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Theorem natded 29656
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 set.mm). 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 Γ𝜓 => Γ𝜓 idi 1 nothing Reiteration is always redundant in Metamath. Definition "new rule" in [Pfenning] p. 18, definition IT in [Clemente] p. 10.
I Γ𝜓 & Γ𝜒 => Γ𝜓𝜒 jca 513 jca 513, pm3.2i 472 Definition I in [Pfenning] p. 18, definition Im,n in [Clemente] p. 10, and definition I in [Indrzejczak] p. 34 (representing both Gentzen's system NK and Jaśkowski)
EL Γ𝜓𝜒 => Γ𝜓 simpld 496 simpld 496, adantr 482 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 Γ𝜓𝜒 => Γ𝜒 simprd 497 simpr 486, adantl 483 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 Γ, 𝜓𝜒 => Γ𝜓𝜒 ex 414 ex 414 Definition I in [Pfenning] p. 18, definition I=>m,n in [Clemente] p. 11, and definition I in [Indrzejczak] p. 33.
E Γ𝜓𝜒 & Γ𝜓 => Γ𝜒 mpd 15 ax-mp 5, mpd 15, mpdan 686, imp 408 Definition E in [Pfenning] p. 18, definition E=>m,n in [Clemente] p. 11, and definition E in [Indrzejczak] p. 33.
IL Γ𝜓 => Γ𝜓𝜒 olcd 873 olc 867, olci 865, olcd 873 Definition I in [Pfenning] p. 18, definition In(1) in [Clemente] p. 12
IR Γ𝜒 => Γ𝜓𝜒 orcd 872 orc 866, orci 864, orcd 872 Definition IR in [Pfenning] p. 18, definition In(2) in [Clemente] p. 12.
E Γ𝜓𝜒 & Γ, 𝜓𝜃 & Γ, 𝜒𝜃 => Γ𝜃 mpjaodan 958 mpjaodan 958, jaodan 957, jaod 858 Definition E in [Pfenning] p. 18, definition Em,n,p in [Clemente] p. 12.
¬I Γ, 𝜓 => Γ¬ 𝜓 inegd 1562 pm2.01d 189
¬I Γ, 𝜓𝜃 & Γ¬ 𝜃 => Γ¬ 𝜓 mtand 815 mtand 815 definition I¬m,n,p in [Clemente] p. 13.
¬I Γ, 𝜓𝜒 & Γ, 𝜓¬ 𝜒 => Γ¬ 𝜓 pm2.65da 816 pm2.65da 816 Contradiction.
¬I Γ, 𝜓¬ 𝜓 => Γ¬ 𝜓 pm2.01da 798 pm2.01d 189, pm2.65da 816, pm2.65d 195 For an alternative falsum-free natural deduction ruleset
¬E Γ𝜓 & Γ¬ 𝜓 => Γ pm2.21fal 1564 pm2.21dd 194
¬E Γ, ¬ 𝜓 => Γ𝜓 pm2.21dd 194 definition E in [Indrzejczak] p. 33.
¬E Γ𝜓 & Γ¬ 𝜓 => Γ𝜃 pm2.21dd 194 pm2.21dd 194, pm2.21d 121, pm2.21 123 For an alternative falsum-free natural deduction ruleset. Definition ¬E in [Pfenning] p. 18.
I Γ trud 1552 tru 1546, trud 1552, mptru 1549 Definition I in [Pfenning] p. 18.
E Γ, ⊥𝜃 falimd 1560 falim 1559 Definition E in [Pfenning] p. 18.
I Γ[𝑎 / 𝑥]𝜓 => Γ𝑥𝜓 alrimiv 1931 alrimiv 1931, ralrimiva 3147 Definition Ia in [Pfenning] p. 18, definition In in [Clemente] p. 32.
E Γ𝑥𝜓 => Γ[𝑡 / 𝑥]𝜓 spsbcd 3792 spcv 3596, rspcv 3609 Definition E in [Pfenning] p. 18, definition En,t in [Clemente] p. 32.
I Γ[𝑡 / 𝑥]𝜓 => Γ𝑥𝜓 spesbcd 3878 spcev 3597, rspcev 3613 Definition I in [Pfenning] p. 18, definition In,t in [Clemente] p. 32.
E Γ𝑥𝜓 & Γ, [𝑎 / 𝑥]𝜓𝜃 => Γ𝜃 exlimddv 1939 exlimddv 1939, exlimdd 2214, exlimdv 1937, rexlimdva 3156 Definition Ea,u in [Pfenning] p. 18, definition Em,n,p,a in [Clemente] p. 32.
C Γ, ¬ 𝜓 => Γ𝜓 efald 1563 efald 1563 Proof by contradiction (classical logic), definition C in [Pfenning] p. 17.
C Γ, ¬ 𝜓𝜓 => Γ𝜓 pm2.18da 799 pm2.18da 799, pm2.18d 127, pm2.18 128 For an alternative falsum-free natural deduction ruleset
¬ ¬C Γ¬ ¬ 𝜓 => Γ𝜓 notnotrd 133 notnotrd 133, notnotr 130 Double negation rule (classical logic), definition NNC in [Pfenning] p. 17, definition E¬n in [Clemente] p. 14.
EM Γ𝜓 ∨ ¬ 𝜓 exmidd 895 exmid 894 Excluded middle (classical logic), definition XM in [Pfenning] p. 17, proof 5.11 in [Clemente] p. 14.
=I Γ𝐴 = 𝐴 eqidd 2734 eqid 2733, eqidd 2734 Introduce equality, definition =I in [Pfenning] p. 127.
=E Γ𝐴 = 𝐵 & Γ[𝐴 / 𝑥]𝜓 => Γ[𝐵 / 𝑥]𝜓 sbceq1dd 3784 sbceq1d 3783, 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 Γ represents the set of (current) hypotheses. We use wff variable names beginning with 𝜓 to provide a closer representation of the Metamath equivalents (which typically use the antedent 𝜑 to represent the context Γ).

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 29657, ex-natded5.3 29660, ex-natded5.5 29663, ex-natded5.7 29664, ex-natded5.8 29666, ex-natded5.13 29668, ex-natded9.20 29670, and ex-natded9.26 29672.

(Contributed by DAW, 4-Feb-2017.) (New usage is discouraged.)

Hypothesis
Ref Expression
natded.1 𝜑
Assertion
Ref Expression
natded 𝜑

Proof of Theorem natded
StepHypRef Expression
1 natded.1 1 𝜑
Colors of variables: wff setvar class
This theorem is referenced by: (None)
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