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	syl3anr1 1301	A syllogism inference. (Contributed by NM, 31-Jul-2007.)
⊢ (𝜑 → 𝜓)    &   ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜃 ∧ 𝜏)) → 𝜂)    ⇒   ⊢ ((𝜒 ∧ (𝜑 ∧ 𝜃 ∧ 𝜏)) → 𝜂)
Theorem	syl3anr2 1302	A syllogism inference. (Contributed by NM, 1-Aug-2007.)
⊢ (𝜑 → 𝜃)    &   ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜃 ∧ 𝜏)) → 𝜂)    ⇒   ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜑 ∧ 𝜏)) → 𝜂)
Theorem	syl3anr3 1303	A syllogism inference. (Contributed by NM, 23-Aug-2007.)
⊢ (𝜑 → 𝜏)    &   ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜃 ∧ 𝜏)) → 𝜂)    ⇒   ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜃 ∧ 𝜑)) → 𝜂)
Theorem	3impdi 1304	Importation inference (undistribute conjunction). (Contributed by NM, 14-Aug-1995.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	3impdir 1305	Importation inference (undistribute conjunction). (Contributed by NM, 20-Aug-1995.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜓)) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜓) → 𝜃)
Theorem	3anidm12 1306	Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.)
⊢ ((𝜑 ∧ 𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	3anidm13 1307	Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜑) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	3anidm23 1308	Inference from idempotent law for conjunction. (Contributed by NM, 1-Feb-2007.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	syl2an3an 1309	syl3an 1291 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 1310	Deduction related to syl3an 1291 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 1311	Infer implication from triple disjunction. (Contributed by NM, 26-Sep-2006.)
⊢ (𝜑 ∨ 𝜓 ∨ 𝜒)    ⇒   ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒)
Theorem	3jao 1312	Disjunction of 3 antecedents. (Contributed by NM, 8-Apr-1994.)
⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓)) → ((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓))
Theorem	3jaob 1313	Disjunction of 3 antecedents. (Contributed by NM, 13-Sep-2011.)
⊢ (((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓)))
Theorem	3jaoi 1314	Disjunction of 3 antecedents (inference). (Contributed by NM, 12-Sep-1995.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 → 𝜓)    &   ⊢ (𝜃 → 𝜓)    ⇒   ⊢ ((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓)
Theorem	3jaod 1315	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 1316	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 1317	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 1318	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 1319	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 1320	Triple disjunction implies negated triple conjunction. (Contributed by Jim Kingdon, 23-Dec-2018.)
⊢ ((¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒) → ¬ (𝜑 ∧ 𝜓 ∧ 𝜒))
Theorem	syl3an9b 1321	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 1322	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 1323	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 1324	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 1325	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 1326	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 1327	Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜃 ∧ 𝜏) ↔ (𝜒 ∧ 𝜃 ∧ 𝜏)))
Theorem	3anbi2d 1328	Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ((𝜃 ∧ 𝜓 ∧ 𝜏) ↔ (𝜃 ∧ 𝜒 ∧ 𝜏)))
Theorem	3anbi3d 1329	Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ((𝜃 ∧ 𝜏 ∧ 𝜓) ↔ (𝜃 ∧ 𝜏 ∧ 𝜒)))
Theorem	3anim123d 1330	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 1331	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 1332	Rearrangement of 6 conjuncts. (Contributed by NM, 13-Mar-1995.)
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏 ∧ 𝜂)) ↔ ((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜏) ∧ (𝜒 ∧ 𝜂)))
Theorem	3an6 1333	Analog of an4 586 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 1334	Analog of or4 772 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.)
⊢ (((𝜑 ∨ 𝜓) ∨ (𝜒 ∨ 𝜃) ∨ (𝜏 ∨ 𝜂)) ↔ ((𝜑 ∨ 𝜒 ∨ 𝜏) ∨ (𝜓 ∨ 𝜃 ∨ 𝜂)))
Theorem	mp3an1 1335	An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
⊢ 𝜑    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
Theorem	mp3an2 1336	An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
⊢ 𝜓    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
Theorem	mp3an3 1337	An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
⊢ 𝜒    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
Theorem	mp3an12 1338	An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜒 → 𝜃)
Theorem	mp3an13 1339	An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.)
⊢ 𝜑    &   ⊢ 𝜒    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜓 → 𝜃)
Theorem	mp3an23 1340	An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.)
⊢ 𝜓    &   ⊢ 𝜒    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	mp3an1i 1341	An inference based on modus ponens. (Contributed by NM, 5-Jul-2005.)
⊢ 𝜓    &   ⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏))    ⇒   ⊢ (𝜑 → ((𝜒 ∧ 𝜃) → 𝜏))
Theorem	mp3anl1 1342	An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
⊢ 𝜑    &   ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)    ⇒   ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)
Theorem	mp3anl2 1343	An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
⊢ 𝜓    &   ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)    ⇒   ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏)
Theorem	mp3anl3 1344	An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
⊢ 𝜒    &   ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)    ⇒   ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜃) → 𝜏)
Theorem	mp3anr1 1345	An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.)
⊢ 𝜓    &   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ (𝜒 ∧ 𝜃)) → 𝜏)
Theorem	mp3anr2 1346	An inference based on modus ponens. (Contributed by NM, 24-Nov-2006.)
⊢ 𝜒    &   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜃)) → 𝜏)
Theorem	mp3anr3 1347	An inference based on modus ponens. (Contributed by NM, 19-Oct-2007.)
⊢ 𝜃    &   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜏)
Theorem	mp3an 1348	An inference based on modus ponens. (Contributed by NM, 14-May-1999.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ 𝜒    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ 𝜃
Theorem	mpd3an3 1349	An inference based on modus ponens. (Contributed by NM, 8-Nov-2007.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
Theorem	mpd3an23 1350	An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	mp3and 1351	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 1352	mp3an 1348 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 1353	mp3an 1348 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 1354	mp3an 1348 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 1355	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 1356	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 1357	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 1358	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 1359	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 1360	Deduction form of disjunctive syllogism. (Contributed by Jim Kingdon, 9-Dec-2017.)
⊢ (𝜑 → ¬ 𝜒)    &   ⊢ (𝜑 → (𝜓 ∨ 𝜒))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	ecase23d 1361	Variation of ecased 1360 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)
1.2.12  True and false constants
1.2.12.1  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 1367 can be checked by the same algorithm that is used for predicate calculus. Its first real use is in Axiom ax-5 1458 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 1372 may be adopted and this subsection moved down to the start of the subsection with wex 1503 below. However, the use of dftru2 1372 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.
Syntax	wal 1362	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
1.2.12.2  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 1367 can be checked by the same algorithm as is used for predicate calculus. Its first real use is in Axiom ax-8 1515 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 1372 may be adopted and this subsection moved down to just above weq 1514 below. However, the use of dftru2 1372 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.
Syntax	cv 1363	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 2175. Since (when 𝑦 is distinct from 𝑥) we have 𝑥 = {𝑦 ∣ 𝑦 ∈ 𝑥} by cvjust 2184, 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 1363 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1363 is intrinsically no different from any other class-building syntax such as cab 2175, cun 3142, or c0 3437. For a general discussion of the theory of classes and the role of cv 1363, see https://us.metamath.org/mpeuni/mmset.html#class 1363. (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 1514 of predicate calculus from the wceq 1364 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 1364	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 1514 of predicate calculus in terms of the wceq 1364 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 1514 or wceq 1364, 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 2182 for more information on the set theory usage of wceq 1364.)
wff 𝐴 = 𝐵
1.2.12.3  Define the true and false constants
Syntax	wtru 1365	⊤ is a wff.
wff ⊤
Theorem	trujust 1366	Soundness justification theorem for df-tru 1367. (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 1367	Definition of the truth value "true", or "verum", denoted by ⊤. This is a tautology, as proved by tru 1368. 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 1368, and other proofs should depend on tru 1368 (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))
Theorem	tru 1368	The truth value ⊤ is provable. (Contributed by Anthony Hart, 13-Oct-2010.)
⊢ ⊤
Syntax	wfal 1369	⊥ is a wff.
wff ⊥
Definition	df-fal 1370	Definition of the truth value "false", or "falsum", denoted by ⊥. See also df-tru 1367. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ (⊥ ↔ ¬ ⊤)
Theorem	fal 1371	The truth value ⊥ is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.)
⊢ ¬ ⊥
Theorem	dftru2 1372	An alternate definition of "true". (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by BJ, 12-Jul-2019.) (New usage is discouraged.)
⊢ (⊤ ↔ (𝜑 → 𝜑))
Theorem	mptru 1373	Eliminate ⊤ as an antecedent. A proposition implied by ⊤ is true. (Contributed by Mario Carneiro, 13-Mar-2014.)
⊢ (⊤ → 𝜑)    ⇒   ⊢ 𝜑
Theorem	tbtru 1374	A proposition is equivalent to itself being equivalent to ⊤. (Contributed by Anthony Hart, 14-Aug-2011.)
⊢ (𝜑 ↔ (𝜑 ↔ ⊤))
Theorem	nbfal 1375	The negation of a proposition is equivalent to itself being equivalent to ⊥. (Contributed by Anthony Hart, 14-Aug-2011.)
⊢ (¬ 𝜑 ↔ (𝜑 ↔ ⊥))
Theorem	bitru 1376	A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.)
⊢ 𝜑    ⇒   ⊢ (𝜑 ↔ ⊤)
Theorem	bifal 1377	A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.)
⊢ ¬ 𝜑    ⇒   ⊢ (𝜑 ↔ ⊥)
Theorem	falim 1378	The truth value ⊥ implies anything. Also called the principle of explosion, or "ex falso quodlibet". (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
⊢ (⊥ → 𝜑)
Theorem	falimd 1379	The truth value ⊥ implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ ((𝜑 ∧ ⊥) → 𝜓)
Theorem	trud 1380	Anything implies ⊤. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
⊢ (𝜑 → ⊤)
Theorem	truan 1381	True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Wolf Lammen, 21-Jul-2019.)
⊢ ((⊤ ∧ 𝜑) ↔ 𝜑)
Theorem	dfnot 1382	Given falsum, we can define the negation of a wff 𝜑 as the statement that a contradiction follows from assuming 𝜑. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 21-Jul-2019.)
⊢ (¬ 𝜑 ↔ (𝜑 → ⊥))
Theorem	inegd 1383	Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ ((𝜑 ∧ 𝜓) → ⊥)    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	pm2.21fal 1384	If a wff and its negation are provable, then falsum is provable. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → ⊥)
Theorem	pclem6 1385	Negation inferred from embedded conjunct. (Contributed by NM, 20-Aug-1993.) (Proof rewritten by Jim Kingdon, 4-May-2018.)
⊢ ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → ¬ 𝜓)
1.2.13  Logical 'xor'
Syntax	wxo 1386	Extend wff definition to include exclusive disjunction ('xor').
wff (𝜑 ⊻ 𝜓)
Definition	df-xor 1387	Define exclusive disjunction (logical 'xor'). Return true if either the left or right, but not both, are true. Contrast with ∧ (wa 104), ∨ (wo 709), and → (wi 4) . (Contributed by FL, 22-Nov-2010.) (Modified by Jim Kingdon, 1-Mar-2018.)
⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
Theorem	xoranor 1388	One way of defining exclusive or. Equivalent to df-xor 1387. (Contributed by Jim Kingdon and Mario Carneiro, 1-Mar-2018.)
⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜓)))
Theorem	excxor 1389	This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.) (Proof rewritten by Jim Kingdon, 5-May-2018.)
⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑 ∧ 𝜓)))
Theorem	xoror 1390	XOR implies OR. (Contributed by BJ, 19-Apr-2019.)
⊢ ((𝜑 ⊻ 𝜓) → (𝜑 ∨ 𝜓))
Theorem	xorbi2d 1391	Deduction joining an equivalence and a left operand to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ((𝜃 ⊻ 𝜓) ↔ (𝜃 ⊻ 𝜒)))
Theorem	xorbi1d 1392	Deduction joining an equivalence and a right operand to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ((𝜓 ⊻ 𝜃) ↔ (𝜒 ⊻ 𝜃)))
Theorem	xorbi12d 1393	Deduction joining two equivalences to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → ((𝜓 ⊻ 𝜃) ↔ (𝜒 ⊻ 𝜏)))
Theorem	xorbi12i 1394	Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ ((𝜑 ⊻ 𝜒) ↔ (𝜓 ⊻ 𝜃))
Theorem	xorbin 1395	A consequence of exclusive or. In classical logic the converse also holds. (Contributed by Jim Kingdon, 8-Mar-2018.)
⊢ ((𝜑 ⊻ 𝜓) → (𝜑 ↔ ¬ 𝜓))
Theorem	pm5.18im 1396	One direction of pm5.18dc 884, which holds for all propositions, not just decidable propositions. (Contributed by Jim Kingdon, 10-Mar-2018.)
⊢ ((𝜑 ↔ 𝜓) → ¬ (𝜑 ↔ ¬ 𝜓))
Theorem	xornbi 1397	A consequence of exclusive or. For decidable propositions this is an equivalence, as seen at xornbidc 1402. (Contributed by Jim Kingdon, 10-Mar-2018.)
⊢ ((𝜑 ⊻ 𝜓) → ¬ (𝜑 ↔ 𝜓))
Theorem	xor3dc 1398	Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 12-Apr-2018.)
⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))))
Theorem	xorcom 1399	⊻ is commutative. (Contributed by David A. Wheeler, 6-Oct-2018.)
⊢ ((𝜑 ⊻ 𝜓) ↔ (𝜓 ⊻ 𝜑))
Theorem	pm5.15dc 1400	A decidable proposition is equivalent to a decidable proposition or its negation. Based on theorem *5.15 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.)
⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ 𝜓) ∨ (𝜑 ↔ ¬ 𝜓))))