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Type | Label | Description | ||||||||||||||||||||||||||||||
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Statement | ||||||||||||||||||||||||||||||||

Theorem | ee323 28301 | e323 28587 without virtual deductions. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | 3ornot23 28302 | If the second and third disjuncts of a true triple disjunction are false, then the first disjunct is true. Automatically derived from 3ornot23VD 28668. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | orbi1r 28303 | orbi1 687 with order of disjuncts reversed. Derived from orbi1rVD 28669. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | bitr3 28304 | Closed nested implication form of bitr3i 243. Derived automatically from bitr3VD 28670. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | 3orbi123 28305 | pm4.39 842 with a 3-conjunct antecedent. This proof is 3orbi123VD 28671 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | syl5imp 28306 | Closed form of syl5 30. Derived automatically from syl5impVD 28684. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | impexp3a 28307 |
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. After the
User's Proof was completed, it was minimized. The completed User's Proof
before minimization is not shown. (Contributed by Alan Sare,
18-Mar-2012.) (Proof modification is discouraged.)
(New usage is discouraged.)
| ||||||||||||||||||||||||||||||

Theorem | com3rgbi 28308 |
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The
completed Virtual Deduction Proof (not shown) was minimized. The
minimized proof is shown.
(Contributed by Alan Sare, 18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
| ||||||||||||||||||||||||||||||

Theorem | impexp3acom3r 28309 |
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The
completed Virtual Deduction Proof (not shown) was minimized. The
minimized proof is shown. (Contributed by Alan Sare, 18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
| ||||||||||||||||||||||||||||||

Theorem | ee1111 28310 |
Non-virtual deduction form of e1111 28485. (Contributed by Alan Sare,
18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
The following User's Proof is a Virtual Deduction proof
completed automatically by the tools program completeusersproof.cmd,
which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof
Assistant. The completed Virtual Deduction Proof (not shown) was
minimized. The minimized proof is shown.
| ||||||||||||||||||||||||||||||

Theorem | pm2.43bgbi 28311 |
Logical equivalence of a 2-left-nested implication and a 1-left-nested
implicated
when two antecedents of the former implication are identical.
(Contributed by Alan Sare, 18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The
completed Virtual
Deduction Proof (not shown) was minimized. The minimized proof is
shown.
| ||||||||||||||||||||||||||||||

Theorem | pm2.43cbi 28312 |
Logical equivalence of a 3-left-nested implication and a 2-left-nested
implicated when two antecedents of the former implication are identical.
(Contributed by Alan Sare, 18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
The following User's Proof is
a Virtual Deduction proof completed automatically by the tools program
completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's
Metamath Proof Assistant. The completed Virtual Deduction Proof (not
shown) was minimized. The minimized proof is shown.
| ||||||||||||||||||||||||||||||

Theorem | ee233 28313 |
Non-virtual deduction form of e233 28586. (Contributed by Alan Sare,
18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The
completed Virtual
Deduction Proof (not shown) was minimized. The minimized proof is
shown.
| ||||||||||||||||||||||||||||||

Theorem | imbi12 28314 | Implication form of imbi12i 317. imbi12 28314 is imbi12VD 28694 without virtual deductions and was automatically derived from imbi12VD 28694 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | imbi13 28315 | Join three logical equivalences to form equivalence of implications. imbi13 28315 is imbi13VD 28695 without virtual deductions and was automatically derived from imbi13VD 28695 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | ee33 28316 |
Non-virtual deduction form of e33 28555. (Contributed by Alan Sare,
18-Mar-2012.) (Proof modification is discouraged.)
(New usage is discouraged.)
The following User's Proof is a Virtual Deduction proof
completed automatically by the tools program completeusersproof.cmd,
which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof
Assistant. The completed Virtual Deduction Proof (not shown) was
minimized. The minimized proof is shown.
| ||||||||||||||||||||||||||||||

Theorem | con5 28317 | Bi-conditional contraposition variation. This proof is con5VD 28721 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | con5i 28318 | Inference form of con5 28317. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | exlimexi 28319 | Inference similar to Theorem 19.23 of [Margaris] p. 90. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | sb5ALT 28320* | Equivalence for substitution. Alternate proof of sb5 2149. This proof is sb5ALTVD 28734 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | eexinst01 28321 | exinst01 28435 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | eexinst11 28322 | exinst11 28436 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | vk15.4j 28323 | Excercise 4j of Unit 15 of "Understanding Symbolic Logic", Fifth Edition (2008), by Virginia Klenk. This proof is the minimized Hilbert-style axiomatic version of the Fitch-style Natural Deduction proof found on page 442 of Klenk and was automatically derived from that proof. vk15.4j 28323 is vk15.4jVD 28735 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | notnot2ALT 28324 | Converse of double negation. Alternate proof of notnot2 106. This proof is notnot2ALTVD 28736 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | con3ALT 28325 | Contraposition. Alternate proof of con3 128. This proof is con3ALTVD 28737 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | ssralv2 28326* |
Quantification restricted to a subclass for two quantifiers. ssralv 3367
for two quantifiers. The proof of ssralv2 28326 was automatically generated
by minimizing the automatically translated proof of ssralv2VD 28687. The
automatic translation is by the tools program
translate_{without}_overwriting.cmd
(Contributed by Alan Sare,
18-Feb-2012.) (Proof modification is discouraged.)
(New usage is discouraged.)
| ||||||||||||||||||||||||||||||

Theorem | sbc3org 28327 | sbcorg 3166 with a 3-disjuncts. This proof is sbc3orgVD 28672 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | alrim3con13v 28328* | Closed form of alrimi 1777 with 2 additional conjuncts having no occurences of the quantifying variable. This proof is 19.21a3con13vVD 28673 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | rspsbc2 28329* | rspsbc 3199 with two quantifying variables. This proof is rspsbc2VD 28676 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | sbcoreleleq 28330* | Substitution of a set variable for another set variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 28680. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | tratrb 28331* | If a class is transitive and any two distinct elements of the class are E-comparable, then every element of that class is transitive. Derived automatically from tratrbVD 28682. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | 3ax5 28332 | ax-5 1563 for a 3 element left-nested implication. Derived automatically from 3ax5VD 28683. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | ordelordALT 28333 | An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 4563 using the Axiom of Regularity indirectly through dford2 7531. dford2 is a weaker definition of ordinal number. Given the Axiom of Regularity, it need not be assumed that because this is inferred by the Axiom of Regularity. ordelordALT 28333 is ordelordALTVD 28688 without virtual deductions and was automatically derived from ordelordALTVD 28688 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | sbcim2g 28334 | Distribution of class substitution over a left-nested implication. Similar to sbcimg 3162. sbcim2g 28334 is sbcim2gVD 28696 without virtual deductions and was automatically derived from sbcim2gVD 28696 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | sbcbi 28335 | Implication form of sbcbiiOLD 3177. sbcbi 28335 is sbcbiVD 28697 without virtual deductions and was automatically derived from sbcbiVD 28697 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | trsbc 28336* | Formula-building inference rule for class substitution, substituting a class variable for the set variable of the transitivity predicate. trsbc 28336 is trsbcVD 28698 without virtual deductions and was automatically derived from trsbcVD 28698 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | truniALT 28337* | The union of a class of transitive sets is transitive. Alternate proof of truni 4276. truniALT 28337 is truniALTVD 28699 without virtual deductions and was automatically derived from truniALTVD 28699 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | sbcssOLD 28338 | Distribute proper substitution through a subclass relation. This theorem was automatically derived from sbcssVD 28704. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | onfrALTlem5 28339* | Lemma for onfrALT 28346. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | onfrALTlem4 28340* | Lemma for onfrALT 28346. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | onfrALTlem3 28341* | Lemma for onfrALT 28346. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | ggen31 28342* | gen31 28431 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | onfrALTlem2 28343* | |||||||||||||||||||||||||||||||

Theorem | cbvexsv 28344* | A theorem pertaining to the substitution for an existentially quantified variable when the substituted variable does not occur in the quantified wff. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | onfrALTlem1 28345* | |||||||||||||||||||||||||||||||

Theorem | onfrALT 28346 | The epsilon relation is foundational on the class of ordinal numbers. onfrALT 28346 is an alternate proof of onfr 4580. onfrALTVD 28712 is the Virtual Deduction proof from which onfrALT 28346 is derived. The Virtual Deduction proof mirrors the working proof of onfr 4580 which is the main part of the proof of Theorem 7.12 of the first edition of TakeutiZaring. The proof of the corresponding Proposition 7.12 of [TakeutiZaring] p. 38 (second edition) does not contain the working proof equivalent of onfrALTVD 28712. This theorem does not rely on the Axiom of Regularity. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | csbeq2g 28347 | Formula-building implication rule for class substitution. Closed form of csbeq2i 3237. csbeq2g 28347 is derived from the virtual deduction proof csbeq2gVD 28713. (Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | 19.41rg 28348 | Closed form of right-to-left implication of 19.41 1896, Theorem 19.41 of [Margaris] p. 90. Derived from 19.41rgVD 28723. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | opelopab4 28349* | Ordered pair membership in a class abstraction of pairs. Compare to elopab 4422. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 6-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | 2pm13.193 28350 | pm13.193 27479 for two variables. pm13.193 27479 is Theorem *13.193 in [WhiteheadRussell] p. 179. Derived from 2pm13.193VD 28724. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | hbntal 28351 | A closed form of hbn 1797. hbnt 1795 is another closed form of hbn 1797. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | hbimpg 28352 | A closed form of hbim 1832. Derived from hbimpgVD 28725. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | hbalg 28353 | Closed form of hbal 1747. Derived from hbalgVD 28726. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | hbexg 28354 | Closed form of nfex 1861. Derived from hbexgVD 28727. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 12-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | a9e2eq 28355* | Alternate form of a9e 1948 for non-distinct , and . a9e2eq 28355 is derived from a9e2eqVD 28728. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | a9e2nd 28356* | If at least two sets exist (dtru 4350) , then the same is true expressed in an alternate form similar to the form of a9e 1948. a9e2nd 28356 is derived from a9e2ndVD 28729. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | a9e2ndeq 28357* | "At least two sets exist" expressed in the form of dtru 4350 is logically equivalent to the same expressed in a form similar to a9e 1948 if dtru 4350 is false implies . a9e2ndeq 28357 is derived from a9e2ndeqVD 28730. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | 2sb5nd 28358* | Equivalence for double substitution 2sb5 2161 without distinct , requirement. 2sb5nd 28358 is derived from 2sb5ndVD 28731. (Contributed by Alan Sare, 30-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | 2uasbanh 28359* | Distribute the unabbreviated form of proper substitution in and out of a conjunction. 2uasbanh 28359 is derived from 2uasbanhVD 28732. (Contributed by Alan Sare, 31-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | 2uasban 28360* | Distribute the unabbreviated form of proper substitution in and out of a conjunction. (Contributed by Alan Sare, 31-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | e2ebind 28361 | Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind 28361 is derived from e2ebindVD 28733. (Contributed by Alan Sare, 27-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | elpwgded 28362 | elpwgdedVD 28738 in conventional notation. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | trelded 28363 | Deduction form of trel 4269. In a transitive class, the membership relation is transitive. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | jaoded 28364 | Deduction form of jao 499. Disjunction of antecedents. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | 3imp31 28365 | The importation inference 3imp 1147 with commutation of the first and third conjuncts of the assertion relative to the hypothesis. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | 3imp21 28366 | The importation inference 3imp 1147 with commutation of the first and second conjuncts of the assertion relative to the hypothesis. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | biimpa21 28367 | biimpa 471 with commutation of the first and second conjuncts of the assertion. (Contributed by Alan Sare, 11-Sep-2016.) | ||||||||||||||||||||||||||||||

Theorem | sbtT 28368 | A substitution into a theorem remains true. sbt 2082 with the existence of no virtual hypotheses for the hypothesis expressed as the empty virtual hypothesis collection. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

19.24.4 What is Virtual Deduction? | ||||||||||||||||||||||||||||||||

Syntax | wvd1 28369 |
A Virtual Deduction proof in a Hilbert-style deductive system is the
analog of a sequent calculus proof. A theorem is proven in a Gentzen
system in order to prove more directly, which may be more intuitive
and easier for some people. The analog of this proof in Metamath's
Hilbert-style system is verified by the Metamath program.
Natural Deduction is a well-known proof method orignally proposed by Gentzen in 1935 and comprehensively summarized by Prawitz in his 1965 monograph "Natural deduction: a proof-theoretic study". Gentzen wished to construct "a formalism that comes as close as possible to natural reasoning". Natural deduction is a response to dissatisfaction with axiomatic proofs such as Hilbert-style axiomatic proofs, which the proofs of Metamath are. In 1926, in Poland, Lukasiewicz advocated a more natural treatment of logic. Jaskowski made the earliest attempts at defining a more natural deduction. Natural deduction in its modern form was independently proposed by Gentzen. Sequent calculus, the chief alternative to Natural Deduction, was created by Gentzen. The following is an except from Stephen Cole Kleene's seminal 1952 book "Introduction to Metamathematics", which contains the first formulation of sequent calculus in the modern style. Kleene states on page 440: . . . the proof of (Gentzen's Hauptsatz) breaks down into a list of cases, each of which is simple to handle. . . . Gentzen's normal form for proofs in the predicate calculus requires a different classification of the deductive steps than is given by the postulates of the formal system of predicate calculus of Chapter IV (Section 19). The implication symbol (the Metamath symbol for implication has been substituted here for the symbol used by Kleene) has to be separated in its role of mediating inferences from its role as a component symbol of the formula being proved. In the former role it will be replaced by a new formal symbol (read "gives" or "entails"), to which properties will be assigned similar to those of the informal symbol in our former derived rules. Gentzen's classification of the deductive operations is made explicit by setting up a new formal system of the predicate calculus. The formal system of propositional and predicate calculus studied previously (Chapters IV ff.) we call now a "Hilbert-type system", and denote by H. Precisely, H denotes any one or a particular one of several systems, according to whether we are considering propositional calculus or predicate calculus, in the classical or the intuitionistic version (Section 23), and according to the sense in which we are using "term" and "formula" (Sections 117,25,31,37,72-76). The same respective choices will apply to the "Gentzen-type system G1" which we introduce now and the G2, G3 and G3a later. The transformation or deductive rules of G1 will apply to objects which are not formulas of the system H, but are built from them by an additional formation rule, so we use a new term "sequent" for these objects. (Gentzen says "Sequenz", which we translate as "sequent", because we have already used "sequence" for any succession of objects, where the German is "Folge".) A sequent is a formal expression of the form , . . . , , . . . , where , . . . , and , . . . , are seqences of a finite number of 0 or more formulas (substituting Metamath notation for Kleene's notation). The part , . . . , is the antecedent, and , . . . , the succedent of the sequent , . . . , , . . . , . When the antecedent and the succedent each have a finite number of 1 or more formulas, the sequent , . . . , , . . . has the same interpretation for G1 as the formula . . . . . . for H. The interpretation extends to the case of an antecedent of 0 formulas by regarding . . . for 0 formulas (the "empty conjunction") as true and . . . for 0 formulas (the "empty disjunction") as false. . . . As in Chapter V, we use Greek capitals . . . to stand for finite sequences of zero or more formulas, but now also as antecedent (succedent), or parts of antecedent (succedent), with separating formal commas included. . . . (End of Kleene excerpt) In chapter V entitled "Formal Deduction" Kleene states, on page 86: Section 20. Formal deduction. Formal proofs of even quite elementary theorems tend to be long. As a price for having analyzed logical deduction into simple steps, more of those steps have to be used. The purpose of formalizing a theory is to get an explicit definition of what constitutes proof in the theory. Having achieved this, there is no need always to appeal directly to the definition. The labor required to establish the formal provability of formulas can be greatly lessened by using metamathematical theorems concerning the existence of formal proofs. If the demonstrations of those theorems do have the finitary character which metamathematics is supposed to have, the demonstrations will indicate, at least implicitly, methods for obtaining the formal proofs. The use of the metamathematical theorems then amounts to abbreviation, often of very great extent, in the presentation of formal proofs. The simpler of such metamathematical theorems we shall call derived rules, since they express principles which can be said to be derived from the postulated rules by showing that the use of them as additional methods of inference does not increase the class of provable formulas. We shall seek by means of derived rules to bring the methods for establishing the facts of formal provability as close as possible to the informal methods of the theory which is being formalized. In setting up the formal system, proof was given the simplest possible structure, consisting of a single sequence of formulas. Some of our derived rules, called "direct rules", will serve to abbreviate for us whole segments of such a sequence; we can then, so to speak, use these segments as prefabricated units in building proofs. But also, in mathematical practice, proofs are common which have a more complicated structure, employing "subsidiary deduction", i.e. deduction under assumptions for the sake of the argument, which assumptions are subsequently discharged. For example, subsidiary deduction is used in a proof by reductio ad absurdum, and less obtrusively when we place the hypothesis of a theorem on a par with proved propositions to deduce the conclusion. Other derived rules, called "subsidiary deduction rules", will give us this kind of procedure. We now introduce, by a metamathematical definition, the notion of "formal deducibility under assumptions". Given a list , . . . of or more (occurences of) formulas, a finite sequence of one or more (occurences of) formulas is called a (formal) deduction from the assumption formulas , . . . , if each formula of the sequence is either one of the formulas , . . . , or an axiom, or an immediate consequence of preceding formulas of a sequence. A deduction is said to be deducible from the assumption formulas (in symbols, ,. . . . ,. ), and is called the conclusion (or endformula) of the deduction. (The symbol may be read "yields".) (End of Kleene excerpt) Gentzen's normal form is a certain direct fashion for proofs and deductions. His sequent calculus, formulated in the modern style by Kleene, is the classical system G1. In this system, the new formal symbol has properties similar to the informal symbol of Kleene's above language of formal deducibility under assumptions. Kleene states on page 440: . . . This leads us to inquire whether there may not be a theorem about the predicate calculus asserting that, if a formula is provable (or deducible from other formulas), it is provable (or deducible) in a certain direct fashion; in other words, a theorem giving a normal form for proofs and deductions, the proofs and deduction in normal form being in some sense direct. (End of Kleene excerpt) There is such a theorem, which was proven by Kleene. Formal proofs in H of even quite elementary theorems tend to be long. As a price for having analyzed logical deduction into simple steps, more of those steps have to be used. The proofs of Metamath are fully detailed formal proofs. We wish to have a means of proving Metamath theorems and deductions in a more direct fashion. Natural Deduction is a system for proving theorems and deductions in a more direct fashion. However, Natural Deduction is not compatible for use with Metamath, which uses a Hilbert-type system. Instead, Kleene's classical system G1 may be used for proving Metamath deductions and theorems in a more direct fashion. The system of Metamath is an H system, not a Gentzen system. Therefore, proofs in Kleene's classical system G1 ("G1") cannot be included in Metamath's system H, which we shall henceforth call "system H" or "H". However, we may translate proofs in G1 into proofs in H. By Kleene's THEOREM 47 (page 446)
By Kleene's COROLLARY of THEOREM 47 (page 448)
denotes the same connective denoted by . " , " , in the context of Virtual Deduction, denotes the same connective denoted by . This Virtual Deduction notation is specified by the following set.mm definitions:
replaces in the analog in H of a sequent in G1 having a non-empty antecedent. If occurs as the outermost connective denoted by or and occurs exactly once, we call the analog in H of a sequent in G1 a "virtual deduction" because the corresponding of the sequent is assigned properties similar to . While sequent calculus proofs (proofs in G1) may have as steps sequents with 0, 1, or more formulas in the succedent, we shall only prove in G1 using sequents with exactly 1 formula in the succedent. The User proves in G1 in order to obtain the benefits of more direct proving using sequent calculus, then translates the proof in G1 into a proof in H. The reference theorems and deductions to be used for proving in G1 are translations of theorems and deductions in set.mm. Each theorem in set.mm corresponds to the theorem in G1. Deductions in G1 corresponding to deductions in H are similarly determined. Theorems in H with one or more occurences of either or may also be translated into theorems in G1 for by replacing the outermost occurence of or of the theorem in H with . Deductions in H may be translated into deductions in G1 in a similar manner. The only theorems and deductions in H useful for proving in G1 for the purpose of obtaining proofs in H are those in which, for each hypothesis or assertion, there are 0 or 1 occurences of and it is the outermost occurence of or . Kleene's THEOREM 46 and its COROLLARY 2 are used for translating from H to G1. By Kleene's THEOREM 46 (page 445)
By Kleene's COROLLARY 2 of THEOREM 46 (page 446)
The procedure for more direct proving of theorems or deductions in H is as follows. The User proves in G1. He(she) uses translated set.mm theorems and deductions as reference theorems and deductions. His(her) proof is only a guess in the sense that he(she) can't verify his(her) proof in G1 because he(she) doesn't have an automated proof checker to use. The proof in G1 is translated into its analog in H for verification by the Metamath program. This proof is called the Virtual Deduction proof. This proof may then be translated into a conventional Metamath proof automatically, removing the unnecessary Virtual Deduction symbols. The translations from H to G1 and G1 to H are trivial. In practice, they may be done without much thought. In principle, they must be done, because the proving is done using sequents, which do not exist in H. The analogs in H of the postulates of G1 are the set.mm postulates. The postulates in G1 corresponding to the Metamath postulates are not the classical system G1 postulates of Kleene (pages 442 and 443). set.mm has the predicate calculus postulates and other posulates. The Kleene classical system G1 postulates correspond to predicate calculus postulates which differ from the Metamath system G1 postulates corresponding to the predicate calculus postulates of Metamath's system H. Metamath's predicate calculus G1 postulates are presumably deducible from the Kleene classical G1 postulates and the Kleene classical G1 postulates are deducible from Metamath's G1 postulates. It should be recognized that, because of the different postulates, the classical G1 system corresponding to Metamath's system H is not identical to Kleene's classical system G1.
Why not create a separate database (setg.mm) of proofs in G1, avoiding the need to translate from H to G1 and from G1 back to H? The translations are trivial. Sequents make the language more complex than is necessary. More direct proving using sequent calculus may be done as a means towards the end of constructing proofs in H. Then, the language may be kept as simple as possible. A system G1 database would be redundant because it would duplicate the information contained in the corresponding system H database. For earlier proofs, each "User's Proof" in the web page description of a Virtual Deduction proof in set.mm is the analog in H of the User's working proof in G1. The User's Proof is automatically completed by completeusersproof.cmd. The completed proof is the Virtual Deduction proof, which is the analog in H of the corresponding fully detailed proof in G1. The completed Virtual Deduction proof of these earlier proofs may be automatically translated into a conventional Metamath proof. In September of 2016 completeusersproof.c was released. The input for completeusersproof.c is a Virtual Deduction User's Proof. Unlike completeusersproof.cmd, the completed proof is in conventional notation. completeusersproof.c eliminates the virtual deduction notation of the User's Proof after utilizing the information it provides. Applying mmj2's unify command is essential to completeusersproof.c. The mmj2 program is invoked within the completeusersproof.c function mmj2Unify(). The original mmj2 program was written by Mel O'Cat. Mario Carneiro has enhanced it. mmj2Unify() is called multiple times during the execution of completeusersproof. A Virtual Deduction proof is the Metamath-specific version of a Natural Deduction Proof. A Virtual Deduction proof generally cannot be directly input on a mmj2 Proof Worksheet and completed by the mmj2 tool because it is usually missing some technical proof steps which are not part of the Virtual Deduction proof but are necessary for a complete Metamath Proof. These missing technical steps may be automatically added by an automated proof assistant. completeusersproof.c is such a proof assistant. completeusersproof.c adds the missing technical steps and finds the reference theorems and deductions in set.mm which unify with the subproofs of the proof. The User may write a Virtual Deduction proof and automatically transform it into a complete Metamath proof using the completeusersproof tool. The completed proof has been checked by the Metamath program. The task of writing a complete Metamath proof is reduced to writing what is essentially a Natural Deduction Proof. Generally, proving using Virtual Deduction and completeusersproof reduces the amount Metamath-specific knowledge required by the User. Often, no knowledge of the specific theorems and deductions in set.mm is required to write some of the subproofs of a Virtual Deduction proof. Often, no knowledge of the Metamath-specific names of reference theorems and deductions in set.mm is required for writing some of the subproofs of a User's Proof. Often, the User may write subproofs of a proof using theorems or deductions commonly used in mathematics and correctly assume that some form of each is contained in set.mm and that completeusersproof will automatically generate the technical steps necessary to utilize them to complete the subproofs. Often, the fraction of the work which may be considered tedious is reduced and the total amount of work is reduced. | ||||||||||||||||||||||||||||||

19.24.5 Virtual Deduction Theorems | ||||||||||||||||||||||||||||||||

Definition | df-vd1 28370 | Definition of virtual deduction. (Contributed by Alan Sare, 21-Apr-2011.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | in1 28371 | Inference form of df-vd1 28370. Virtual deduction introduction rule of converting the virtual hypothesis of a 1-virtual hypothesis virtual deduction into an antecedent. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | iin1 28372 | in1 28371 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | dfvd1ir 28373 | Inference form of df-vd1 28370 with the virtual deduction as the assertion. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | idn1 28374 | Virtual deduction identity rule which is id 20 with virtual deduction symbols. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | ax172 28375* | Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | dfvd1imp 28376 | Left-to-right part of definition of virtual deduction. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | dfvd1impr 28377 | Right-to-left part of definition of virtual deduction. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Syntax | wvd2 28378 | Syntax for a 2-hypothesis virtual deduction. (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Definition | df-vd2 28379 | Definition of a 2-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | dfvd2 28380 | Definition of a 2-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Syntax | wvhc2 28381 | Syntax for a 2-virtual hypotheses collection. (Contributed by Alan Sare, 23-Apr-2015.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Definition | df-vhc2 28382 | Definition of a 2-virtual hypotheses collection. (Contributed by Alan Sare, 23-Apr-2015.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | dfvd2an 28383 | Definition of a 2-hypothesis virtual deduction in vd conjunction form. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | dfvd2ani 28384 | Inference form of dfvd2an 28383. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | dfvd2anir 28385 | Right-to-left inference form of dfvd2an 28383. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | dfvd2i 28386 | Inference form of dfvd2 28380. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | dfvd2ir 28387 | Right-to-left inference form of dfvd2 28380. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Syntax | wvd3 28388 | Syntax for a 3-hypothesis virtual deduction. (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Syntax | wvhc3 28389 | Syntax for a 3-virtual hypotheses collection. (Contributed by Alan Sare, 13-Jun-2015.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Definition | df-vhc3 28390 | Definition of a 3-virtual hypotheses collection. (Contributed by Alan Sare, 13-Jun-2015.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Definition | df-vd3 28391 | Definition of a 3-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | dfvd3 28392 | Definition of a 3-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | dfvd3i 28393 | Inference form of dfvd3 28392. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | dfvd3ir 28394 | Right-to-left inference form of dfvd3 28392. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | dfvd3an 28395 | Definition of a 3-hypothesis virtual deduction in vd conjunction form. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | dfvd3ani 28396 | Inference form of dfvd3an 28395. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | dfvd3anir 28397 | Right-to-left inference form of dfvd3an 28395. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Syntax | wvhc4 28398 | Syntax for a 4-virtual hypotheses collection. (Contributed by Alan Sare, 17-Oct-2017.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Syntax | wvhc5 28399 | Syntax for a 5-virtual hypotheses collection. (Contributed by Alan Sare, 17-Oct-2017.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Syntax | wvhc6 28400 | Syntax for a 6-element virtual hypotheses collection. (Contributed by Alan Sare, 17-Oct-2017.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

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