Theorem natded 30427

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.

Name	Natural Deduction Rule	Translation	Recommendation	Comments
IT	Γ⊢ 𝜓 => Γ⊢ 𝜓	idi 1	nothing	Reiteration is always redundant in Metamath. Definition "new rule" in [Pfenning] p. 18, definition IT in [Clemente] p. 10.
∧I	Γ⊢ 𝜓 & Γ⊢ 𝜒 => Γ⊢ 𝜓 ∧ 𝜒	jca 511	jca 511, pm3.2i 470	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	Γ⊢ 𝜓 ∧ 𝜒 => Γ⊢ 𝜓	simpld 494	simpld 494, adantr 480	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	Γ⊢ 𝜓 ∧ 𝜒 => Γ⊢ 𝜒	simprd 495	simpr 484, adantl 481	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, 𝜓⊢ 𝜒 => Γ⊢ 𝜓 → 𝜒	ex 412	ex 412	Definition →I in [Pfenning] p. 18, definition I=>m,n in [Clemente] p. 11, and definition →I in [Indrzejczak] p. 33.
→E	Γ⊢ 𝜓 → 𝜒 & Γ⊢ 𝜓 => Γ⊢ 𝜒	mpd 15	ax-mp 5, mpd 15, mpdan 687, imp 406	Definition →E in [Pfenning] p. 18, definition E=>m,n in [Clemente] p. 11, and definition →E in [Indrzejczak] p. 33.
∨IL	Γ⊢ 𝜓 => Γ⊢ 𝜓 ∨ 𝜒	olcd 874	olc 868, olci 866, olcd 874	Definition ∨I in [Pfenning] p. 18, definition I∨n(1) in [Clemente] p. 12
∨IR	Γ⊢ 𝜒 => Γ⊢ 𝜓 ∨ 𝜒	orcd 873	orc 867, orci 865, orcd 873	Definition ∨IR in [Pfenning] p. 18, definition I∨n(2) in [Clemente] p. 12.
∨E	Γ⊢ 𝜓 ∨ 𝜒 & Γ, 𝜓⊢ 𝜃 & Γ, 𝜒⊢ 𝜃 => Γ⊢ 𝜃	mpjaodan 960	mpjaodan 960, jaodan 959, jaod 859	Definition ∨E in [Pfenning] p. 18, definition E∨m,n,p in [Clemente] p. 12.
¬I	Γ, 𝜓⊢ ⊥ => Γ⊢ ¬ 𝜓	inegd 1561	pm2.01d 190
¬I	Γ, 𝜓⊢ 𝜃 & Γ⊢ ¬ 𝜃 => Γ⊢ ¬ 𝜓	mtand 815	mtand 815	definition I¬m,n,p in [Clemente] p. 13.
¬I	Γ, 𝜓⊢ 𝜒 & Γ, 𝜓⊢ ¬ 𝜒 => Γ⊢ ¬ 𝜓	pm2.65da 816	pm2.65da 816	Contradiction.
¬I	Γ, 𝜓⊢ ¬ 𝜓 => Γ⊢ ¬ 𝜓	pm2.01da 798	pm2.01d 190, pm2.65da 816, pm2.65d 196	For an alternative falsum-free natural deduction ruleset
¬E	Γ⊢ 𝜓 & Γ⊢ ¬ 𝜓 => Γ⊢ ⊥	pm2.21fal 1563	pm2.21dd 195
¬E	Γ, ¬ 𝜓⊢ ⊥ => Γ⊢ 𝜓		pm2.21dd 195	definition →E in [Indrzejczak] p. 33.
¬E	Γ⊢ 𝜓 & Γ⊢ ¬ 𝜓 => Γ⊢ 𝜃	pm2.21dd 195	pm2.21dd 195, pm2.21d 121, pm2.21 123	For an alternative falsum-free natural deduction ruleset. Definition ¬E in [Pfenning] p. 18.
⊤I	Γ⊢ ⊤	trud 1551	tru 1545, trud 1551, mptru 1548	Definition ⊤I in [Pfenning] p. 18.
⊥E	Γ, ⊥⊢ 𝜃	falimd 1559	falim 1558	Definition ⊥E in [Pfenning] p. 18.
∀I	Γ⊢ [𝑎 / 𝑥]𝜓 => Γ⊢ ∀𝑥𝜓	alrimiv 1928	alrimiv 1928, ralrimiva 3126	Definition ∀Ia in [Pfenning] p. 18, definition I∀n in [Clemente] p. 32.
∀E	Γ⊢ ∀𝑥𝜓 => Γ⊢ [𝑡 / 𝑥]𝜓	spsbcd 3752	spcv 3557, rspcv 3570	Definition ∀E in [Pfenning] p. 18, definition E∀n,t in [Clemente] p. 32.
∃I	Γ⊢ [𝑡 / 𝑥]𝜓 => Γ⊢ ∃𝑥𝜓	spesbcd 3831	spcev 3558, rspcev 3574	Definition ∃I in [Pfenning] p. 18, definition I∃n,t in [Clemente] p. 32.
∃E	Γ⊢ ∃𝑥𝜓 & Γ, [𝑎 / 𝑥]𝜓⊢ 𝜃 => Γ⊢ 𝜃	exlimddv 1936	exlimddv 1936, exlimdd 2225, exlimdv 1934, rexlimdva 3135	Definition ∃Ea,u in [Pfenning] p. 18, definition E∃m,n,p,a in [Clemente] p. 32.
⊥C	Γ, ¬ 𝜓⊢ ⊥ => Γ⊢ 𝜓	efald 1562	efald 1562	Proof by contradiction (classical logic), definition ⊥C in [Pfenning] p. 17.
⊥C	Γ, ¬ 𝜓⊢ 𝜓 => Γ⊢ 𝜓	pm2.18da 799	pm2.18da 799, pm2.18d 127, pm2.18 128	For an alternative falsum-free natural deduction ruleset
¬ ¬C	Γ⊢ ¬ ¬ 𝜓 => Γ⊢ 𝜓	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	Γ⊢ 𝜓 ∨ ¬ 𝜓	exmidd 895	exmid 894	Excluded middle (classical logic), definition XM in [Pfenning] p. 17, proof 5.11 in [Clemente] p. 14.
=I	Γ⊢ 𝐴 = 𝐴	eqidd 2735	eqid 2734, eqidd 2735	Introduce equality, definition =I in [Pfenning] p. 127.
=E	Γ⊢ 𝐴 = 𝐵 & Γ[𝐴 / 𝑥]𝜓 => Γ⊢ [𝐵 / 𝑥]𝜓	sbceq1dd 3744	sbceq1d 3743, 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 antecedent 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 30428, ex-natded5.3 30431, ex-natded5.5 30434, ex-natded5.7 30435, ex-natded5.8 30437, ex-natded5.13 30439, ex-natded9.20 30441, and ex-natded9.26 30443.

(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

Step	Hyp	Ref	Expression
1		natded.1	1 ⊢ 𝜑

This theorem is referenced by: (None)