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

Theorem | ordelordALT 28301 | An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 4414 using the Axiom of Regularity indirectly through dford2 7321. 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 28301 is ordelordALTVD 28643 without virtual deductions and was automatically derived from ordelordALTVD 28643 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 28302 | Distribution of class substitution over a left-nested implication. Similar to sbcimg 3032. sbcim2g 28302 is sbcim2gVD 28651 without virtual deductions and was automatically derived from sbcim2gVD 28651 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 28303 | Implication form of sbcbiiOLD 3047. sbcbi 28303 is sbcbiVD 28652 without virtual deductions and was automatically derived from sbcbiVD 28652 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 28304* | Formula-building inference rule for class substitution, substituting a class variable for the set variable of the transitivity predicate. trsbc 28304 is trsbcVD 28653 without virtual deductions and was automatically derived from trsbcVD 28653 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 28305* | The union of a class of transitive sets is transitive. Alternate proof of truni 4127. truniALT 28305 is truniALTVD 28654 without virtual deductions and was automatically derived from truniALTVD 28654 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 28306 | Distribute proper substitution through a subclass relation. This theorem was automatically derived from sbcssVD 28659. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | onfrALTlem5 28307* | Lemma for onfrALT 28314. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | onfrALTlem4 28308* | Lemma for onfrALT 28314. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | onfrALTlem3 28309* | Lemma for onfrALT 28314. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | ggen31 28310* | gen31 28393 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | onfrALTlem2 28311* | |||||||||||||||

Theorem | cbvexsv 28312* | 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 28313* | |||||||||||||||

Theorem | onfrALT 28314 | The epsilon relation is foundational on the class of ordinal numbers. onfrALT 28314 is an alternate proof of onfr 4431. onfrALTVD 28667 is the Virtual Deduction proof from which onfrALT 28314 is derived. The Virtual Deduction proof mirrors the working proof of onfr 4431 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 28667. 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 28315 | Formula-building implication rule for class substitution. Closed form of csbeq2i 3107. csbeq2g 28315 is derived from the virtual deduction proof csbeq2gVD 28668. (Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | 19.41rg 28316 | Closed form of right-to-left implication of 19.41 1815, Theorem 19.41 of [Margaris] p. 90. Derived from 19.41rgVD 28678. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | opelopab4 28317* | Ordered pair membership in a class abstraction of pairs. Compare to elopab 4272. (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 28318 | pm13.193 27611 for two variables. pm13.193 27611 is Theorem *13.193 in [WhiteheadRussell] p. 179. Derived from 2pm13.193VD 28679. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | hbntal 28319 | A closed form of hbn 1720. hbnt 1724 is another closed form of hbn 1720. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | hbimpg 28320 | A closed form of hbim 1725. Derived from hbimpgVD 28680. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | hbalg 28321 | Closed form of hbal 1710. Derived from hbalgVD 28681. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | hbexg 28322 | Closed form of nfex 1767. Derived from hbexgVD 28682. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 12-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | a9e2eq 28323* | Alternate form of a9e 1891 for non-distinct , and . a9e2eq 28323 is derived from a9e2eqVD 28683. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | a9e2nd 28324* | If at least two sets exist (dtru 4201) , then the same is true expressed in an alternate form similar to the form of a9e 1891. a9e2nd 28324 is derived from a9e2ndVD 28684. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | a9e2ndeq 28325* | "At least two sets exist" expressed in the form of dtru 4201 is logically equivalent to the same expressed in a form similar to a9e 1891 if dtru 4201 is false implies . a9e2ndeq 28325 is derived from a9e2ndeqVD 28685. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | 2sb5nd 28326* | Equivalence for double substitution 2sb5 2051 without distinct , requirement. 2sb5nd 28326 is derived from 2sb5ndVD 28686. (Contributed by Alan Sare, 30-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | 2uasbanh 28327* | Distribute the unabbreviated form of proper substitution in and out of a conjunction. 2uasbanh 28327 is derived from 2uasbanhVD 28687. (Contributed by Alan Sare, 31-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | 2uasban 28328* | 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 28329 | Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind 28329 is derived from e2ebindVD 28688. (Contributed by Alan Sare, 27-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | elpwgded 28330 | elpwgdedVD 28693 in conventional notation. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | trelded 28331 | Deduction form of trel 4120. 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 28332 | Deduction form of jao 498. Disjunction of antecedents. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | 3imp31 28333 | The importation inference 3imp 1145 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 28334 | The importation inference 3imp 1145 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 28335 | biimpa 470 with commutation of the first and second conjuncts of the assertion. (Contributed by Alan Sare, 11-Sep-2016.) | ||||||||||||||

Theorem | sbtT 28336 | A substitution into a theorem remains true. sbt 1973 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.) | ||||||||||||||

18.25.2 What is Virtual Deduction? | ||||||||||||||||

Syntax | wvd1 28337 |
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. | ||||||||||||||

18.25.3 Virtual Deduction Theorems | ||||||||||||||||

Definition | df-vd1 28338 | Definition of virtual deduction. (Contributed by Alan Sare, 21-Apr-2011.) (New usage is discouraged.) | ||||||||||||||

Theorem | in1 28339 | Inference form of df-vd1 28338. 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 28340 | in1 28339 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | dfvd1ir 28341 | Inference form of df-vd1 28338 with the virtual deduction as the assertion. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | idn1 28342 | Virtual deduction identity rule which is id 19 with virtual deduction symbols. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | ax172 28343* | 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 28344 | 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 28345 | 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 28346 | Syntax for a 2-hypothesis virtual deduction. (New usage is discouraged.) | ||||||||||||||

Definition | df-vd2 28347 | Definition of a 2-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (New usage is discouraged.) | ||||||||||||||

Theorem | dfvd2 28348 | Definition of a 2-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Syntax | wvhc2 28349 | Syntax for a 2-virtual hypotheses collection. (Contributed by Alan Sare, 23-Apr-2015.) (New usage is discouraged.) | ||||||||||||||

Definition | df-vhc2 28350 | Definition of a 2-virtual hypotheses collection. (Contributed by Alan Sare, 23-Apr-2015.) (New usage is discouraged.) | ||||||||||||||

Theorem | dfvd2an 28351 | 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 28352 | Inference form of dfvd2an 28351. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | dfvd2anir 28353 | Right-to-left inference form of dfvd2an 28351. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | dfvd2i 28354 | Inference form of dfvd2 28348. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | dfvd2ir 28355 | Right-to-left inference form of dfvd2 28348. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Syntax | wvd3 28356 | Syntax for a 3-hypothesis virtual deduction. (New usage is discouraged.) | ||||||||||||||

Syntax | wvhc3 28357 | Syntax for a 3-virtual hypotheses collection. (Contributed by Alan Sare, 13-Jun-2015.) (New usage is discouraged.) | ||||||||||||||

Definition | df-vhc3 28358 | Definition of a 3-virtual hypotheses collection. (Contributed by Alan Sare, 13-Jun-2015.) (New usage is discouraged.) | ||||||||||||||

Definition | df-vd3 28359 | Definition of a 3-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (New usage is discouraged.) | ||||||||||||||

Theorem | dfvd3 28360 | Definition of a 3-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | dfvd3i 28361 | Inference form of dfvd3 28360. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | dfvd3ir 28362 | Right-to-left inference form of dfvd3 28360. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | dfvd3an 28363 | 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 28364 | Inference form of dfvd3an 28363. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | dfvd3anir 28365 | Right-to-left inference form of dfvd3an 28363. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Syntax | wvhc4 28366 | Syntax for a 4-virtual hypotheses collection. (Contributed by Alan Sare, 12-Sep-2017.) (New usage is discouraged.) | ||||||||||||||

Syntax | wvhc5 28367 | Syntax for a 5-virtual hypotheses collection. (Contributed by Alan Sare, 12-Sep-2017.) (New usage is discouraged.) | ||||||||||||||

Syntax | w5a 28368 | Extend wff definition to include 5-way conjunction ('and'). (Contributed by Alan Sare, 12-Sep-2017.) (New usage is discouraged.) | ||||||||||||||

Theorem | vd01 28369 | A virtual hypothesis virtually infers a theorem. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | vd02 28370 | 2 virtual hypotheses virtually infer a theorem. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | vd03 28371 | A theorem is virtually inferred by the 3 virtual hypotheses. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | vd12 28372 | A virtual deduction with 1 virtual hypothesis virtually inferring a virtual conclusion infers that the same conclusion is virtually inferred by the same virtual hypothesis and an additional hypothesis. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | vd13 28373 | A virtual deduction with 1 virtual hypothesis virtually inferring a virtual conclusion infers that the same conclusion is virtually inferred by the same virtual hypothesis and a two additional hypotheses. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | vd23 28374 | A virtual deduction with 2 virtual hypotheses virtually inferring a virtual conclusion infers that the same conclusion is virtually inferred by the same 2 virtual hypotheses and a third hypothesis. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | dfvd2imp 28375 | The virtual deduction form of a 2-antecedent nested implication implies the 2-antecedent nested implication. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | dfvd2impr 28376 | A 2-antecedent nested implication implies its virtual deduction form. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | in2 28377 | The virtual deduction introduction rule of converting the end virtual hypothesis of 2 virtual hypotheses into an antecedent. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | int2 28378 | The virtual deduction introduction rule of converting the end virtual hypothesis of 2 virtual hypotheses into an antecedent. Conventional form of int2 28378 is ex 423. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | iin2 28379 | in2 28377 without virtual deductions. (Contributed by Alan Sare, 20-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | in2an 28380 | The virtual deduction introduction rule converting the second conjunct of the second virtual hypothesis into the antecedent of the conclusion. exp3a 425 is the non-virtual deduction form of in2an 28380. (Contributed by Alan Sare, 30-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | in3 28381 | The virtual deduction introduction rule of converting the end virtual hypothesis of 3 virtual hypotheses into an antecedent. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | iin3 28382 | in3 28381 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | in3an 28383 | The virtual deduction introduction rule converting the second conjunct of the third virtual hypothesis into the antecedent of the conclusion. exp4a 589 is the non-virtual deduction form of in3an 28383. (Contributed by Alan Sare, 25-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | int3 28384 | The virtual deduction introduction rule of converting the end virtual hypothesis of 3 virtual hypotheses into an antecedent. Conventional form of int3 28384 is 3expia 1153. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | idn2 28385 | Virtual deduction identity rule which is idd 21 with virtual deduction symbols. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | iden2 28386 | Virtual deduction identity rule. simpr 447 in conjunction form Virtual Deduction notation. (Contributed by Alan Sare, 5-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | idn3 28387 | Virtual deduction identity rule for 3 virtual hypotheses. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | gen11 28388* | Virtual deduction generalizing rule for 1 quantifying variable and 1 virtual hypothesis. alrimiv 1617 is gen11 28388 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | gen11nv 28389 | Virtual deduction generalizing rule for 1 quantifying variable and 1 virtual hypothesis without distinct variables. alrimih 1552 is gen11nv 28389 without virtual deductions. (Contributed by Alan Sare, 12-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | gen12 28390* | Virtual deduction generalizing rule for 2 quantifying variables and 1 virtual hypothesis. gen12 28390 is alrimivv 1618 with virtual deductions. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | gen21 28391* | Virtual deduction generalizing rule for 1 quantifying variables and 2 virtual hypothesis. gen21 28391 is alrimdv 1619 with virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | gen21nv 28392 | Virtual deduction form of alrimdh 1574. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | gen31 28393* | Virtual deduction generalizing rule for 1 quantifying variable and 3 virtual hypothesis. gen31 28393 is ggen31 28310 with virtual deductions. (Contributed by Alan Sare, 22-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | gen22 28394* | Virtual deduction generalizing rule for 2 quantifying variables and 2 virtual hypothesis. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | ggen22 28395* | gen22 28394 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | exinst 28396 | Existential Instantiation. Virtual deduction form of exlimexi 28287. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | exinst01 28397 | Existential Instantiation. Virtual Deduction rule corresponding to a special case of the Natural Deduction Sequent Calculus rule called Rule C in [Margaris] p. 79 and E in Table 1 on page 4 of the paper "Extracting information from intermediate T-systems" (2000) presented at IMLA99 by Mauro Ferrari, Camillo Fiorentini, and Pierangelo Miglioli. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | exinst11 28398 | Existential Instantiation. Virtual Deduction rule corresponding to a special case of the Natural Deduction Sequent Calculus rule called Rule C in [Margaris] p. 79 and E in Table 1 on page 4 of the paper "Extracting information from intermediate T-systems" (2000) presented at IMLA99 by Mauro Ferrari, Camillo Fiorentini, and Pierangelo Miglioli. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | e1_ 28399 | A Virtual deduction elimination rule. syl 15 is e1_ 28399 without virtual deductions. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

Theorem | el1 28400 | A Virtual deduction elimination rule. syl 15 is el1 28400 without virtual deductions. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||

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