P. 14 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 1301-1400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	syl322anc 1301	Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → 𝜃)    &   ⊢ (𝜑 → 𝜏)    &   ⊢ (𝜑 → 𝜂)    &   ⊢ (𝜑 → 𝜁)    &   ⊢ (𝜑 → 𝜎)    &   ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂) ∧ (𝜁 ∧ 𝜎)) → 𝜌)    ⇒   ⊢ (𝜑 → 𝜌)
Theorem	syl233anc 1302	Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → 𝜃)    &   ⊢ (𝜑 → 𝜏)    &   ⊢ (𝜑 → 𝜂)    &   ⊢ (𝜑 → 𝜁)    &   ⊢ (𝜑 → 𝜎)    &   ⊢ (𝜑 → 𝜌)    &   ⊢ (((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏 ∧ 𝜂) ∧ (𝜁 ∧ 𝜎 ∧ 𝜌)) → 𝜇)    ⇒   ⊢ (𝜑 → 𝜇)
Theorem	syl323anc 1303	Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → 𝜃)    &   ⊢ (𝜑 → 𝜏)    &   ⊢ (𝜑 → 𝜂)    &   ⊢ (𝜑 → 𝜁)    &   ⊢ (𝜑 → 𝜎)    &   ⊢ (𝜑 → 𝜌)    &   ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂) ∧ (𝜁 ∧ 𝜎 ∧ 𝜌)) → 𝜇)    ⇒   ⊢ (𝜑 → 𝜇)
Theorem	syl332anc 1304	Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → 𝜃)    &   ⊢ (𝜑 → 𝜏)    &   ⊢ (𝜑 → 𝜂)    &   ⊢ (𝜑 → 𝜁)    &   ⊢ (𝜑 → 𝜎)    &   ⊢ (𝜑 → 𝜌)    &   ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂 ∧ 𝜁) ∧ (𝜎 ∧ 𝜌)) → 𝜇)    ⇒   ⊢ (𝜑 → 𝜇)
Theorem	syl333anc 1305	A syllogism inference combined with contraction. (Contributed by NM, 10-Mar-2012.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → 𝜃)    &   ⊢ (𝜑 → 𝜏)    &   ⊢ (𝜑 → 𝜂)    &   ⊢ (𝜑 → 𝜁)    &   ⊢ (𝜑 → 𝜎)    &   ⊢ (𝜑 → 𝜌)    &   ⊢ (𝜑 → 𝜇)    &   ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂 ∧ 𝜁) ∧ (𝜎 ∧ 𝜌 ∧ 𝜇)) → 𝜆)    ⇒   ⊢ (𝜑 → 𝜆)
Theorem	syl3an1 1306	A syllogism inference. (Contributed by NM, 22-Aug-1995.)
⊢ (𝜑 → 𝜓)    &   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏)
Theorem	syl3an2 1307	A syllogism inference. (Contributed by NM, 22-Aug-1995.)
⊢ (𝜑 → 𝜒)    &   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → 𝜏)
Theorem	syl3an3 1308	A syllogism inference. (Contributed by NM, 22-Aug-1995.)
⊢ (𝜑 → 𝜃)    &   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜏)
Theorem	syl3an1b 1309	A syllogism inference. (Contributed by NM, 22-Aug-1995.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏)
Theorem	syl3an2b 1310	A syllogism inference. (Contributed by NM, 22-Aug-1995.)
⊢ (𝜑 ↔ 𝜒)    &   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → 𝜏)
Theorem	syl3an3b 1311	A syllogism inference. (Contributed by NM, 22-Aug-1995.)
⊢ (𝜑 ↔ 𝜃)    &   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜏)
Theorem	syl3an1br 1312	A syllogism inference. (Contributed by NM, 22-Aug-1995.)
⊢ (𝜓 ↔ 𝜑)    &   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏)
Theorem	syl3an2br 1313	A syllogism inference. (Contributed by NM, 22-Aug-1995.)
⊢ (𝜒 ↔ 𝜑)    &   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → 𝜏)
Theorem	syl3an3br 1314	A syllogism inference. (Contributed by NM, 22-Aug-1995.)
⊢ (𝜃 ↔ 𝜑)    &   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜏)
Theorem	syl3an 1315	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 1316	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 1317	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	syld3an3 1318	A syllogism inference. (Contributed by NM, 20-May-2007.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜏)
Theorem	syld3an1 1319	A syllogism inference. (Contributed by NM, 7-Jul-2008.)
⊢ ((𝜒 ∧ 𝜓 ∧ 𝜃) → 𝜑)    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜒 ∧ 𝜓 ∧ 𝜃) → 𝜏)
Theorem	syld3an2 1320	A syllogism inference. (Contributed by NM, 20-May-2007.)
⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜓)    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏)
Theorem	syl3anl1 1321	A syllogism inference. (Contributed by NM, 24-Feb-2005.)
⊢ (𝜑 → 𝜓)    &   ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂)    ⇒   ⊢ (((𝜑 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂)
Theorem	syl3anl2 1322	A syllogism inference. (Contributed by NM, 24-Feb-2005.)
⊢ (𝜑 → 𝜒)    &   ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂)    ⇒   ⊢ (((𝜓 ∧ 𝜑 ∧ 𝜃) ∧ 𝜏) → 𝜂)
Theorem	syl3anl3 1323	A syllogism inference. (Contributed by NM, 24-Feb-2005.)
⊢ (𝜑 → 𝜃)    &   ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂)    ⇒   ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜑) ∧ 𝜏) → 𝜂)
Theorem	syl3anl 1324	A triple syllogism inference. (Contributed by NM, 24-Dec-2006.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 → 𝜃)    &   ⊢ (𝜏 → 𝜂)    &   ⊢ (((𝜓 ∧ 𝜃 ∧ 𝜂) ∧ 𝜁) → 𝜎)    ⇒   ⊢ (((𝜑 ∧ 𝜒 ∧ 𝜏) ∧ 𝜁) → 𝜎)
Theorem	syl3anr1 1325	A syllogism inference. (Contributed by NM, 31-Jul-2007.)
⊢ (𝜑 → 𝜓)    &   ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜃 ∧ 𝜏)) → 𝜂)    ⇒   ⊢ ((𝜒 ∧ (𝜑 ∧ 𝜃 ∧ 𝜏)) → 𝜂)
Theorem	syl3anr2 1326	A syllogism inference. (Contributed by NM, 1-Aug-2007.)
⊢ (𝜑 → 𝜃)    &   ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜃 ∧ 𝜏)) → 𝜂)    ⇒   ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜑 ∧ 𝜏)) → 𝜂)
Theorem	syl3anr3 1327	A syllogism inference. (Contributed by NM, 23-Aug-2007.)
⊢ (𝜑 → 𝜏)    &   ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜃 ∧ 𝜏)) → 𝜂)    ⇒   ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜃 ∧ 𝜑)) → 𝜂)
Theorem	syldbl2 1328	Stacked hypotheseis implies goal. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ ((𝜑 ∧ 𝜓) → (𝜓 → 𝜃))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
Theorem	3impdi 1329	Importation inference (undistribute conjunction). (Contributed by NM, 14-Aug-1995.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	3impdir 1330	Importation inference (undistribute conjunction). (Contributed by NM, 20-Aug-1995.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜓)) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜓) → 𝜃)
Theorem	3anidm12 1331	Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.)
⊢ ((𝜑 ∧ 𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	3anidm13 1332	Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜑) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	3anidm23 1333	Inference from idempotent law for conjunction. (Contributed by NM, 1-Feb-2007.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	syl2an3an 1334	syl3an 1315 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜃 → 𝜏)    &   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜏) → 𝜂)    ⇒   ⊢ ((𝜑 ∧ 𝜃) → 𝜂)
Theorem	syl2an23an 1335	Deduction related to syl3an 1315 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ ((𝜃 ∧ 𝜑) → 𝜏)    &   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜏) → 𝜂)    ⇒   ⊢ ((𝜃 ∧ 𝜑) → 𝜂)
Theorem	3ori 1336	Infer implication from triple disjunction. (Contributed by NM, 26-Sep-2006.)
⊢ (𝜑 ∨ 𝜓 ∨ 𝜒)    ⇒   ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒)
Theorem	3jao 1337	Disjunction of 3 antecedents. (Contributed by NM, 8-Apr-1994.)
⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓)) → ((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓))
Theorem	3jaob 1338	Disjunction of 3 antecedents. (Contributed by NM, 13-Sep-2011.)
⊢ (((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓)))
Theorem	3jaoi 1339	Disjunction of 3 antecedents (inference). (Contributed by NM, 12-Sep-1995.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 → 𝜓)    &   ⊢ (𝜃 → 𝜓)    ⇒   ⊢ ((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓)
Theorem	3jaod 1340	Disjunction of 3 antecedents (deduction). (Contributed by NM, 14-Oct-2005.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜃 → 𝜒))    &   ⊢ (𝜑 → (𝜏 → 𝜒))    ⇒   ⊢ (𝜑 → ((𝜓 ∨ 𝜃 ∨ 𝜏) → 𝜒))
Theorem	3jaoian 1341	Disjunction of 3 antecedents (inference). (Contributed by NM, 14-Oct-2005.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜃 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜏 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (((𝜑 ∨ 𝜃 ∨ 𝜏) ∧ 𝜓) → 𝜒)
Theorem	3jaodan 1342	Disjunction of 3 antecedents (deduction). (Contributed by NM, 14-Oct-2005.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜃) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜏) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜃 ∨ 𝜏)) → 𝜒)
Theorem	mpjao3dan 1343	Eliminate a 3-way disjunction in a deduction. (Contributed by Thierry Arnoux, 13-Apr-2018.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜃) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜏) → 𝜒)    &   ⊢ (𝜑 → (𝜓 ∨ 𝜃 ∨ 𝜏))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	3jaao 1344	Inference conjoining and disjoining the antecedents of three implications. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜃 → (𝜏 → 𝜒))    &   ⊢ (𝜂 → (𝜁 → 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜂) → ((𝜓 ∨ 𝜏 ∨ 𝜁) → 𝜒))
Theorem	3ianorr 1345	Triple disjunction implies negated triple conjunction. (Contributed by Jim Kingdon, 23-Dec-2018.)
⊢ ((¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒) → ¬ (𝜑 ∧ 𝜓 ∧ 𝜒))
Theorem	syl3an9b 1346	Nested syllogism inference conjoining 3 dissimilar antecedents. (Contributed by NM, 1-May-1995.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜃 → (𝜒 ↔ 𝜏))    &   ⊢ (𝜂 → (𝜏 ↔ 𝜁))    ⇒   ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜂) → (𝜓 ↔ 𝜁))
Theorem	3orbi123d 1347	Deduction joining 3 equivalences to form equivalence of disjunctions. (Contributed by NM, 20-Apr-1994.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜏))    &   ⊢ (𝜑 → (𝜂 ↔ 𝜁))    ⇒   ⊢ (𝜑 → ((𝜓 ∨ 𝜃 ∨ 𝜂) ↔ (𝜒 ∨ 𝜏 ∨ 𝜁)))
Theorem	3anbi123d 1348	Deduction joining 3 equivalences to form equivalence of conjunctions. (Contributed by NM, 22-Apr-1994.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜏))    &   ⊢ (𝜑 → (𝜂 ↔ 𝜁))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜃 ∧ 𝜂) ↔ (𝜒 ∧ 𝜏 ∧ 𝜁)))
Theorem	3anbi12d 1349	Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜃 ∧ 𝜂) ↔ (𝜒 ∧ 𝜏 ∧ 𝜂)))
Theorem	3anbi13d 1350	Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜂 ∧ 𝜃) ↔ (𝜒 ∧ 𝜂 ∧ 𝜏)))
Theorem	3anbi23d 1351	Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → ((𝜂 ∧ 𝜓 ∧ 𝜃) ↔ (𝜂 ∧ 𝜒 ∧ 𝜏)))
Theorem	3anbi1d 1352	Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜃 ∧ 𝜏) ↔ (𝜒 ∧ 𝜃 ∧ 𝜏)))
Theorem	3anbi2d 1353	Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ((𝜃 ∧ 𝜓 ∧ 𝜏) ↔ (𝜃 ∧ 𝜒 ∧ 𝜏)))
Theorem	3anbi3d 1354	Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ((𝜃 ∧ 𝜏 ∧ 𝜓) ↔ (𝜃 ∧ 𝜏 ∧ 𝜒)))
Theorem	3anim123d 1355	Deduction joining 3 implications to form implication of conjunctions. (Contributed by NM, 24-Feb-2005.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜃 → 𝜏))    &   ⊢ (𝜑 → (𝜂 → 𝜁))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜃 ∧ 𝜂) → (𝜒 ∧ 𝜏 ∧ 𝜁)))
Theorem	3orim123d 1356	Deduction joining 3 implications to form implication of disjunctions. (Contributed by NM, 4-Apr-1997.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜃 → 𝜏))    &   ⊢ (𝜑 → (𝜂 → 𝜁))    ⇒   ⊢ (𝜑 → ((𝜓 ∨ 𝜃 ∨ 𝜂) → (𝜒 ∨ 𝜏 ∨ 𝜁)))
Theorem	an6 1357	Rearrangement of 6 conjuncts. (Contributed by NM, 13-Mar-1995.)
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏 ∧ 𝜂)) ↔ ((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜏) ∧ (𝜒 ∧ 𝜂)))
Theorem	3an6 1358	Analog of an4 588 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂)) ↔ ((𝜑 ∧ 𝜒 ∧ 𝜏) ∧ (𝜓 ∧ 𝜃 ∧ 𝜂)))
Theorem	3or6 1359	Analog of or4 778 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.)
⊢ (((𝜑 ∨ 𝜓) ∨ (𝜒 ∨ 𝜃) ∨ (𝜏 ∨ 𝜂)) ↔ ((𝜑 ∨ 𝜒 ∨ 𝜏) ∨ (𝜓 ∨ 𝜃 ∨ 𝜂)))
Theorem	mp3an1 1360	An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
⊢ 𝜑    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
Theorem	mp3an2 1361	An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
⊢ 𝜓    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
Theorem	mp3an3 1362	An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
⊢ 𝜒    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
Theorem	mp3an12 1363	An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜒 → 𝜃)
Theorem	mp3an13 1364	An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.)
⊢ 𝜑    &   ⊢ 𝜒    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜓 → 𝜃)
Theorem	mp3an23 1365	An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.)
⊢ 𝜓    &   ⊢ 𝜒    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	mp3an1i 1366	An inference based on modus ponens. (Contributed by NM, 5-Jul-2005.)
⊢ 𝜓    &   ⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏))    ⇒   ⊢ (𝜑 → ((𝜒 ∧ 𝜃) → 𝜏))
Theorem	mp3anl1 1367	An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
⊢ 𝜑    &   ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)    ⇒   ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)
Theorem	mp3anl2 1368	An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
⊢ 𝜓    &   ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)    ⇒   ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏)
Theorem	mp3anl3 1369	An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
⊢ 𝜒    &   ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)    ⇒   ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜃) → 𝜏)
Theorem	mp3anr1 1370	An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.)
⊢ 𝜓    &   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ (𝜒 ∧ 𝜃)) → 𝜏)
Theorem	mp3anr2 1371	An inference based on modus ponens. (Contributed by NM, 24-Nov-2006.)
⊢ 𝜒    &   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜃)) → 𝜏)
Theorem	mp3anr3 1372	An inference based on modus ponens. (Contributed by NM, 19-Oct-2007.)
⊢ 𝜃    &   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜏)
Theorem	mp3an 1373	An inference based on modus ponens. (Contributed by NM, 14-May-1999.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ 𝜒    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ 𝜃
Theorem	mpd3an3 1374	An inference based on modus ponens. (Contributed by NM, 8-Nov-2007.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
Theorem	mpd3an23 1375	An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	mp3and 1376	A deduction based on modus ponens. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → 𝜃)    &   ⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏))    ⇒   ⊢ (𝜑 → 𝜏)
Theorem	mp3an12i 1377	mp3an 1373 with antecedents in standard conjunction form and with one hypothesis an implication. (Contributed by Alan Sare, 28-Aug-2016.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ (𝜒 → 𝜃)    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)    ⇒   ⊢ (𝜒 → 𝜏)
Theorem	mp3an2i 1378	mp3an 1373 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.)
⊢ 𝜑    &   ⊢ (𝜓 → 𝜒)    &   ⊢ (𝜓 → 𝜃)    &   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ (𝜓 → 𝜏)
Theorem	mp3an3an 1379	mp3an 1373 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.)
⊢ 𝜑    &   ⊢ (𝜓 → 𝜒)    &   ⊢ (𝜃 → 𝜏)    &   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂)    ⇒   ⊢ ((𝜓 ∧ 𝜃) → 𝜂)
Theorem	mp3an2ani 1380	An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
⊢ 𝜑    &   ⊢ (𝜓 → 𝜒)    &   ⊢ ((𝜓 ∧ 𝜃) → 𝜏)    &   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂)    ⇒   ⊢ ((𝜓 ∧ 𝜃) → 𝜂)
Theorem	biimp3a 1381	Infer implication from a logical equivalence. Similar to biimpa 296. (Contributed by NM, 4-Sep-2005.)
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	biimp3ar 1382	Infer implication from a logical equivalence. Similar to biimpar 297. (Contributed by NM, 2-Jan-2009.)
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜒)
Theorem	3anandis 1383	Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 18-Apr-2007.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜃)) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏)
Theorem	3anandirs 1384	Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 25-Jul-2006.) (Revised by NM, 18-Apr-2007.)
⊢ (((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜃) ∧ (𝜒 ∧ 𝜃)) → 𝜏)    ⇒   ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)
Theorem	ecased 1385	Deduction form of disjunctive syllogism. (Contributed by Jim Kingdon, 9-Dec-2017.)
⊢ (𝜑 → ¬ 𝜒)    &   ⊢ (𝜑 → (𝜓 ∨ 𝜒))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	ecase23d 1386	Variation of ecased 1385 with three disjuncts instead of two. (Contributed by NM, 22-Apr-1994.) (Revised by Jim Kingdon, 9-Dec-2017.)
⊢ (𝜑 → ¬ 𝜒)    &   ⊢ (𝜑 → ¬ 𝜃)    &   ⊢ (𝜑 → (𝜓 ∨ 𝜒 ∨ 𝜃))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	ecase2d 1387	Deduction for elimination by cases. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Sep-2024.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → ¬ (𝜓 ∧ 𝜒))    &   ⊢ (𝜑 → ¬ (𝜓 ∧ 𝜃))    &   ⊢ (𝜑 → (𝜏 ∨ (𝜒 ∨ 𝜃)))    ⇒   ⊢ (𝜑 → 𝜏)
Theorem	3bior1fd 1388	A disjunction is equivalent to a threefold disjunction with single falsehood, analogous to biorf 751. (Contributed by Alexander van der Vekens, 8-Sep-2017.)
⊢ (𝜑 → ¬ 𝜃)    ⇒   ⊢ (𝜑 → ((𝜒 ∨ 𝜓) ↔ (𝜃 ∨ 𝜒 ∨ 𝜓)))
Theorem	3bior1fand 1389	A disjunction is equivalent to a threefold disjunction with single falsehood of a conjunction. (Contributed by Alexander van der Vekens, 8-Sep-2017.)
⊢ (𝜑 → ¬ 𝜃)    ⇒   ⊢ (𝜑 → ((𝜒 ∨ 𝜓) ↔ ((𝜃 ∧ 𝜏) ∨ 𝜒 ∨ 𝜓)))
Theorem	3bior2fd 1390	A wff is equivalent to its threefold disjunction with double falsehood, analogous to biorf 751. (Contributed by Alexander van der Vekens, 8-Sep-2017.)
⊢ (𝜑 → ¬ 𝜃)    &   ⊢ (𝜑 → ¬ 𝜒)    ⇒   ⊢ (𝜑 → (𝜓 ↔ (𝜃 ∨ 𝜒 ∨ 𝜓)))
Theorem	3biant1d 1391	A conjunction is equivalent to a threefold conjunction with single truth, analogous to biantrud 304. (Contributed by Alexander van der Vekens, 26-Sep-2017.)
⊢ (𝜑 → 𝜃)    ⇒   ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜃 ∧ 𝜒 ∧ 𝜓)))
Theorem	intn3an1d 1392	Introduction of a triple conjunct inside a contradiction. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → ¬ (𝜓 ∧ 𝜒 ∧ 𝜃))
Theorem	intn3an2d 1393	Introduction of a triple conjunct inside a contradiction. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → ¬ (𝜒 ∧ 𝜓 ∧ 𝜃))
Theorem	intn3an3d 1394	Introduction of a triple conjunct inside a contradiction. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → ¬ (𝜒 ∧ 𝜃 ∧ 𝜓))
1.2.13  True and false constants
1.2.13.1  Universal quantifier for use by df-tru Even though it is not ordinarily part of propositional calculus, the universal quantifier ∀ is introduced here so that the soundness of Definition df-tru 1400 can be checked by the same algorithm that is used for predicate calculus. Its first real use is in Axiom ax-5 1495 in the predicate calculus section below. For those who want propositional calculus to be self-contained, i.e., to use wff variables only, the alternate Definition dftru2 1405 may be adopted and this subsection moved down to the start of the subsection with wex 1540 below. However, the use of dftru2 1405 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.
Syntax	wal 1395	Extend wff definition to include the universal quantifier ("for all"). ∀𝑥𝜑 is read "𝜑 (phi) is true for all 𝑥". Typically, in its final application 𝜑 would be replaced with a wff containing a (free) occurrence of the variable 𝑥, for example 𝑥 = 𝑦. In a universe with a finite number of objects, "for all" is equivalent to a big conjunction (AND) with one wff for each possible case of 𝑥. When the universe is infinite (as with set theory), such a propositional-calculus equivalent is not possible because an infinitely long formula has no meaning, but conceptually the idea is the same.
wff ∀𝑥𝜑
1.2.13.2  Equality predicate for use by df-tru Even though it is not ordinarily part of propositional calculus, the equality predicate = is introduced here so that the soundness of definition df-tru 1400 can be checked by the same algorithm as is used for predicate calculus. Its first real use is in Axiom ax-8 1552 in the predicate calculus section below. For those who want propositional calculus to be self-contained, i.e., to use wff variables only, the alternate definition dftru2 1405 may be adopted and this subsection moved down to just above weq 1551 below. However, the use of dftru2 1405 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.
Syntax	cv 1396	This syntax construction states that a variable 𝑥, which has been declared to be a setvar variable by $f statement vx, is also a class expression. This can be justified informally as follows. We know that the class builder {𝑦 ∣ 𝑦 ∈ 𝑥} is a class by cab 2217. Since (when 𝑦 is distinct from 𝑥) we have 𝑥 = {𝑦 ∣ 𝑦 ∈ 𝑥} by cvjust 2226, we can argue that the syntax "class 𝑥 " can be viewed as an abbreviation for "class {𝑦 ∣ 𝑦 ∈ 𝑥}". See the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class." While it is tempting and perhaps occasionally useful to view cv 1396 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1396 is intrinsically no different from any other class-building syntax such as cab 2217, cun 3198, or c0 3494. For a general discussion of the theory of classes and the role of cv 1396, see https://us.metamath.org/mpeuni/mmset.html#class 1396. (The description above applies to set theory, not predicate calculus. The purpose of introducing class 𝑥 here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1551 of predicate calculus from the wceq 1397 of set theory, so that we don't overload the = connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers.)
class 𝑥
Syntax	wceq 1397	Extend wff definition to include class equality. For a general discussion of the theory of classes, see https://us.metamath.org/mpeuni/mmset.html#class. (The purpose of introducing wff 𝐴 = 𝐵 here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1551 of predicate calculus in terms of the wceq 1397 of set theory, so that we don't "overload" the = connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. For example, some parsers - although not the Metamath program - stumble on the fact that the = in 𝑥 = 𝑦 could be the = of either weq 1551 or wceq 1397, although mathematically it makes no difference. The class variables 𝐴 and 𝐵 are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-cleq 2224 for more information on the set theory usage of wceq 1397.)
wff 𝐴 = 𝐵
1.2.13.3  Define the true and false constants
Syntax	wtru 1398	⊤ is a wff.
wff ⊤
Theorem	trujust 1399	Soundness justification theorem for df-tru 1400. (Contributed by Mario Carneiro, 17-Nov-2013.) (Revised by NM, 11-Jul-2019.)
⊢ ((∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥) ↔ (∀𝑦 𝑦 = 𝑦 → ∀𝑦 𝑦 = 𝑦))
Definition	df-tru 1400	Definition of the truth value "true", or "verum", denoted by ⊤. This is a tautology, as proved by tru 1401. In this definition, an instance of id 19 is used as the definiens, although any tautology, such as an axiom, can be used in its place. This particular id 19 instance was chosen so this definition can be checked by the same algorithm that is used for predicate calculus. This definition should be referenced directly only by tru 1401, and other proofs should depend on tru 1401 (directly or indirectly) instead of this definition, since there are many alternate ways to define ⊤. (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by NM, 11-Jul-2019.) (New usage is discouraged.)
⊢ (⊤ ↔ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))