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Theorem List for Metamath Proof Explorer - 1501-1600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsyl32anc 1501 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂)) → 𝜁)       (𝜑𝜁)
 
Theoremsyl122anc 1502 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   ((𝜓 ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → 𝜁)       (𝜑𝜁)
 
Theoremsyl212anc 1503 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (((𝜓𝜒) ∧ 𝜃 ∧ (𝜏𝜂)) → 𝜁)       (𝜑𝜁)
 
Theoremsyl221anc 1504 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (((𝜓𝜒) ∧ (𝜃𝜏) ∧ 𝜂) → 𝜁)       (𝜑𝜁)
 
Theoremsyl113anc 1505 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   ((𝜓𝜒 ∧ (𝜃𝜏𝜂)) → 𝜁)       (𝜑𝜁)
 
Theoremsyl131anc 1506 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   ((𝜓 ∧ (𝜒𝜃𝜏) ∧ 𝜂) → 𝜁)       (𝜑𝜁)
 
Theoremsyl311anc 1507 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (((𝜓𝜒𝜃) ∧ 𝜏𝜂) → 𝜁)       (𝜑𝜁)
 
Theoremsyl33anc 1508 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁)) → 𝜎)       (𝜑𝜎)
 
Theoremsyl222anc 1509 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (((𝜓𝜒) ∧ (𝜃𝜏) ∧ (𝜂𝜁)) → 𝜎)       (𝜑𝜎)
 
Theoremsyl123anc 1510 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   ((𝜓 ∧ (𝜒𝜃) ∧ (𝜏𝜂𝜁)) → 𝜎)       (𝜑𝜎)
 
Theoremsyl132anc 1511 Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   ((𝜓 ∧ (𝜒𝜃𝜏) ∧ (𝜂𝜁)) → 𝜎)       (𝜑𝜎)
 
Theoremsyl213anc 1512 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (((𝜓𝜒) ∧ 𝜃 ∧ (𝜏𝜂𝜁)) → 𝜎)       (𝜑𝜎)
 
Theoremsyl231anc 1513 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (((𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ 𝜁) → 𝜎)       (𝜑𝜎)
 
Theoremsyl312anc 1514 Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (((𝜓𝜒𝜃) ∧ 𝜏 ∧ (𝜂𝜁)) → 𝜎)       (𝜑𝜎)
 
Theoremsyl321anc 1515 Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂) ∧ 𝜁) → 𝜎)       (𝜑𝜎)
 
Theoremsyl133anc 1516 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   ((𝜓 ∧ (𝜒𝜃𝜏) ∧ (𝜂𝜁𝜎)) → 𝜌)       (𝜑𝜌)
 
Theoremsyl313anc 1517 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (((𝜓𝜒𝜃) ∧ 𝜏 ∧ (𝜂𝜁𝜎)) → 𝜌)       (𝜑𝜌)
 
Theoremsyl331anc 1518 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁) ∧ 𝜎) → 𝜌)       (𝜑𝜌)
 
Theoremsyl223anc 1519 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (((𝜓𝜒) ∧ (𝜃𝜏) ∧ (𝜂𝜁𝜎)) → 𝜌)       (𝜑𝜌)
 
Theoremsyl232anc 1520 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (((𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ (𝜁𝜎)) → 𝜌)       (𝜑𝜌)
 
Theoremsyl322anc 1521 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂) ∧ (𝜁𝜎)) → 𝜌)       (𝜑𝜌)
 
Theoremsyl233anc 1522 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (𝜑𝜌)    &   (((𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ (𝜁𝜎𝜌)) → 𝜇)       (𝜑𝜇)
 
Theoremsyl323anc 1523 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (𝜑𝜌)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂) ∧ (𝜁𝜎𝜌)) → 𝜇)       (𝜑𝜇)
 
Theoremsyl332anc 1524 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (𝜑𝜌)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁) ∧ (𝜎𝜌)) → 𝜇)       (𝜑𝜇)
 
Theoremsyl333anc 1525 A syllogism inference combined with contraction. (Contributed by NM, 10-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (𝜑𝜌)    &   (𝜑𝜇)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁) ∧ (𝜎𝜌𝜇)) → 𝜆)       (𝜑𝜆)
 
Theoremsyl3an1b 1526 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜑𝜓)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜑𝜒𝜃) → 𝜏)
 
Theoremsyl3an2b 1527 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜑𝜒)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜓𝜑𝜃) → 𝜏)
 
Theoremsyl3an3b 1528 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜑𝜃)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜓𝜒𝜑) → 𝜏)
 
Theoremsyl3an1br 1529 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜓𝜑)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜑𝜒𝜃) → 𝜏)
 
Theoremsyl3an2br 1530 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜒𝜑)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜓𝜑𝜃) → 𝜏)
 
Theoremsyl3an3br 1531 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜃𝜑)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜓𝜒𝜑) → 𝜏)
 
Theoremsyld3an3 1532 A syllogism inference. (Contributed by NM, 20-May-2007.)
((𝜑𝜓𝜒) → 𝜃)    &   ((𝜑𝜓𝜃) → 𝜏)       ((𝜑𝜓𝜒) → 𝜏)
 
Theoremsyld3an1 1533 A syllogism inference. (Contributed by NM, 7-Jul-2008.) (Proof shortened by Wolf Lammen, 26-Jun-2022.)
((𝜒𝜓𝜃) → 𝜑)    &   ((𝜑𝜓𝜃) → 𝜏)       ((𝜒𝜓𝜃) → 𝜏)
 
Theoremsyld3an1OLD 1534 Obsolete version of syld3an1 1533 as of 26-Jun-2022. (Contributed by NM, 7-Jul-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜒𝜓𝜃) → 𝜑)    &   ((𝜑𝜓𝜃) → 𝜏)       ((𝜒𝜓𝜃) → 𝜏)
 
Theoremsyld3an2 1535 A syllogism inference. (Contributed by NM, 20-May-2007.)
((𝜑𝜒𝜃) → 𝜓)    &   ((𝜑𝜓𝜃) → 𝜏)       ((𝜑𝜒𝜃) → 𝜏)
 
Theoremsyld3an2OLD 1536 Obsolete version of syld3an2 1535 as of 26-Jun-2022. (Contributed by NM, 7-Jul-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜒𝜃) → 𝜓)    &   ((𝜑𝜓𝜃) → 𝜏)       ((𝜑𝜒𝜃) → 𝜏)
 
Theoremsyl3anl1 1537 A syllogism inference. (Contributed by NM, 24-Feb-2005.)
(𝜑𝜓)    &   (((𝜓𝜒𝜃) ∧ 𝜏) → 𝜂)       (((𝜑𝜒𝜃) ∧ 𝜏) → 𝜂)
 
Theoremsyl3anl2 1538 A syllogism inference. (Contributed by NM, 24-Feb-2005.) (Proof shortened by Wolf Lammen, 27-Jun-2022.)
(𝜑𝜒)    &   (((𝜓𝜒𝜃) ∧ 𝜏) → 𝜂)       (((𝜓𝜑𝜃) ∧ 𝜏) → 𝜂)
 
Theoremsyl3anl2OLD 1539 Obsolete version of syl3anl2 1538 as of 27-Jun-2022. (Contributed by NM, 24-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜒)    &   (((𝜓𝜒𝜃) ∧ 𝜏) → 𝜂)       (((𝜓𝜑𝜃) ∧ 𝜏) → 𝜂)
 
Theoremsyl3anl3 1540 A syllogism inference. (Contributed by NM, 24-Feb-2005.)
(𝜑𝜃)    &   (((𝜓𝜒𝜃) ∧ 𝜏) → 𝜂)       (((𝜓𝜒𝜑) ∧ 𝜏) → 𝜂)
 
Theoremsyl3anl 1541 A triple syllogism inference. (Contributed by NM, 24-Dec-2006.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)    &   (((𝜓𝜃𝜂) ∧ 𝜁) → 𝜎)       (((𝜑𝜒𝜏) ∧ 𝜁) → 𝜎)
 
Theoremsyl3anr1 1542 A syllogism inference. (Contributed by NM, 31-Jul-2007.)
(𝜑𝜓)    &   ((𝜒 ∧ (𝜓𝜃𝜏)) → 𝜂)       ((𝜒 ∧ (𝜑𝜃𝜏)) → 𝜂)
 
Theoremsyl3anr2 1543 A syllogism inference. (Contributed by NM, 1-Aug-2007.) (Proof shortened by Wolf Lammen, 27-Jun-2022.)
(𝜑𝜃)    &   ((𝜒 ∧ (𝜓𝜃𝜏)) → 𝜂)       ((𝜒 ∧ (𝜓𝜑𝜏)) → 𝜂)
 
Theoremsyl3anr2OLD 1544 Obsolete version of syl3anr2 1543 as of 27-Jun-2022. (Contributed by NM, 1-Aug-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜃)    &   ((𝜒 ∧ (𝜓𝜃𝜏)) → 𝜂)       ((𝜒 ∧ (𝜓𝜑𝜏)) → 𝜂)
 
Theoremsyl3anr3 1545 A syllogism inference. (Contributed by NM, 23-Aug-2007.)
(𝜑𝜏)    &   ((𝜒 ∧ (𝜓𝜃𝜏)) → 𝜂)       ((𝜒 ∧ (𝜓𝜃𝜑)) → 𝜂)
 
Theorem3anidm12 1546 Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.)
((𝜑𝜑𝜓) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theorem3anidm13 1547 Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.)
((𝜑𝜓𝜑) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theorem3anidm23 1548 Inference from idempotent law for conjunction. (Contributed by NM, 1-Feb-2007.)
((𝜑𝜓𝜓) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremsyl2an3an 1549 syl3an 1203 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜃𝜏)    &   ((𝜓𝜒𝜏) → 𝜂)       ((𝜑𝜃) → 𝜂)
 
Theoremsyl2an23an 1550 Deduction related to syl3an 1203 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.) (Proof shortened by Wolf Lammen, 28-Jun-2022.)
(𝜑𝜓)    &   (𝜑𝜒)    &   ((𝜃𝜑) → 𝜏)    &   ((𝜓𝜒𝜏) → 𝜂)       ((𝜃𝜑) → 𝜂)
 
Theoremsyl2an23anOLD 1551 Obsolete version of syl2an23an 1550 as of 28-Jun-2022. (Contributed by Alan Sare, 31-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜑𝜒)    &   ((𝜃𝜑) → 𝜏)    &   ((𝜓𝜒𝜏) → 𝜂)       ((𝜃𝜑) → 𝜂)
 
Theorem3ori 1552 Infer implication from triple disjunction. (Contributed by NM, 26-Sep-2006.)
(𝜑𝜓𝜒)       ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒)
 
Theorem3jao 1553 Disjunction of three antecedents. (Contributed by NM, 8-Apr-1994.)
(((𝜑𝜓) ∧ (𝜒𝜓) ∧ (𝜃𝜓)) → ((𝜑𝜒𝜃) → 𝜓))
 
Theorem3jaoOLD 1554 Obsolete version of 3jao 1553 as of 28-Jun-2022. (Contributed by NM, 8-Apr-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) ∧ (𝜒𝜓) ∧ (𝜃𝜓)) → ((𝜑𝜒𝜃) → 𝜓))
 
Theorem3jaob 1555 Disjunction of three antecedents. (Contributed by NM, 13-Sep-2011.)
(((𝜑𝜒𝜃) → 𝜓) ↔ ((𝜑𝜓) ∧ (𝜒𝜓) ∧ (𝜃𝜓)))
 
Theorem3jaoi 1556 Disjunction of three antecedents (inference). (Contributed by NM, 12-Sep-1995.)
(𝜑𝜓)    &   (𝜒𝜓)    &   (𝜃𝜓)       ((𝜑𝜒𝜃) → 𝜓)
 
Theorem3jaod 1557 Disjunction of three antecedents (deduction). (Contributed by NM, 14-Oct-2005.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))    &   (𝜑 → (𝜏𝜒))       (𝜑 → ((𝜓𝜃𝜏) → 𝜒))
 
Theorem3jaoian 1558 Disjunction of three antecedents (inference). (Contributed by NM, 14-Oct-2005.)
((𝜑𝜓) → 𝜒)    &   ((𝜃𝜓) → 𝜒)    &   ((𝜏𝜓) → 𝜒)       (((𝜑𝜃𝜏) ∧ 𝜓) → 𝜒)
 
Theorem3jaodan 1559 Disjunction of three antecedents (deduction). (Contributed by NM, 14-Oct-2005.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜃) → 𝜒)    &   ((𝜑𝜏) → 𝜒)       ((𝜑 ∧ (𝜓𝜃𝜏)) → 𝜒)
 
Theoremmpjao3dan 1560 Eliminate a three-way disjunction in a deduction. (Contributed by Thierry Arnoux, 13-Apr-2018.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜃) → 𝜒)    &   ((𝜑𝜏) → 𝜒)    &   (𝜑 → (𝜓𝜃𝜏))       (𝜑𝜒)
 
Theorem3jaao 1561 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 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜒))    &   (𝜂 → (𝜁𝜒))       ((𝜑𝜃𝜂) → ((𝜓𝜏𝜁) → 𝜒))
 
Theoremsyl3an9b 1562 Nested syllogism inference conjoining 3 dissimilar antecedents. (Contributed by NM, 1-May-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))    &   (𝜂 → (𝜏𝜁))       ((𝜑𝜃𝜂) → (𝜓𝜁))
 
Theorem3orbi123d 1563 Deduction joining 3 equivalences to form equivalence of disjunctions. (Contributed by NM, 20-Apr-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))    &   (𝜑 → (𝜂𝜁))       (𝜑 → ((𝜓𝜃𝜂) ↔ (𝜒𝜏𝜁)))
 
Theorem3anbi123d 1564 Deduction joining 3 equivalences to form equivalence of conjunctions. (Contributed by NM, 22-Apr-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))    &   (𝜑 → (𝜂𝜁))       (𝜑 → ((𝜓𝜃𝜂) ↔ (𝜒𝜏𝜁)))
 
Theorem3anbi12d 1565 Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃𝜂) ↔ (𝜒𝜏𝜂)))
 
Theorem3anbi13d 1566 Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜂𝜃) ↔ (𝜒𝜂𝜏)))
 
Theorem3anbi23d 1567 Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜂𝜓𝜃) ↔ (𝜂𝜒𝜏)))
 
Theorem3anbi1d 1568 Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃𝜏) ↔ (𝜒𝜃𝜏)))
 
Theorem3anbi2d 1569 Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓𝜏) ↔ (𝜃𝜒𝜏)))
 
Theorem3anbi3d 1570 Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜏𝜓) ↔ (𝜃𝜏𝜒)))
 
Theorem3anim123d 1571 Deduction joining 3 implications to form implication of conjunctions. (Contributed by NM, 24-Feb-2005.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))    &   (𝜑 → (𝜂𝜁))       (𝜑 → ((𝜓𝜃𝜂) → (𝜒𝜏𝜁)))
 
Theorem3orim123d 1572 Deduction joining 3 implications to form implication of disjunctions. (Contributed by NM, 4-Apr-1997.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))    &   (𝜑 → (𝜂𝜁))       (𝜑 → ((𝜓𝜃𝜂) → (𝜒𝜏𝜁)))
 
Theoreman6 1573 Rearrangement of 6 conjuncts. (Contributed by NM, 13-Mar-1995.)
(((𝜑𝜓𝜒) ∧ (𝜃𝜏𝜂)) ↔ ((𝜑𝜃) ∧ (𝜓𝜏) ∧ (𝜒𝜂)))
 
Theorem3an6 1574 Analogue of an4 646 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) ↔ ((𝜑𝜒𝜏) ∧ (𝜓𝜃𝜂)))
 
Theorem3or6 1575 Analogue of or4 955 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.)
(((𝜑𝜓) ∨ (𝜒𝜃) ∨ (𝜏𝜂)) ↔ ((𝜑𝜒𝜏) ∨ (𝜓𝜃𝜂)))
 
Theoremmp3an1 1576 An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
𝜑    &   ((𝜑𝜓𝜒) → 𝜃)       ((𝜓𝜒) → 𝜃)
 
Theoremmp3an2 1577 An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
𝜓    &   ((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜒) → 𝜃)
 
Theoremmp3an3 1578 An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
𝜒    &   ((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)
 
Theoremmp3an12 1579 An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)
𝜑    &   𝜓    &   ((𝜑𝜓𝜒) → 𝜃)       (𝜒𝜃)
 
Theoremmp3an13 1580 An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.)
𝜑    &   𝜒    &   ((𝜑𝜓𝜒) → 𝜃)       (𝜓𝜃)
 
Theoremmp3an23 1581 An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.)
𝜓    &   𝜒    &   ((𝜑𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
Theoremmp3an1i 1582 An inference based on modus ponens. (Contributed by NM, 5-Jul-2005.)
𝜓    &   (𝜑 → ((𝜓𝜒𝜃) → 𝜏))       (𝜑 → ((𝜒𝜃) → 𝜏))
 
Theoremmp3anl1 1583 An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
𝜑    &   (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜓𝜒) ∧ 𝜃) → 𝜏)
 
Theoremmp3anl2 1584 An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
𝜓    &   (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜑𝜒) ∧ 𝜃) → 𝜏)
 
Theoremmp3anl3 1585 An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
𝜒    &   (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜑𝜓) ∧ 𝜃) → 𝜏)
 
Theoremmp3anr1 1586 An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.)
𝜓    &   ((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)       ((𝜑 ∧ (𝜒𝜃)) → 𝜏)
 
Theoremmp3anr2 1587 An inference based on modus ponens. (Contributed by NM, 24-Nov-2006.)
𝜒    &   ((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)       ((𝜑 ∧ (𝜓𝜃)) → 𝜏)
 
Theoremmp3anr3 1588 An inference based on modus ponens. (Contributed by NM, 19-Oct-2007.)
𝜃    &   ((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)       ((𝜑 ∧ (𝜓𝜒)) → 𝜏)
 
Theoremmp3an 1589 An inference based on modus ponens. (Contributed by NM, 14-May-1999.)
𝜑    &   𝜓    &   𝜒    &   ((𝜑𝜓𝜒) → 𝜃)       𝜃
 
Theoremmpd3an3 1590 An inference based on modus ponens. (Contributed by NM, 8-Nov-2007.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)
 
Theoremmpd3an23 1591 An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.)
(𝜑𝜓)    &   (𝜑𝜒)    &   ((𝜑𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
Theoremmp3and 1592 A deduction based on modus ponens. (Contributed by Mario Carneiro, 24-Dec-2016.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑 → ((𝜓𝜒𝜃) → 𝜏))       (𝜑𝜏)
 
Theoremmp3an12i 1593 mp3an 1589 with antecedents in standard conjunction form and with one hypothesis an implication. (Contributed by Alan Sare, 28-Aug-2016.)
𝜑    &   𝜓    &   (𝜒𝜃)    &   ((𝜑𝜓𝜃) → 𝜏)       (𝜒𝜏)
 
Theoremmp3an2i 1594 mp3an 1589 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.)
𝜑    &   (𝜓𝜒)    &   (𝜓𝜃)    &   ((𝜑𝜒𝜃) → 𝜏)       (𝜓𝜏)
 
Theoremmp3an3an 1595 mp3an 1589 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.)
𝜑    &   (𝜓𝜒)    &   (𝜃𝜏)    &   ((𝜑𝜒𝜏) → 𝜂)       ((𝜓𝜃) → 𝜂)
 
Theoremmp3an2ani 1596 An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
𝜑    &   (𝜓𝜒)    &   ((𝜓𝜃) → 𝜏)    &   ((𝜑𝜒𝜏) → 𝜂)       ((𝜓𝜃) → 𝜂)
 
Theorembiimp3a 1597 Infer implication from a logical equivalence. Similar to biimpa 470. (Contributed by NM, 4-Sep-2005.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑𝜓𝜒) → 𝜃)
 
Theorembiimp3ar 1598 Infer implication from a logical equivalence. Similar to biimpar 471. (Contributed by NM, 2-Jan-2009.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑𝜓𝜃) → 𝜒)
 
Theorem3anandis 1599 Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 18-Apr-2007.)
(((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜑𝜃)) → 𝜏)       ((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)
 
Theorem3anandirs 1600 Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 25-Jul-2006.)
(((𝜑𝜃) ∧ (𝜓𝜃) ∧ (𝜒𝜃)) → 𝜏)       (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)
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