝜂 ∧ (𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜏)) → 𝜑) | ||
| Theorem | simpr22 1278 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.) |
| ⊢ ((𝜂 ∧ (𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜏)) → 𝜓) | ||
| Theorem | simpr23 1279 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.) |
| ⊢ ((𝜂 ∧ (𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜏)) → 𝜒) | ||
| Theorem | simpr31 1280 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.) |
| ⊢ ((𝜂 ∧ (𝜃 ∧ 𝜏 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒))) → 𝜑) | ||
| Theorem | simpr32 1281 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.) |
| ⊢ ((𝜂 ∧ (𝜃 ∧ 𝜏 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒))) → 𝜓) | ||
| Theorem | simpr33 1282 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.) |
| ⊢ ((𝜂 ∧ (𝜃 ∧ 𝜏 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒))) → 𝜒) | ||
| Theorem | simp1l1 1283 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ ((((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏 ∧ 𝜂) → 𝜑) | ||
| Theorem | simp1l2 1284 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ ((((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏 ∧ 𝜂) → 𝜓) | ||
| Theorem | simp1l3 1285 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ ((((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏 ∧ 𝜂) → 𝜒) | ||
| Theorem | simp1r1 1286 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ (((𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒)) ∧ 𝜏 ∧ 𝜂) → 𝜑) | ||
| Theorem | simp1r2 1287 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ (((𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒)) ∧ 𝜏 ∧ 𝜂) → 𝜓) | ||
| Theorem | simp1r3 1288 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ (((𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒)) ∧ 𝜏 ∧ 𝜂) → 𝜒) | ||
| Theorem | simp2l1 1289 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ ((𝜏 ∧ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜂) → 𝜑) | ||
| Theorem | simp2l2 1290 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ ((𝜏 ∧ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜂) → 𝜓) | ||
| Theorem | simp2l3 1291 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ ((𝜏 ∧ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜂) → 𝜒) | ||
| Theorem | simp2r1 1292 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ ((𝜏 ∧ (𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒)) ∧ 𝜂) → 𝜑) | ||
| Theorem | simp2r2 1293 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ ((𝜏 ∧ (𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒)) ∧ 𝜂) → 𝜓) | ||
| Theorem | simp2r3 1294 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ ((𝜏 ∧ (𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒)) ∧ 𝜂) → 𝜒) | ||
| Theorem | simp3l1 1295 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ ((𝜏 ∧ 𝜂 ∧ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃)) → 𝜑) | ||
| Theorem | simp3l2 1296 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ ((𝜏 ∧ 𝜂 ∧ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃)) → 𝜓) | ||
| Theorem | simp3l3 1297 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ ((𝜏 ∧ 𝜂 ∧ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃)) → 𝜒) | ||
| Theorem | simp3r1 1298 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ ((𝜏 ∧ 𝜂 ∧ (𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒))) → 𝜑) | ||
| Theorem | simp3r2 1299 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ ((𝜏 ∧ 𝜂 ∧ (𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒))) → 𝜓) | ||
| Theorem | simp3r3 1300 | Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) |
| ⊢ ((𝜏 ∧ 𝜂 ∧ (𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒))) → 𝜒) | ||
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