P. 12 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 1101-1200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	3imp231 1101	Importation inference. (Contributed by Alan Sare, 17-Oct-2017.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    ⇒   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜃)
Theorem	3imp21 1102	The importation inference 3imp 1098 with commutation of the first and second conjuncts of the assertion relative to the hypothesis. (Contributed by Alan Sare, 11-Sep-2016.) (Revised to shorten 3com12 1114 by Wolf Lammen, 23-Jun-2022.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    ⇒   ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜃)
Theorem	3impaOLD 1103	Obsolete version of 3impa 1097 as of 20-Jun-2022. (Contributed by NM, 20-Aug-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	3impb 1104	Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	3impib 1105	Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.)
⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	3impia 1106	Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.) (Proof shortened by Wolf Lammen, 21-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	3impiaOLD 1107	Obsolete version of 3impia 1106 as of 21-Jun-2022. (Contributed by NM, 13-Jun-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	3expa 1108	Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.) (Revised to shorten 3exp 1109 and pm3.2an3 1396 by Wolf Lammen, 22-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)
Theorem	3exp 1109	Exportation inference. (Contributed by NM, 30-May-1994.) (Proof shortened by Wolf Lammen, 22-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
Theorem	3expb 1110	Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)
Theorem	3expia 1111	Exportation from triple conjunction. (Contributed by NM, 19-May-2007.) (Proof shortened by Wolf Lammen, 22-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))
Theorem	3expiaOLD 1112	Obsolete version of 3expia 1111 as of 22-Jun-2022. (Contributed by NM, 19-May-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))
Theorem	3expib 1113	Exportation from triple conjunction. (Contributed by NM, 19-May-2007.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))
Theorem	3com12 1114	Commutation in antecedent. Swap 1st and 2nd. (Contributed by NM, 28-Jan-1996.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 22-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜃)
Theorem	3com13 1115	Commutation in antecedent. Swap 1st and 3rd. (Contributed by NM, 28-Jan-1996.) (Proof shortened by Wolf Lammen, 22-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜒 ∧ 𝜓 ∧ 𝜑) → 𝜃)
Theorem	3comr 1116	Commutation in antecedent. Rotate right. (Contributed by NM, 28-Jan-1996.) (Revised by Wolf Lammen, 9-Apr-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜓) → 𝜃)
Theorem	3com23 1117	Commutation in antecedent. Swap 2nd and 3rd. (Contributed by NM, 28-Jan-1996.) (Proof shortened by Wolf Lammen, 9-Apr-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜓) → 𝜃)
Theorem	3coml 1118	Commutation in antecedent. Rotate left. (Contributed by NM, 28-Jan-1996.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜃)
Theorem	3jca 1119	Join consequents with conjunction. (Contributed by NM, 9-Apr-1994.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))
Theorem	3jcad 1120	Deduction conjoining the consequents of three implications. (Contributed by NM, 25-Sep-2005.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜓 → 𝜃))    &   ⊢ (𝜑 → (𝜓 → 𝜏))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃 ∧ 𝜏)))
Theorem	3adant1 1121	Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.) (Proof shortened by Wolf Lammen, 21-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜃 ∧ 𝜑 ∧ 𝜓) → 𝜒)
Theorem	3adant2 1122	Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜓) → 𝜒)
Theorem	3adant3 1123	Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.) (Proof shortened by Wolf Lammen, 21-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜒)
Theorem	3ad2ant1 1124	Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
⊢ (𝜑 → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜒)
Theorem	3ad2ant2 1125	Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
⊢ (𝜑 → 𝜒)    ⇒   ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → 𝜒)
Theorem	3ad2ant3 1126	Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
⊢ (𝜑 → 𝜒)    ⇒   ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜑) → 𝜒)
Theorem	simp1 1127	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜑)
Theorem	simp2 1128	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜓)
Theorem	simp3 1129	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜒)
Theorem	simp1i 1130	Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
⊢ (𝜑 ∧ 𝜓 ∧ 𝜒)    ⇒   ⊢ 𝜑
Theorem	simp2i 1131	Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
⊢ (𝜑 ∧ 𝜓 ∧ 𝜒)    ⇒   ⊢ 𝜓
Theorem	simp3i 1132	Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
⊢ (𝜑 ∧ 𝜓 ∧ 𝜒)    ⇒   ⊢ 𝜒
Theorem	simp1d 1133	Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	simp2d 1134	Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	simp3d 1135	Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	simp1bi 1136	Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	simp2bi 1137	Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	simp3bi 1138	Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	3simpa 1139	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 21-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜑 ∧ 𝜓))
Theorem	3simpaOLD 1140	Obsolete version of 3simpa 1139 as of 21-Jun-2022. (Contributed by NM, 21-Apr-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜑 ∧ 𝜓))
Theorem	3simpb 1141	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 21-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜑 ∧ 𝜒))
Theorem	3simpbOLD 1142	Obsolete version of 3simpb 1141 as of 21-Jun-2022. (Contributed by NM, 21-Apr-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜑 ∧ 𝜒))
Theorem	3simpc 1143	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 21-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜒))
Theorem	3simpcOLD 1144	Obsolete version of 3simpc 1143 as of 21-Jun-2022. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜒))
Theorem	simp1OLD 1145	Obsolete version of simp1 1127 as of 22-Jun-2022. (Contributed by NM, 21-Apr-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜑)
Theorem	simp2OLD 1146	Obsolete version of simp2 1128 as of 22-Jun-2022. (Contributed by NM, 21-Apr-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜓)
Theorem	simp3OLD 1147	Obsolete version of simp3 1129 as of 22-Jun-2022. (Contributed by NM, 21-Apr-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜒)
Theorem	3adant1OLD 1148	Obsolete version of 3adant1 1121 as of 21-Jun-2022. (Contributed by NM, 16-Jul-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜃 ∧ 𝜑 ∧ 𝜓) → 𝜒)
Theorem	3adant2OLD 1149	Obsolete version of 3adant1 1121 as of 21-Jun-2022. (Contributed by NM, 16-Jul-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜓) → 𝜒)
Theorem	3adant3OLD 1150	Obsolete version of 3adant3 1123 as of 21-Jun-2022. (Contributed by NM, 16-Jul-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜒)
Theorem	3anim123i 1151	Join antecedents and consequents with conjunction. (Contributed by NM, 8-Apr-1994.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 → 𝜃)    &   ⊢ (𝜏 → 𝜂)    ⇒   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → (𝜓 ∧ 𝜃 ∧ 𝜂))
Theorem	3anim1i 1152	Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 16-Aug-2009.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → (𝜓 ∧ 𝜒 ∧ 𝜃))
Theorem	3anim2i 1153	Add two conjuncts to antecedent and consequent. (Contributed by AV, 21-Nov-2019.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜃) → (𝜒 ∧ 𝜓 ∧ 𝜃))
Theorem	3anim3i 1154	Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 19-Aug-2009.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜒 ∧ 𝜃 ∧ 𝜑) → (𝜒 ∧ 𝜃 ∧ 𝜓))
Theorem	3anbi123i 1155	Join 3 biconditionals with conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜃)    &   ⊢ (𝜏 ↔ 𝜂)    ⇒   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) ↔ (𝜓 ∧ 𝜃 ∧ 𝜂))
Theorem	3orbi123i 1156	Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜃)    &   ⊢ (𝜏 ↔ 𝜂)    ⇒   ⊢ ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂))
Theorem	3anbi1i 1157	Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))
Theorem	3anbi2i 1158	Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜃) ↔ (𝜒 ∧ 𝜓 ∧ 𝜃))
Theorem	3anbi3i 1159	Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ((𝜒 ∧ 𝜃 ∧ 𝜑) ↔ (𝜒 ∧ 𝜃 ∧ 𝜓))
Theorem	syl3an 1160	A triple syllogism inference. (Contributed by NM, 13-May-2004.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 → 𝜃)    &   ⊢ (𝜏 → 𝜂)    &   ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁)    ⇒   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁)
Theorem	syl3anb 1161	A triple syllogism inference. (Contributed by NM, 15-Oct-2005.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜃)    &   ⊢ (𝜏 ↔ 𝜂)    &   ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁)    ⇒   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁)
Theorem	syl3anbr 1162	A triple syllogism inference. (Contributed by NM, 29-Dec-2011.)
⊢ (𝜓 ↔ 𝜑)    &   ⊢ (𝜃 ↔ 𝜒)    &   ⊢ (𝜂 ↔ 𝜏)    &   ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁)    ⇒   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁)
Theorem	syl3an1 1163	A syllogism inference. (Contributed by NM, 22-Aug-1995.)
⊢ (𝜑 → 𝜓)    &   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏)
Theorem	syl3an2 1164	A syllogism inference. (Contributed by NM, 22-Aug-1995.) (Proof shortened by Wolf Lammen, 26-Jun-2022.)
⊢ (𝜑 → 𝜒)    &   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → 𝜏)
Theorem	syl3an2OLD 1165	Obsolete version of syl3an2 1164 as of 26-Jun-2022. (Contributed by NM, 22-Aug-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝜒)    &   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → 𝜏)
Theorem	syl3an3 1166	A syllogism inference. (Contributed by NM, 22-Aug-1995.) (Proof shortened by Wolf Lammen, 26-Jun-2022.)
⊢ (𝜑 → 𝜃)    &   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜏)
Theorem	syl3an3OLD 1167	Obsolete version of syl3an3 1166 as of 26-Jun-2022. (Contributed by NM, 22-Aug-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝜃)    &   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜏)
Theorem	3adantl1 1168	Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜏 ∧ 𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)
Theorem	3adantl2 1169	Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜑 ∧ 𝜏 ∧ 𝜓) ∧ 𝜒) → 𝜃)
Theorem	3adantl3 1170	Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜏) ∧ 𝜒) → 𝜃)
Theorem	3adantr1 1171	Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜏 ∧ 𝜓 ∧ 𝜒)) → 𝜃)
Theorem	3adantr2 1172	Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜏 ∧ 𝜒)) → 𝜃)
Theorem	3adantr3 1173	Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜏)) → 𝜃)
Theorem	ad4ant123 1174	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜃)
Theorem	ad4ant124 1175	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜏) ∧ 𝜒) → 𝜃)
Theorem	ad4ant134 1176	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((((𝜑 ∧ 𝜏) ∧ 𝜓) ∧ 𝜒) → 𝜃)
Theorem	ad4ant234 1177	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((((𝜏 ∧ 𝜑) ∧ 𝜓) ∧ 𝜒) → 𝜃)
Theorem	3adant1l 1178	Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (Proof shortened by Wolf Lammen, 23-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜏 ∧ 𝜑) ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	3adant1lOLD 1179	Obsolete version of 3adant1l 1178 as of 23-Jun-2022. (Contributed by NM, 8-Jan-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜏 ∧ 𝜑) ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	3adant1r 1180	Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (Proof shortened by Wolf Lammen, 23-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜑 ∧ 𝜏) ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	3adant1rOLD 1181	Obsolete version of 3adant1r 1180 as of 23-Jun-2022. (Contributed by NM, 8-Jan-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜑 ∧ 𝜏) ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	3adant2l 1182	Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (Proof shortened by Wolf Lammen, 25-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜏 ∧ 𝜓) ∧ 𝜒) → 𝜃)
Theorem	3adant2lOLD 1183	Obsolete version of 3adant2l 1182 as of 25-Jun-2022. (Contributed by NM, 8-Jan-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜏 ∧ 𝜓) ∧ 𝜒) → 𝜃)
Theorem	3adant2r 1184	Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (Proof shortened by Wolf Lammen, 25-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜏) ∧ 𝜒) → 𝜃)
Theorem	3adant2rOLD 1185	Obsolete version of 3adant2r 1184 as of 25-Jun-2022. (Contributed by NM, 8-Jan-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜏) ∧ 𝜒) → 𝜃)
Theorem	3adant3l 1186	Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (Proof shortened by Wolf Lammen, 25-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ (𝜏 ∧ 𝜒)) → 𝜃)
Theorem	3adant3lOLD 1187	Obsolete version of 3adant3l 1186 as of 25-Jun-2022. (Contributed by NM, 8-Jan-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ (𝜏 ∧ 𝜒)) → 𝜃)
Theorem	3adant3r 1188	Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (Proof shortened by Wolf Lammen, 25-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ (𝜒 ∧ 𝜏)) → 𝜃)
Theorem	3adant3rOLD 1189	Obsolete version of 3adant3r 1188 as of 25-Jun-2022. (Contributed by NM, 8-Jan-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ (𝜒 ∧ 𝜏)) → 𝜃)
Theorem	3adant3r1 1190	Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Feb-2008.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜏 ∧ 𝜓 ∧ 𝜒)) → 𝜃)
Theorem	3adant3r2 1191	Deduction adding a conjunct to antecedent. (Contributed by NM, 17-Feb-2008.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜏 ∧ 𝜒)) → 𝜃)
Theorem	3adant3r3 1192	Deduction adding a conjunct to antecedent. (Contributed by NM, 18-Feb-2008.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜏)) → 𝜃)
Theorem	3ad2antl1 1193	Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
⊢ ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜏) ∧ 𝜒) → 𝜃)
Theorem	3ad2antl2 1194	Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
⊢ ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜓 ∧ 𝜑 ∧ 𝜏) ∧ 𝜒) → 𝜃)
Theorem	3ad2antl3 1195	Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
⊢ ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜓 ∧ 𝜏 ∧ 𝜑) ∧ 𝜒) → 𝜃)
Theorem	3ad2antr1 1196	Deduction adding conjuncts to antecedent. (Contributed by NM, 25-Dec-2007.)
⊢ ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜒 ∧ 𝜓 ∧ 𝜏)) → 𝜃)
Theorem	3ad2antr2 1197	Deduction adding conjuncts to antecedent. (Contributed by NM, 27-Dec-2007.)
⊢ ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜏)) → 𝜃)
Theorem	3ad2antr3 1198	Deduction adding conjuncts to antecedent. (Contributed by NM, 30-Dec-2007.)
⊢ ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜏 ∧ 𝜒)) → 𝜃)
Theorem	simpl1 1199	Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.)
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜑)
Theorem	simpl1OLD 1200	Obsolete version of simpl1 1199 as of 23-Jun-2022. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜑)