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Theorem natded 21699
Description: Here are typical natural deduction (ND) rules in the style of Gentzen and Jaśkowski, along with MPE translations of them. This also shows the recommended theorems when you find yourself needing these rules (the recommendations encourage a slightly different proof style that works more naturally with metamath). A decent list of the standard rules of natural deduction can be found beginning with definition /\I in [Pfenning] p. 18. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. Many more citations could be added.

NameNatural Deduction RuleTranslation RecommendationComments
IT  _G |-  ps =>  _G |-  ps idi 2 nothing Reiteration is always redundant in Metamath. Definition "new rule" in [Pfenning] p. 18, definition IT in [Clemente] p. 10.
 /\I  _G |-  ps &  _G |-  ch =>  _G |-  ps  /\  ch jca 519 jca 519, pm3.2i 442 Definition  /\I in [Pfenning] p. 18, definition I /\m,n in [Clemente] p. 10, and definition  /\I in [Indrzejczak] p. 34 (representing both Gentzen's system NK and Jaśkowski)
 /\EL  _G |-  ps  /\  ch =>  _G |-  ps simpld 446 simpld 446, adantr 452 Definition  /\EL in [Pfenning] p. 18, definition E /\(1) in [Clemente] p. 11, and definition  /\E in [Indrzejczak] p. 34 (representing both Gentzen's system NK and Jaśkowski)
 /\ER  _G |-  ps  /\  ch =>  _G |-  ch simprd 450 simpr 448, adantl 453 Definition  /\ER in [Pfenning] p. 18, definition E /\(2) in [Clemente] p. 11, and definition  /\E in [Indrzejczak] p. 34 (representing both Gentzen's system NK and Jaśkowski)
 ->I  _G ,  ps |-  ch =>  _G |-  ps  ->  ch ex 424 ex 424 Definition  ->I in [Pfenning] p. 18, definition I=>m,n in [Clemente] p. 11, and definition  ->I in [Indrzejczak] p. 33.
 ->E  _G |-  ps  ->  ch &  _G |-  ps =>  _G |-  ch mpd 15 ax-mp 8, mpd 15, mpdan 650, imp 419 Definition  ->E in [Pfenning] p. 18, definition E=>m,n in [Clemente] p. 11, and definition  ->E in [Indrzejczak] p. 33.
 \/IL  _G |-  ps =>  _G |-  ps  \/  ch olcd 383 olc 374, olci 381, olcd 383 Definition  \/I in [Pfenning] p. 18, definition I \/n(1) in [Clemente] p. 12
 \/IR  _G |-  ch =>  _G |-  ps  \/  ch orcd 382 orc 375, orci 380, orcd 382 Definition  \/IR in [Pfenning] p. 18, definition I \/n(2) in [Clemente] p. 12.
 \/E  _G |-  ps  \/  ch &  _G ,  ps |-  th &  _G ,  ch |-  th =>  _G |-  th mpjaodan 762 mpjaodan 762, jaodan 761, jaod 370 Definition  \/E in [Pfenning] p. 18, definition E \/m,n,p in [Clemente] p. 12.
 -.I  _G ,  ps |-  F. =>  _G |-  -.  ps inegd 1342 pm2.01d 163
 -.I  _G ,  ps |-  th &  _G |-  -.  th =>  _G |-  -.  ps mtand 641 mtand 641 definition I -.m,n,p in [Clemente] p. 13.
 -.I  _G ,  ps |-  ch &  _G ,  ps |-  -.  ch =>  _G |-  -.  ps pm2.65da 560 pm2.65da 560 Contradiction.
 -.I  _G ,  ps |-  -.  ps =>  _G |-  -.  ps pm2.01da 430 pm2.01d 163, pm2.65da 560, pm2.65d 168 For an alternative falsum-free natural deduction ruleset
 -.E  _G |-  ps &  _G |-  -.  ps =>  _G |-  F. pm2.21fal 1344 pm2.21dd 101
 -.E  _G ,  -.  ps |-  F. =>  _G |-  ps pm2.21dd 101 definition  ->E in [Indrzejczak] p. 33.
 -.E  _G |-  ps &  _G |-  -.  ps =>  _G |-  th pm2.21dd 101 pm2.21dd 101, pm2.21d 100, pm2.21 102 For an alternative falsum-free natural deduction ruleset. Definition  -.E in [Pfenning] p. 18.
 T.I  _G |-  T. a1tru 1339 tru 1330, a1tru 1339, trud 1332 Definition  T.I in [Pfenning] p. 18.
 F.E  _G ,  F.  |-  th falimd 1338 falim 1337 Definition  F.E in [Pfenning] p. 18.
 A.I  _G |-  [ a  /  x ] ps =>  _G |-  A. x ps alrimiv 1641 alrimiv 1641, ralrimiva 2781 Definition  A.Ia in [Pfenning] p. 18, definition I A.n in [Clemente] p. 32.
 A.E  _G |-  A. x ps =>  _G |-  [ t  /  x ] ps spsbcd 3166 spcv 3034, rspcv 3040 Definition  A.E in [Pfenning] p. 18, definition E A.n,t in [Clemente] p. 32.
 E.I  _G |-  [ t  /  x ] ps =>  _G |-  E. x ps spesbcd 3235 spcev 3035, rspcev 3044 Definition  E.I in [Pfenning] p. 18, definition I E.n,t in [Clemente] p. 32.
 E.E  _G |-  E. x ps &  _G ,  [ a  /  x ] ps |-  th =>  _G |-  th exlimddv 1648 exlimddv 1648, exlimdd 1912, exlimdv 1646, rexlimdva 2822 Definition  E.Ea,u in [Pfenning] p. 18, definition E E.m,n,p,a in [Clemente] p. 32.
 F.C  _G ,  -.  ps |-  F. =>  _G |-  ps efald 1343 efald 1343 Proof by contradiction (classical logic), definition  F.C in [Pfenning] p. 17.
 F.C  _G ,  -.  ps |-  ps =>  _G |-  ps pm2.18da 431 pm2.18da 431, pm2.18d 105, pm2.18 104 For an alternative falsum-free natural deduction ruleset
 -.  -.C  _G |-  -.  -.  ps =>  _G |-  ps notnotrd 107 notnotrd 107, notnot2 106 Double negation rule (classical logic), definition NNC in [Pfenning] p. 17, definition E -.n in [Clemente] p. 14.
EM  _G |-  ps  \/  -.  ps exmidd 406 exmid 405 Excluded middle (classical logic), definition XM in [Pfenning] p. 17, proof 5.11 in [Clemente] p. 14.
 =I  _G |-  A  =  A eqidd 2436 eqid 2435, eqidd 2436 Introduce equality, definition =I in [Pfenning] p. 127.
 =E  _G |-  A  =  B &  _G [. A  /  x ]. ps =>  _G |-  [. B  /  x ]. ps sbceq1dd 3159 sbceq1d 3158, equality theorems Eliminate equality, definition =E in [Pfenning] p. 127. (Both E1 and E2.)

Note that MPE uses classical logic, not intuitionist logic. As is conventional, the "I" rules are introduction rules, "E" rules are elimination rules, the "C" rules are conversion rules, and  _G represents the set of (current) hypotheses. We use wff variable names beginning with  ps to provide a closer representation of the Metamath equivalents (which typically use the antedent  ph to represent the context  _G).

Most of this information was developed by Mario Carneiro and posted on 3-Feb-2017. For more information, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer.

For annotated examples where some traditional ND rules are directly applied in MPE, see ex-natded5.2 21700, ex-natded5.3 21703, ex-natded5.5 21706, ex-natded5.7 21707, ex-natded5.8 21709, ex-natded5.13 21711, ex-natded9.20 21713, and ex-natded9.26 21715.

(Contributed by DAW, 4-Feb-2017.)

Hypothesis
Ref Expression
natded.1  |-  ph
Assertion
Ref Expression
natded  |-  ph

Proof of Theorem natded
StepHypRef Expression
1 natded.1 1  |-  ph
Colors of variables: wff set class
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