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	anandi3r 1101	Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜓)))
Theorem	3anidm 1102	Idempotent law for conjunction. (Contributed by Peter Mazsa, 17-Oct-2023.)
⊢ ((𝜑 ∧ 𝜑 ∧ 𝜑) ↔ 𝜑)
Theorem	3an4anass 1103	Associative law for four conjunctions with a triple conjunction. (Contributed by Alexander van der Vekens, 24-Jun-2018.)
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))
Theorem	3ioran 1104	Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011.)
⊢ (¬ (𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒))
Theorem	3ianor 1105	Negated triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Revised by Wolf Lammen, 8-Apr-2022.)
⊢ (¬ (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))
Theorem	3anor 1106	Triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Wolf Lammen, 8-Apr-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))
Theorem	3oran 1107	Triple disjunction in terms of triple conjunction. (Contributed by NM, 8-Oct-2012.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒))
Theorem	3impa 1108	Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.) (Revised to shorten 3imp 1109 by Wolf Lammen, 20-Jun-2022.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	3imp 1109	Importation inference. (Contributed by NM, 8-Apr-1994.) (Proof shortened by Wolf Lammen, 20-Jun-2022.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	3imp31 1110	The importation inference 3imp 1109 with commutation of the first and third conjuncts of the assertion relative to the hypothesis. (Contributed by Alan Sare, 11-Sep-2016.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    ⇒   ⊢ ((𝜒 ∧ 𝜓 ∧ 𝜑) → 𝜃)
Theorem	3imp231 1111	Importation inference. (Contributed by Alan Sare, 17-Oct-2017.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    ⇒   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜃)
Theorem	3imp21 1112	The importation inference 3imp 1109 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 1121 by Wolf Lammen, 23-Jun-2022.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    ⇒   ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜃)
Theorem	3impb 1113	Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	3impib 1114	Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.)
⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	3impia 1115	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	3expa 1116	Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.) (Revised to shorten 3exp 1117 and pm3.2an3 1338 by Wolf Lammen, 22-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)
Theorem	3exp 1117	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 1118	Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)
Theorem	3expia 1119	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	3expib 1120	Exportation from triple conjunction. (Contributed by NM, 19-May-2007.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))
Theorem	3com12 1121	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 1122	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 1123	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 1124	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 1125	Commutation in antecedent. Rotate left. (Contributed by NM, 28-Jan-1996.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜃)
Theorem	3jca 1126	Join consequents with conjunction. (Contributed by NM, 9-Apr-1994.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))
Theorem	3jcad 1127	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 1128	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 1129	Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜓) → 𝜒)
Theorem	3adant3 1130	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 1131	Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
⊢ (𝜑 → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜒)
Theorem	3ad2ant2 1132	Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
⊢ (𝜑 → 𝜒)    ⇒   ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → 𝜒)
Theorem	3ad2ant3 1133	Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
⊢ (𝜑 → 𝜒)    ⇒   ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜑) → 𝜒)
Theorem	simp1 1134	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜑)
Theorem	simp2 1135	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜓)
Theorem	simp3 1136	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜒)
Theorem	simp1i 1137	Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
⊢ (𝜑 ∧ 𝜓 ∧ 𝜒)    ⇒   ⊢ 𝜑
Theorem	simp2i 1138	Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
⊢ (𝜑 ∧ 𝜓 ∧ 𝜒)    ⇒   ⊢ 𝜓
Theorem	simp3i 1139	Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
⊢ (𝜑 ∧ 𝜓 ∧ 𝜒)    ⇒   ⊢ 𝜒
Theorem	simp1d 1140	Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	simp2d 1141	Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	simp3d 1142	Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	simp1bi 1143	Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	simp2bi 1144	Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	simp3bi 1145	Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	3simpa 1146	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 21-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜑 ∧ 𝜓))
Theorem	3simpb 1147	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 21-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜑 ∧ 𝜒))
Theorem	3simpc 1148	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	3anim123i 1149	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 1150	Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 16-Aug-2009.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → (𝜓 ∧ 𝜒 ∧ 𝜃))
Theorem	3anim2i 1151	Add two conjuncts to antecedent and consequent. (Contributed by AV, 21-Nov-2019.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜃) → (𝜒 ∧ 𝜓 ∧ 𝜃))
Theorem	3anim3i 1152	Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 19-Aug-2009.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜒 ∧ 𝜃 ∧ 𝜑) → (𝜒 ∧ 𝜃 ∧ 𝜓))
Theorem	3anbi123i 1153	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 1154	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 1155	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 1156	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 1157	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 1158	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 1159	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 1160	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 1161	A syllogism inference. (Contributed by NM, 22-Aug-1995.)
⊢ (𝜑 → 𝜓)    &   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏)
Theorem	syl3an2 1162	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	syl3an3 1163	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	3adantl1 1164	Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜏 ∧ 𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)
Theorem	3adantl2 1165	Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜑 ∧ 𝜏 ∧ 𝜓) ∧ 𝜒) → 𝜃)
Theorem	3adantl3 1166	Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜏) ∧ 𝜒) → 𝜃)
Theorem	3adantr1 1167	Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜏 ∧ 𝜓 ∧ 𝜒)) → 𝜃)
Theorem	3adantr2 1168	Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜏 ∧ 𝜒)) → 𝜃)
Theorem	3adantr3 1169	Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜏)) → 𝜃)
Theorem	ad4ant123 1170	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 1171	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 1172	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 1173	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 1174	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	3adant1r 1175	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	3adant2l 1176	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	3adant2r 1177	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	3adant3l 1178	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	3adant3r 1179	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	3adant3r1 1180	Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Feb-2008.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜏 ∧ 𝜓 ∧ 𝜒)) → 𝜃)
Theorem	3adant3r2 1181	Deduction adding a conjunct to antecedent. (Contributed by NM, 17-Feb-2008.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜏 ∧ 𝜒)) → 𝜃)
Theorem	3adant3r3 1182	Deduction adding a conjunct to antecedent. (Contributed by NM, 18-Feb-2008.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜏)) → 𝜃)
Theorem	3ad2antl1 1183	Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
⊢ ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜏) ∧ 𝜒) → 𝜃)
Theorem	3ad2antl2 1184	Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
⊢ ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜓 ∧ 𝜑 ∧ 𝜏) ∧ 𝜒) → 𝜃)
Theorem	3ad2antl3 1185	Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
⊢ ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜓 ∧ 𝜏 ∧ 𝜑) ∧ 𝜒) → 𝜃)
Theorem	3ad2antr1 1186	Deduction adding conjuncts to antecedent. (Contributed by NM, 25-Dec-2007.)
⊢ ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜒 ∧ 𝜓 ∧ 𝜏)) → 𝜃)
Theorem	3ad2antr2 1187	Deduction adding conjuncts to antecedent. (Contributed by NM, 27-Dec-2007.)
⊢ ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜏)) → 𝜃)
Theorem	3ad2antr3 1188	Deduction adding conjuncts to antecedent. (Contributed by NM, 30-Dec-2007.)
⊢ ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜏 ∧ 𝜒)) → 𝜃)
Theorem	simpl1 1189	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 1190	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 1191	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 1192	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 1193	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 1194	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 1195	Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃) → 𝜑)
Theorem	simp1r 1196	Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃) → 𝜓)
Theorem	simp2l 1197	Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜓)
Theorem	simp2r 1198	Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜒)
Theorem	simp3l 1199	Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
⊢ ((𝜑 ∧ 𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜒)
Theorem	simp3r 1200	Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
⊢ ((𝜑 ∧ 𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜃)