Theorem List for Intuitionistic Logic Explorer - 1001-1100 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | simpl2 1001 |
Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
|
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜓) |
|
Theorem | simpl3 1002 |
Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
|
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜒) |
|
Theorem | simpr1 1003 |
Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
|
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜓) |
|
Theorem | simpr2 1004 |
Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
|
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜒) |
|
Theorem | simpr3 1005 |
Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
|
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜃) |
|
Theorem | simp1i 1006 |
Infer a conjunct from a triple conjunction. (Contributed by NM,
19-Apr-2005.)
|
⊢ (𝜑 ∧ 𝜓 ∧ 𝜒) ⇒ ⊢ 𝜑 |
|
Theorem | simp2i 1007 |
Infer a conjunct from a triple conjunction. (Contributed by NM,
19-Apr-2005.)
|
⊢ (𝜑 ∧ 𝜓 ∧ 𝜒) ⇒ ⊢ 𝜓 |
|
Theorem | simp3i 1008 |
Infer a conjunct from a triple conjunction. (Contributed by NM,
19-Apr-2005.)
|
⊢ (𝜑 ∧ 𝜓 ∧ 𝜒) ⇒ ⊢ 𝜒 |
|
Theorem | simp1d 1009 |
Deduce a conjunct from a triple conjunction. (Contributed by NM,
4-Sep-2005.)
|
⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃)) ⇒ ⊢ (𝜑 → 𝜓) |
|
Theorem | simp2d 1010 |
Deduce a conjunct from a triple conjunction. (Contributed by NM,
4-Sep-2005.)
|
⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃)) ⇒ ⊢ (𝜑 → 𝜒) |
|
Theorem | simp3d 1011 |
Deduce a conjunct from a triple conjunction. (Contributed by NM,
4-Sep-2005.)
|
⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃)) ⇒ ⊢ (𝜑 → 𝜃) |
|
Theorem | simp1bi 1012 |
Deduce a conjunct from a triple conjunction. (Contributed by Jonathan
Ben-Naim, 3-Jun-2011.)
|
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)) ⇒ ⊢ (𝜑 → 𝜓) |
|
Theorem | simp2bi 1013 |
Deduce a conjunct from a triple conjunction. (Contributed by Jonathan
Ben-Naim, 3-Jun-2011.)
|
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)) ⇒ ⊢ (𝜑 → 𝜒) |
|
Theorem | simp3bi 1014 |
Deduce a conjunct from a triple conjunction. (Contributed by Jonathan
Ben-Naim, 3-Jun-2011.)
|
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)) ⇒ ⊢ (𝜑 → 𝜃) |
|
Theorem | 3adant1 1015 |
Deduction adding a conjunct to antecedent. (Contributed by NM,
16-Jul-1995.)
|
⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ ((𝜃 ∧ 𝜑 ∧ 𝜓) → 𝜒) |
|
Theorem | 3adant2 1016 |
Deduction adding a conjunct to antecedent. (Contributed by NM,
16-Jul-1995.)
|
⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜓) → 𝜒) |
|
Theorem | 3adant3 1017 |
Deduction adding a conjunct to antecedent. (Contributed by NM,
16-Jul-1995.)
|
⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜒) |
|
Theorem | 3ad2ant1 1018 |
Deduction adding conjuncts to an antecedent. (Contributed by NM,
21-Apr-2005.)
|
⊢ (𝜑 → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜒) |
|
Theorem | 3ad2ant2 1019 |
Deduction adding conjuncts to an antecedent. (Contributed by NM,
21-Apr-2005.)
|
⊢ (𝜑 → 𝜒) ⇒ ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → 𝜒) |
|
Theorem | 3ad2ant3 1020 |
Deduction adding conjuncts to an antecedent. (Contributed by NM,
21-Apr-2005.)
|
⊢ (𝜑 → 𝜒) ⇒ ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜑) → 𝜒) |
|
Theorem | simp1l 1021 |
Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
|
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃) → 𝜑) |
|
Theorem | simp1r 1022 |
Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
|
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃) → 𝜓) |
|
Theorem | simp2l 1023 |
Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
|
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜓) |
|
Theorem | simp2r 1024 |
Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
|
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜒) |
|
Theorem | simp3l 1025 |
Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
|
⊢ ((𝜑 ∧ 𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜒) |
|
Theorem | simp3r 1026 |
Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
|
⊢ ((𝜑 ∧ 𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜃) |
|
Theorem | simp11 1027 |
Simplification of doubly triple conjunction. (Contributed by NM,
17-Nov-2011.)
|
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃 ∧ 𝜏) → 𝜑) |
|
Theorem | simp12 1028 |
Simplification of doubly triple conjunction. (Contributed by NM,
17-Nov-2011.)
|
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃 ∧ 𝜏) → 𝜓) |
|
Theorem | simp13 1029 |
Simplification of doubly triple conjunction. (Contributed by NM,
17-Nov-2011.)
|
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃 ∧ 𝜏) → 𝜒) |
|
Theorem | simp21 1030 |
Simplification of doubly triple conjunction. (Contributed by NM,
17-Nov-2011.)
|
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜓) |
|
Theorem | simp22 1031 |
Simplification of doubly triple conjunction. (Contributed by NM,
17-Nov-2011.)
|
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜒) |
|
Theorem | simp23 1032 |
Simplification of doubly triple conjunction. (Contributed by NM,
17-Nov-2011.)
|
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜃) |
|
Theorem | simp31 1033 |
Simplification of doubly triple conjunction. (Contributed by NM,
17-Nov-2011.)
|
⊢ ((𝜑 ∧ 𝜓 ∧ (𝜒 ∧ 𝜃 ∧ 𝜏)) → 𝜒) |
|
Theorem | simp32 1034 |
Simplification of doubly triple conjunction. (Contributed by NM,
17-Nov-2011.)
|
⊢ ((𝜑 ∧ 𝜓 ∧ (𝜒 ∧ 𝜃 ∧ 𝜏)) → 𝜃) |
|
Theorem | simp33 1035 |
Simplification of doubly triple conjunction. (Contributed by NM,
17-Nov-2011.)
|
⊢ ((𝜑 ∧ 𝜓 ∧ (𝜒 ∧ 𝜃 ∧ 𝜏)) → 𝜏) |
|
Theorem | simpll1 1036 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜑) |
|
Theorem | simpll2 1037 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜓) |
|
Theorem | simpll3 1038 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜒) |
|
Theorem | simplr1 1039 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ (((𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒)) ∧ 𝜏) → 𝜑) |
|
Theorem | simplr2 1040 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ (((𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒)) ∧ 𝜏) → 𝜓) |
|
Theorem | simplr3 1041 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ (((𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒)) ∧ 𝜏) → 𝜒) |
|
Theorem | simprl1 1042 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜏 ∧ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃)) → 𝜑) |
|
Theorem | simprl2 1043 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜏 ∧ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃)) → 𝜓) |
|
Theorem | simprl3 1044 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜏 ∧ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃)) → 𝜒) |
|
Theorem | simprr1 1045 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜏 ∧ (𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒))) → 𝜑) |
|
Theorem | simprr2 1046 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜏 ∧ (𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒))) → 𝜓) |
|
Theorem | simprr3 1047 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜏 ∧ (𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒))) → 𝜒) |
|
Theorem | simpl1l 1048 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜑) |
|
Theorem | simpl1r 1049 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜓) |
|
Theorem | simpl2l 1050 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ (((𝜒 ∧ (𝜑 ∧ 𝜓) ∧ 𝜃) ∧ 𝜏) → 𝜑) |
|
Theorem | simpl2r 1051 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ (((𝜒 ∧ (𝜑 ∧ 𝜓) ∧ 𝜃) ∧ 𝜏) → 𝜓) |
|
Theorem | simpl3l 1052 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ (((𝜒 ∧ 𝜃 ∧ (𝜑 ∧ 𝜓)) ∧ 𝜏) → 𝜑) |
|
Theorem | simpl3r 1053 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ (((𝜒 ∧ 𝜃 ∧ (𝜑 ∧ 𝜓)) ∧ 𝜏) → 𝜓) |
|
Theorem | simpr1l 1054 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜏 ∧ ((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃)) → 𝜑) |
|
Theorem | simpr1r 1055 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜏 ∧ ((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃)) → 𝜓) |
|
Theorem | simpr2l 1056 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜏 ∧ (𝜒 ∧ (𝜑 ∧ 𝜓) ∧ 𝜃)) → 𝜑) |
|
Theorem | simpr2r 1057 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜏 ∧ (𝜒 ∧ (𝜑 ∧ 𝜓) ∧ 𝜃)) → 𝜓) |
|
Theorem | simpr3l 1058 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜏 ∧ (𝜒 ∧ 𝜃 ∧ (𝜑 ∧ 𝜓))) → 𝜑) |
|
Theorem | simpr3r 1059 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜏 ∧ (𝜒 ∧ 𝜃 ∧ (𝜑 ∧ 𝜓))) → 𝜓) |
|
Theorem | simp1ll 1060 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃 ∧ 𝜏) → 𝜑) |
|
Theorem | simp1lr 1061 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃 ∧ 𝜏) → 𝜓) |
|
Theorem | simp1rl 1062 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ (((𝜒 ∧ (𝜑 ∧ 𝜓)) ∧ 𝜃 ∧ 𝜏) → 𝜑) |
|
Theorem | simp1rr 1063 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ (((𝜒 ∧ (𝜑 ∧ 𝜓)) ∧ 𝜃 ∧ 𝜏) → 𝜓) |
|
Theorem | simp2ll 1064 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜃 ∧ ((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜑) |
|
Theorem | simp2lr 1065 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜃 ∧ ((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜓) |
|
Theorem | simp2rl 1066 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜃 ∧ (𝜒 ∧ (𝜑 ∧ 𝜓)) ∧ 𝜏) → 𝜑) |
|
Theorem | simp2rr 1067 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜃 ∧ (𝜒 ∧ (𝜑 ∧ 𝜓)) ∧ 𝜏) → 𝜓) |
|
Theorem | simp3ll 1068 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜃 ∧ 𝜏 ∧ ((𝜑 ∧ 𝜓) ∧ 𝜒)) → 𝜑) |
|
Theorem | simp3lr 1069 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜃 ∧ 𝜏 ∧ ((𝜑 ∧ 𝜓) ∧ 𝜒)) → 𝜓) |
|
Theorem | simp3rl 1070 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜃 ∧ 𝜏 ∧ (𝜒 ∧ (𝜑 ∧ 𝜓))) → 𝜑) |
|
Theorem | simp3rr 1071 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜃 ∧ 𝜏 ∧ (𝜒 ∧ (𝜑 ∧ 𝜓))) → 𝜓) |
|
Theorem | simpl11 1072 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃 ∧ 𝜏) ∧ 𝜂) → 𝜑) |
|
Theorem | simpl12 1073 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃 ∧ 𝜏) ∧ 𝜂) → 𝜓) |
|
Theorem | simpl13 1074 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃 ∧ 𝜏) ∧ 𝜂) → 𝜒) |
|
Theorem | simpl21 1075 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ (((𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜏) ∧ 𝜂) → 𝜑) |
|
Theorem | simpl22 1076 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ (((𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜏) ∧ 𝜂) → 𝜓) |
|
Theorem | simpl23 1077 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ (((𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜏) ∧ 𝜂) → 𝜒) |
|
Theorem | simpl31 1078 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ (((𝜃 ∧ 𝜏 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒)) ∧ 𝜂) → 𝜑) |
|
Theorem | simpl32 1079 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ (((𝜃 ∧ 𝜏 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒)) ∧ 𝜂) → 𝜓) |
|
Theorem | simpl33 1080 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ (((𝜃 ∧ 𝜏 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒)) ∧ 𝜂) → 𝜒) |
|
Theorem | simpr11 1081 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜂 ∧ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃 ∧ 𝜏)) → 𝜑) |
|
Theorem | simpr12 1082 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜂 ∧ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃 ∧ 𝜏)) → 𝜓) |
|
Theorem | simpr13 1083 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜂 ∧ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃 ∧ 𝜏)) → 𝜒) |
|
Theorem | simpr21 1084 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜂 ∧ (𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜏)) → 𝜑) |
|
Theorem | simpr22 1085 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜂 ∧ (𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜏)) → 𝜓) |
|
Theorem | simpr23 1086 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜂 ∧ (𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜏)) → 𝜒) |
|
Theorem | simpr31 1087 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜂 ∧ (𝜃 ∧ 𝜏 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒))) → 𝜑) |
|
Theorem | simpr32 1088 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜂 ∧ (𝜃 ∧ 𝜏 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒))) → 𝜓) |
|
Theorem | simpr33 1089 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((𝜂 ∧ (𝜃 ∧ 𝜏 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒))) → 𝜒) |
|
Theorem | simp1l1 1090 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏 ∧ 𝜂) → 𝜑) |
|
Theorem | simp1l2 1091 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏 ∧ 𝜂) → 𝜓) |
|
Theorem | simp1l3 1092 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ ((((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏 ∧ 𝜂) → 𝜒) |
|
Theorem | simp1r1 1093 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ (((𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒)) ∧ 𝜏 ∧ 𝜂) → 𝜑) |
|
Theorem | simp1r2 1094 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ (((𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒)) ∧ 𝜏 ∧ 𝜂) → 𝜓) |
|
Theorem | simp1r3 1095 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|
⊢ (((𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒)) ∧ 𝜏 ∧ 𝜂) → 𝜒) |
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Theorem | simp2l1 1096 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
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⊢ ((𝜏 ∧ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜂) → 𝜑) |
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Theorem | simp2l2 1097 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
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⊢ ((𝜏 ∧ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜂) → 𝜓) |
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Theorem | simp2l3 1098 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
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⊢ ((𝜏 ∧ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜂) → 𝜒) |
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Theorem | simp2r1 1099 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
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⊢ ((𝜏 ∧ (𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒)) ∧ 𝜂) → 𝜑) |
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Theorem | simp2r2 1100 |
Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
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⊢ ((𝜏 ∧ (𝜃 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒)) ∧ 𝜂) → 𝜓) |