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Theorem List for Metamath Proof Explorer - 28601-28700   *Has distinct variable group(s)
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Statement

Theoremee1111 28601 Non-virtual deduction form of e1111 28778. (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.
 h1:: h2:: h3:: h4:: h5:: 6:1,5: 7:6: 8:2,7: 9:8: 10:9: 11:3,10: 12:11: 13:12: 14:4,13: qed:14:

Theorempm2.43bgbi 28602 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.
 1:: 2:: 3:1,2: 4:: 5:3,4: 6:: qed:5,6:

Theorempm2.43cbi 28603 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.
 1:: 2:: 3:1,2: 4:: 5:3,4: 6:: qed:5,6:

Theoremee233 28604 Non-virtual deduction form of e233 28879. (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.
 h1:: h2:: h3:: h4:: 5:1,4: 6:5: 7:2,6: 8:7: 9:8: 10:9: 11:10: 12:3,11: 13:12: 14:13: qed:14:

Theoremimbi12 28605 Implication form of imbi12i 318. imbi12 28605 is imbi12VD 28987 without virtual deductions and was automatically derived from imbi12VD 28987 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.)

Theoremimbi13 28606 Join three logical equivalences to form equivalence of implications. imbi13 28606 is imbi13VD 28988 without virtual deductions and was automatically derived from imbi13VD 28988 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.)

Theoremee33 28607 Non-virtual deduction form of e33 28848. (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.
 h1:: h2:: h3:: 4:1,3: 5:4: 6:2,5: 7:6: 8:7: qed:8:

Theoremcon5 28608 Bi-conditional contraposition variation. This proof is con5VD 29014 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremcon5i 28609 Inference form of con5 28608. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremexlimexi 28610 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.)

Theoremsb5ALT 28611* Equivalence for substitution. Alternate proof of sb5 2178. This proof is sb5ALTVD 29027 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremeexinst01 28612 exinst01 28728 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremeexinst11 28613 exinst11 28729 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremvk15.4j 28614 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 28614 is vk15.4jVD 29028 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnotnot2ALT 28615 Converse of double negation. Alternate proof of notnot2 107. This proof is notnot2ALTVD 29029 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremcon3ALT 28616 Contraposition. Alternate proof of con3 129. This proof is con3ALTVD 29030 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremssralv2 28617* Quantification restricted to a subclass for two quantifiers. ssralv 3409 for two quantifiers. The proof of ssralv2 28617 was automatically generated by minimizing the automatically translated proof of ssralv2VD 28980. The automatic translation is by the tools program translatewithout_overwriting.cmd (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsbc3org 28618 sbcorg 3208 with a 3-disjuncts. This proof is sbc3orgVD 28965 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremalrim3con13v 28619* Closed form of alrimi 1782 with 2 additional conjuncts having no occurences of the quantifying variable. This proof is 19.21a3con13vVD 28966 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremrspsbc2 28620* rspsbc 3241 with two quantifying variables. This proof is rspsbc2VD 28969 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsbcoreleleq 28621* Substitution of a set variable for another set variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 28973. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremtratrb 28622* 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 28975. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem3ax5 28623 ax-5 1567 for a 3 element left-nested implication. Derived automatically from 3ax5VD 28976. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremordelordALT 28624 An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 4605 using the Axiom of Regularity indirectly through dford2 7577. 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 28624 is ordelordALTVD 28981 without virtual deductions and was automatically derived from ordelordALTVD 28981 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.)

Theoremsbcim2g 28625 Distribution of class substitution over a left-nested implication. Similar to sbcimg 3204. sbcim2g 28625 is sbcim2gVD 28989 without virtual deductions and was automatically derived from sbcim2gVD 28989 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.)

Theoremsbcbi 28626 Implication form of sbcbiiOLD 3219. sbcbi 28626 is sbcbiVD 28990 without virtual deductions and was automatically derived from sbcbiVD 28990 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.)

Theoremtrsbc 28627* Formula-building inference rule for class substitution, substituting a class variable for the set variable of the transitivity predicate. trsbc 28627 is trsbcVD 28991 without virtual deductions and was automatically derived from trsbcVD 28991 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.)

TheoremtruniALT 28628* The union of a class of transitive sets is transitive. Alternate proof of truni 4318. truniALT 28628 is truniALTVD 28992 without virtual deductions and was automatically derived from truniALTVD 28992 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.)

TheoremsbcssOLD 28629 Distribute proper substitution through a subclass relation. This theorem was automatically derived from sbcssVD 28997. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremonfrALTlem5 28630* Lemma for onfrALT 28637. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremonfrALTlem4 28631* Lemma for onfrALT 28637. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremonfrALTlem3 28632* Lemma for onfrALT 28637. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremggen31 28633* gen31 28724 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremonfrALTlem2 28634* Lemma for onfrALT 28637. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremcbvexsv 28635* 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.)

TheoremonfrALTlem1 28636* Lemma for onfrALT 28637. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremonfrALT 28637 The epsilon relation is foundational on the class of ordinal numbers. onfrALT 28637 is an alternate proof of onfr 4622. onfrALTVD 29005 is the Virtual Deduction proof from which onfrALT 28637 is derived. The Virtual Deduction proof mirrors the working proof of onfr 4622 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 29005. 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.)

Theoremcsbeq2g 28638 Formula-building implication rule for class substitution. Closed form of csbeq2i 3279. csbeq2g 28638 is derived from the virtual deduction proof csbeq2gVD 29006. (Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem19.41rg 28639 Closed form of right-to-left implication of 19.41 1901, Theorem 19.41 of [Margaris] p. 90. Derived from 19.41rgVD 29016. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremopelopab4 28640* Ordered pair membership in a class abstraction of pairs. Compare to elopab 4464. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 6-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem2pm13.193 28641 pm13.193 27590 for two variables. pm13.193 27590 is Theorem *13.193 in [WhiteheadRussell] p. 179. Derived from 2pm13.193VD 29017. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremhbntal 28642 A closed form of hbn 1802. hbnt 1800 is another closed form of hbn 1802. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremhbimpg 28643 A closed form of hbim 1837. Derived from hbimpgVD 29018. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremhbalg 28644 Closed form of hbal 1752. Derived from hbalgVD 29019. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremhbexg 28645 Closed form of nfex 1866. Derived from hbexgVD 29020. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 12-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorema9e2eq 28646* Alternate form of a9e 1953 for non-distinct , and . a9e2eq 28646 is derived from a9e2eqVD 29021. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorema9e2nd 28647* If at least two sets exist (dtru 4392) , then the same is true expressed in an alternate form similar to the form of a9e 1953. a9e2nd 28647 is derived from a9e2ndVD 29022. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorema9e2ndeq 28648* "At least two sets exist" expressed in the form of dtru 4392 is logically equivalent to the same expressed in a form similar to a9e 1953 if dtru 4392 is false implies . a9e2ndeq 28648 is derived from a9e2ndeqVD 29023. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem2sb5nd 28649* Equivalence for double substitution 2sb5 2190 without distinct , requirement. 2sb5nd 28649 is derived from 2sb5ndVD 29024. (Contributed by Alan Sare, 30-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem2uasbanh 28650* Distribute the unabbreviated form of proper substitution in and out of a conjunction. 2uasbanh 28650 is derived from 2uasbanhVD 29025. (Contributed by Alan Sare, 31-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem2uasban 28651* 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.)

Theoreme2ebind 28652 Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind 28652 is derived from e2ebindVD 29026. (Contributed by Alan Sare, 27-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremelpwgded 28653 elpwgdedVD 29031 in conventional notation. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremtrelded 28654 Deduction form of trel 4311. In a transitive class, the membership relation is transitive. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremjaoded 28655 Deduction form of jao 500. Disjunction of antecedents. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem3imp31 28656 The importation inference 3imp 1148 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.)

Theorem3imp21 28657 The importation inference 3imp 1148 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.)

Theorembiimpa21 28658 biimpa 472 with commutation of the first and second conjuncts of the assertion. (Contributed by Alan Sare, 11-Sep-2016.)

TheoremsbtT 28659 A substitution into a theorem remains true. sbt 2093 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.)

Theoremex3 28660 Apply ex 425 to a hypothesis with a 3-right-nested conjunction antecedent, with the antecedent of the assertion being a triple conjunction rather than a 2-right-nested conjunction. (Contributed by Alan Sare, 22-Apr-2018.)

Theoremnot12an2impnot1 28661 If a double conjunction is false and the second conjunct is true, then the first conjunct is false. http://www.virtualdeduction.com/not12an2impnot1vd.html is the Virtual Deduction proof verified by automatically transforming it into the Metamath proof of not12an2impnot1 28661 using completeusersproof, which is verified by the Metamath program. http://www.virtualdeduction.com/not12an2impnot1ro.html is a form of the completed proof which preserves the Virtual Deduction proof's step numbers and their ordering. (Contributed by Alan Sare, 13-Jun-2018.)

19.24.4  What is Virtual Deduction?

Syntaxwvd1 28662 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 writing rigorously verifiable mathematical proofs 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)

 if in G1 then in H

By Kleene's COROLLARY of THEOREM 47 (page 448)

 if in G1 then in H if in G1 then in H if in G1 then in H

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:

 df-vd1 28663 dfvd2an 28676 dfvd3an 28688

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)

 if in H then in G1

By Kleene's COROLLARY 2 of THEOREM 46 (page 446)

 if in H then in G1 if in H then in G1 if in H then in G1

To prove in H the User simply proves in G1 and translates each G1 proof step into a H proof step. The translation is trivial and immediate. The proof in H is in Virtual Deduction notation. It is a working proof in the sense that, if it has no errors, each theorem and deduction of the proof is true, but may or may not, after being translated into conventional notation, unify with any theorem or deduction schemata in set.mm. Each theorem or deduction schemata in set.mm has a particular form. The working proof written by the User (the "User's Proof" or "Virtual Deduction Proof") may contain theorems and deductions which would unify with a variant of a theorem or deduction schemata in set.mm, but not with any particular form of that theorem or deduction schemata in set.mm.

The computer program completeusersproof.c may be applied to a Virtual Deduction proof to automatically add steps to the proof ("technical steps") which, if possible, transforms the form of a theorem or deduction of the Virtual Deduction proof not unifiable with a theorem or deduction schemata in set.mm into a variant form which is. For theorems and deductions of the Virtual Deduction proof which are completable in this way, completeusersproof saves the User the extra work involved in satisfying the constraint that the theorem or deduction is in a form which unifies with a theorem or deduction schemata in set.mm. mmj2, which is invoked by completeusersproof, automatically finds one of the reference theorems or deductions in set.mm which unifies with each theorem and deduction in the proof satisfying this constraint and labels the theorem or the assertion step of the deduction.

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 (superseded by completeusersproof.c in September of 2016). 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.

The input for completeusersproof.c is a Virtual Deduction proof. Unlike completeusersproof.cmd, the completed proof is in conventional notation. completeusersproof.c eliminates the virtual deduction notation of the Virtual Deduction proof after utilizing the information it provides.

Applying mmj2's unify command is essential to completeusersproof. 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 a Metamath-specific version of a Natural Deduction Proof. In order for mmj2 to complete a Virtual Deduction proof it is necessary that each theorem or deduction of the proof is in a form which unifies with a theorem or deduction schemata in set.mm. completeusersproof weakens this constraint.

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.

The completeusersproof program and all associated files necessary to use it may be downloaded from the Metamath web site. All syntax definitions, theorems, and deductions necessary to create Virtual Deduction proofs are contained in set.mm. Examples of Virtual Deduction proofs in mmj2 Proof Worksheet .txt format are included in the completeusersproof download.

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

Definitiondf-vd1 28663 Definition of virtual deduction. (Contributed by Alan Sare, 21-Apr-2011.) (New usage is discouraged.)

Theoremin1 28664 Inference form of df-vd1 28663. 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.)

Theoremiin1 28665 in1 28664 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdfvd1ir 28666 Inference form of df-vd1 28663 with the virtual deduction as the assertion. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremidn1 28667 Virtual deduction identity rule which is id 21 with virtual deduction symbols. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax172 28668* 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.)

Theoremdfvd1imp 28669 Left-to-right part of definition of virtual deduction. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdfvd1impr 28670 Right-to-left part of definition of virtual deduction. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Syntaxwvd2 28671 Syntax for a 2-hypothesis virtual deduction. (New usage is discouraged.)

Definitiondf-vd2 28672 Definition of a 2-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (New usage is discouraged.)

Theoremdfvd2 28673 Definition of a 2-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Syntaxwvhc2 28674 Syntax for a 2-virtual hypotheses collection. (Contributed by Alan Sare, 23-Apr-2015.) (New usage is discouraged.)

Definitiondf-vhc2 28675 Definition of a 2-virtual hypotheses collection. (Contributed by Alan Sare, 23-Apr-2015.) (New usage is discouraged.)

Theoremdfvd2an 28676 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.)

Theoremdfvd2ani 28677 Inference form of dfvd2an 28676. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdfvd2anir 28678 Right-to-left inference form of dfvd2an 28676. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdfvd2i 28679 Inference form of dfvd2 28673. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdfvd2ir 28680 Right-to-left inference form of dfvd2 28673. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Syntaxwvd3 28681 Syntax for a 3-hypothesis virtual deduction. (New usage is discouraged.)

Syntaxwvhc3 28682 Syntax for a 3-virtual hypotheses collection. (Contributed by Alan Sare, 13-Jun-2015.) (New usage is discouraged.)

Definitiondf-vhc3 28683 Definition of a 3-virtual hypotheses collection. (Contributed by Alan Sare, 13-Jun-2015.) (New usage is discouraged.)

Definitiondf-vd3 28684 Definition of a 3-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (New usage is discouraged.)

Theoremdfvd3 28685 Definition of a 3-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdfvd3i 28686 Inference form of dfvd3 28685. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdfvd3ir 28687 Right-to-left inference form of dfvd3 28685. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdfvd3an 28688 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.)

Theoremdfvd3ani 28689 Inference form of dfvd3an 28688. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdfvd3anir 28690 Right-to-left inference form of dfvd3an 28688. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Syntaxwvhc4 28691 Syntax for a 4-virtual hypotheses collection. (Contributed by Alan Sare, 17-Oct-2017.) (New usage is discouraged.)

Syntaxwvhc5 28692 Syntax for a 5-virtual hypotheses collection. (Contributed by Alan Sare, 17-Oct-2017.) (New usage is discouraged.)

Syntaxwvhc6 28693 Syntax for a 6-element virtual hypotheses collection. (Contributed by Alan Sare, 17-Oct-2017.) (New usage is discouraged.)

Syntaxwvhc7 28694 Syntax for a 7-element virtual hypotheses collection. (Contributed by Alan Sare, 17-Oct-2017.) (New usage is discouraged.)

Syntaxwvhc8 28695 Syntax for an 8-element virtual hypotheses collection. (Contributed by Alan Sare, 17-Oct-2017.) (New usage is discouraged.)

Syntaxwvhc9 28696 Syntax for a 9-element virtual hypotheses collection. (Contributed by Alan Sare, 17-Oct-2017.) (New usage is discouraged.)

Syntaxwvhc10 28697 Syntax for a 10-element virtual hypotheses collection. (Contributed by Alan Sare, 17-Oct-2017.) (New usage is discouraged.)

Syntaxwvhc11 28698 Syntax for an 11-element virtual hypotheses collection. (Contributed by Alan Sare, 17-Oct-2017.) (New usage is discouraged.)

Syntaxwvhc12 28699 Syntax for an 12-element virtual hypotheses collection. (Contributed by Alan Sare, 17-Oct-2017.) (New usage is discouraged.)

Theoremvd01 28700 A virtual hypothesis virtually infers a theorem. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

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