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Type | Label | Description | ||||||||||||||
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Statement | ||||||||||||||||
Theorem | sbc3or 41601 | sbcor 3747 with a 3-disjuncts. This proof is sbc3orgVD 41920 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Revised by NM, 24-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ([𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓 ∨ [𝐴 / 𝑥]𝜒)) | ||||||||||||||||
Theorem | alrim3con13v 41602* | Closed form of alrimi 2212 with 2 additional conjuncts having no occurrences of the quantifying variable. This proof is 19.21a3con13vVD 41921 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒))) | ||||||||||||||||
Theorem | rspsbc2 41603* | rspsbc 3786 with two quantifying variables. This proof is rspsbc2VD 41924 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑))) | ||||||||||||||||
Theorem | sbcoreleleq 41604* | Substitution of a setvar variable for another setvar variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 41928. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴))) | ||||||||||||||||
Theorem | tratrb 41605* | 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 41930. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → Tr 𝐵) | ||||||||||||||||
Theorem | ordelordALT 41606 | An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 6189 using the Axiom of Regularity indirectly through dford2 9106. dford2 is a weaker definition of ordinal number. Given the Axiom of Regularity, it need not be assumed that E Fr 𝐴 because this is inferred by the Axiom of Regularity. ordelordALT 41606 is ordelordALTVD 41936 without virtual deductions and was automatically derived from ordelordALTVD 41936 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.) | ||||||||||||||
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵) | ||||||||||||||||
Theorem | sbcim2g 41607 | Distribution of class substitution over a left-nested implication. Similar to sbcimg 3745. sbcim2g 41607 is sbcim2gVD 41944 without virtual deductions and was automatically derived from sbcim2gVD 41944 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.) | ||||||||||||||
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))) | ||||||||||||||||
Theorem | sbcbi 41608 | Implication form of sbcbii 3754. sbcbi 41608 is sbcbiVD 41945 without virtual deductions and was automatically derived from sbcbiVD 41945 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.) | ||||||||||||||
⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) | ||||||||||||||||
Theorem | trsbc 41609* | Formula-building inference rule for class substitution, substituting a class variable for the setvar variable of the transitivity predicate. trsbc 41609 is trsbcVD 41946 without virtual deductions and was automatically derived from trsbcVD 41946 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.) | ||||||||||||||
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴)) | ||||||||||||||||
Theorem | truniALT 41610* | The union of a class of transitive sets is transitive. Alternate proof of truni 5150. truniALT 41610 is truniALTVD 41947 without virtual deductions and was automatically derived from truniALTVD 41947 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.) | ||||||||||||||
⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪ 𝐴) | ||||||||||||||||
Theorem | onfrALTlem5 41611* | Lemma for onfrALT 41618. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ([(𝑎 ∩ 𝑥) / 𝑏]((𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦 ∈ 𝑏 (𝑏 ∩ 𝑦) = ∅) ↔ (((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩ 𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)) | ||||||||||||||||
Theorem | onfrALTlem4 41612* | Lemma for onfrALT 41618. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) | ||||||||||||||||
Theorem | onfrALTlem3 41613* | Lemma for onfrALT 41618. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)) | ||||||||||||||||
Theorem | ggen31 41614* | gen31 41690 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → ∀𝑥𝜃))) | ||||||||||||||||
Theorem | onfrALTlem2 41615* | Lemma for onfrALT 41618. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)) | ||||||||||||||||
Theorem | cbvexsv 41616* | 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.) | ||||||||||||||
⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) | ||||||||||||||||
Theorem | onfrALTlem1 41617* | Lemma for onfrALT 41618. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)) | ||||||||||||||||
Theorem | onfrALT 41618 | The membership relation is foundational on the class of ordinal numbers. onfrALT 41618 is an alternate proof of onfr 6206. onfrALTVD 41960 is the Virtual Deduction proof from which onfrALT 41618 is derived. The Virtual Deduction proof mirrors the working proof of onfr 6206 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 41960. 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.) | ||||||||||||||
⊢ E Fr On | ||||||||||||||||
Theorem | 19.41rg 41619 | Closed form of right-to-left implication of 19.41 2236, Theorem 19.41 of [Margaris] p. 90. Derived from 19.41rgVD 41971. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓))) | ||||||||||||||||
Theorem | opelopab4 41620* | Ordered pair membership in a class abstraction of ordered pairs. Compare to elopab 5382. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 6-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (⟨𝑢, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)) | ||||||||||||||||
Theorem | 2pm13.193 41621 | pm13.193 41478 for two variables. pm13.193 41478 is Theorem *13.193 in [WhiteheadRussell] p. 179. Derived from 2pm13.193VD 41972. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)) | ||||||||||||||||
Theorem | hbntal 41622 | A closed form of hbn 2300. hbnt 2299 is another closed form of hbn 2300. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑)) | ||||||||||||||||
Theorem | hbimpg 41623 | A closed form of hbim 2304. Derived from hbimpgVD 41973. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ((∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓)) → ∀𝑥((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))) | ||||||||||||||||
Theorem | hbalg 41624 | Closed form of hbal 2172. Derived from hbalgVD 41974. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)) | ||||||||||||||||
Theorem | hbexg 41625 | Closed form of nfex 2333. Derived from hbexgVD 41975. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 12-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)) | ||||||||||||||||
Theorem | ax6e2eq 41626* | Alternate form of ax6e 2391 for non-distinct 𝑥, 𝑦 and 𝑢 = 𝑣. ax6e2eq 41626 is derived from ax6e2eqVD 41976. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))) | ||||||||||||||||
Theorem | ax6e2nd 41627* | If at least two sets exist (dtru 5237) , then the same is true expressed in an alternate form similar to the form of ax6e 2391. ax6e2nd 41627 is derived from ax6e2ndVD 41977. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) | ||||||||||||||||
Theorem | ax6e2ndeq 41628* | "At least two sets exist" expressed in the form of dtru 5237 is logically equivalent to the same expressed in a form similar to ax6e 2391 if dtru 5237 is false implies 𝑢 = 𝑣. ax6e2ndeq 41628 is derived from ax6e2ndeqVD 41978. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) ↔ ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) | ||||||||||||||||
Theorem | 2sb5nd 41629* | Equivalence for double substitution 2sb5 2279 without distinct 𝑥, 𝑦 requirement. 2sb5nd 41629 is derived from 2sb5ndVD 41979. (Contributed by Alan Sare, 30-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))) | ||||||||||||||||
Theorem | 2uasbanh 41630* | Distribute the unabbreviated form of proper substitution in and out of a conjunction. 2uasbanh 41630 is derived from 2uasbanhVD 41980. (Contributed by Alan Sare, 31-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (𝜒 ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))) ⇒ ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)) ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))) | ||||||||||||||||
Theorem | 2uasban 41631* | 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.) | ||||||||||||||
⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)) ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))) | ||||||||||||||||
Theorem | e2ebind 41632 | Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind 41632 is derived from e2ebindVD 41981. (Contributed by Alan Sare, 27-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑)) | ||||||||||||||||
Theorem | elpwgded 41633 | elpwgdedVD 41986 in conventional notation. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜓 → 𝐴 ⊆ 𝐵) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝒫 𝐵) | ||||||||||||||||
Theorem | trelded 41634 | Deduction form of trel 5143. In a transitive class, the membership relation is transitive. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (𝜑 → Tr 𝐴) & ⊢ (𝜓 → 𝐵 ∈ 𝐶) & ⊢ (𝜒 → 𝐶 ∈ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝐵 ∈ 𝐴) | ||||||||||||||||
Theorem | jaoded 41635 | Deduction form of jao 959. Disjunction of antecedents. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜃 → (𝜏 → 𝜒)) & ⊢ (𝜂 → (𝜓 ∨ 𝜏)) ⇒ ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜂) → 𝜒) | ||||||||||||||||
Theorem | sbtT 41636 | A substitution into a theorem remains true. sbt 2072 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.) | ||||||||||||||
⊢ (⊤ → 𝜑) ⇒ ⊢ [𝑦 / 𝑥]𝜑 | ||||||||||||||||
Theorem | not12an2impnot1 41637 | If a double conjunction is false and the second conjunct is true, then the first conjunct is false. https://us.metamath.org/other/completeusersproof/not12an2impnot1vd.html is the Virtual Deduction proof verified by automatically transforming it into the Metamath proof of not12an2impnot1 41637 using completeusersproof, which is verified by the Metamath program. https://us.metamath.org/other/completeusersproof/not12an2impnot1ro.html 41637 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.) | ||||||||||||||
⊢ ((¬ (𝜑 ∧ 𝜓) ∧ 𝜓) → ¬ 𝜑) | ||||||||||||||||
Syntax | wvd1 41638 |
A Virtual Deduction proof in a Hilbert-style deductive system is the
analogue 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 analogue 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-theoretical 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 excerpt 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 𝜑, . . . , 𝜓 → 𝜒, . . . , 𝜃 where 𝜑 , . . . , 𝜓 and 𝜒, . . . , 𝜃 are seqences of a finite number of 0 or more formulas (substituting Metamath notation for Kleene's notation). The part 𝜑, . . . , 𝜓 is the antecedent, and 𝜒, . . . , 𝜃 the succedent of the sequent 𝜑, . . . , 𝜓 → 𝜒, . . . , 𝜃. When the antecedent and the succedent each have a finite number of 1 or more formulas, the sequent 𝜑, . . . , 𝜓 → 𝜒, . . . 𝜃 has the same interpretation for G1 as the formula ((𝜑 ∧. . . ∧ 𝜓) → (𝜒 ∨. . . ∨ 𝜃)) for H. The interpretation extends to the case of an antecedent of 0 formulas by regarding (𝜑 ∧. . . ∧ 𝜓) for 0 formulas (the "empty conjunction") as true and (𝜒 ∨. . . ∨ 𝜃) 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 𝜑, . . . 𝜓 of 0 or more (occurrences of) formulas, a finite sequence of one or more (occurrences of) formulas is called a (formal) deduction from the assumption formulas 𝜑, . . . 𝜓, if each formula of the sequence is either one of the formulas 𝜑, . . . 𝜓, 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, 𝜑,. . . . ,. 𝜓⊢ 𝜒), 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)
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 analogue in H of a sequent in G1 having a nonempty antecedent. If ▶ occurs as the outermost connective denoted by ▶ or → and occurs exactly once, we call the analogue 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 ⊢ 𝜑 in set.mm corresponds to the theorem ⊢ → 𝜑 in G1. Deductions in G1 corresponding to deductions in H are similarly determined. Theorems in H with one or more occurrences of either ▶ or → may also be translated into theorems in G1 for by replacing the outermost occurrence 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 occurrences of ▶ and it is the outermost occurrence 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)
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 scheme in set.mm. Each theorem or deduction scheme 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 scheme in set.mm, but not with any particular form of that theorem or deduction scheme 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 scheme 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 scheme 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 analogue 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 analogue 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 L. 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 scheme 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. https://us.metamath.org/other/completeusersproof/suctrvd.html 41663, https://us.metamath.org/other/completeusersproof/sineq0altvd.html 41663, https://us.metamath.org/other/completeusersproof/iunconlem2vd.html 41663, https://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html 41663, and https://us.metamath.org/other/completeusersproof/chordthmaltvd.html 41663 are examples of Virtual Deduction proofs. Generally, proving using Virtual Deduction and completeusersproof reduces the amount of 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. | ||||||||||||||
wff ( 𝜑 ▶ 𝜓 ) | ||||||||||||||||
Definition | df-vd1 41639 | Definition of virtual deduction. (Contributed by Alan Sare, 21-Apr-2011.) (New usage is discouraged.) | ||||||||||||||
⊢ (( 𝜑 ▶ 𝜓 ) ↔ (𝜑 → 𝜓)) | ||||||||||||||||
Theorem | in1 41640 | Inference form of df-vd1 41639. 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.) | ||||||||||||||
⊢ ( 𝜑 ▶ 𝜓 ) ⇒ ⊢ (𝜑 → 𝜓) | ||||||||||||||||
Theorem | iin1 41641 | in1 41640 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||||||||||||||||
Theorem | dfvd1ir 41642 | Inference form of df-vd1 41639 with the virtual deduction as the assertion. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (𝜑 → 𝜓) ⇒ ⊢ ( 𝜑 ▶ 𝜓 ) | ||||||||||||||||
Theorem | idn1 41643 | Virtual deduction identity rule which is id 22 with virtual deduction symbols. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( 𝜑 ▶ 𝜑 ) | ||||||||||||||||
Theorem | dfvd1imp 41644 | Left-to-right part of definition of virtual deduction. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (( 𝜑 ▶ 𝜓 ) → (𝜑 → 𝜓)) | ||||||||||||||||
Theorem | dfvd1impr 41645 | Right-to-left part of definition of virtual deduction. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ((𝜑 → 𝜓) → ( 𝜑 ▶ 𝜓 )) | ||||||||||||||||
Syntax | wvd2 41646 | Syntax for a 2-hypothesis virtual deduction. (New usage is discouraged.) | ||||||||||||||
wff ( 𝜑 , 𝜓 ▶ 𝜒 ) | ||||||||||||||||
Definition | df-vd2 41647 | Definition of a 2-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (New usage is discouraged.) | ||||||||||||||
⊢ (( 𝜑 , 𝜓 ▶ 𝜒 ) ↔ ((𝜑 ∧ 𝜓) → 𝜒)) | ||||||||||||||||
Theorem | dfvd2 41648 | Definition of a 2-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (( 𝜑 , 𝜓 ▶ 𝜒 ) ↔ (𝜑 → (𝜓 → 𝜒))) | ||||||||||||||||
Syntax | wvhc2 41649 | Syntax for a 2-element virtual hypotheses collection. (Contributed by Alan Sare, 23-Apr-2015.) (New usage is discouraged.) | ||||||||||||||
wff ( 𝜑 , 𝜓 ) | ||||||||||||||||
Definition | df-vhc2 41650 | Definition of a 2-element virtual hypotheses collection. (Contributed by Alan Sare, 23-Apr-2015.) (New usage is discouraged.) | ||||||||||||||
⊢ (( 𝜑 , 𝜓 ) ↔ (𝜑 ∧ 𝜓)) | ||||||||||||||||
Theorem | dfvd2an 41651 | 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.) | ||||||||||||||
⊢ (( ( 𝜑 , 𝜓 ) ▶ 𝜒 ) ↔ ((𝜑 ∧ 𝜓) → 𝜒)) | ||||||||||||||||
Theorem | dfvd2ani 41652 | Inference form of dfvd2an 41651. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( ( 𝜑 , 𝜓 ) ▶ 𝜒 ) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||||||||||||||||
Theorem | dfvd2anir 41653 | Right-to-left inference form of dfvd2an 41651. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ ( ( 𝜑 , 𝜓 ) ▶ 𝜒 ) | ||||||||||||||||
Theorem | dfvd2i 41654 | Inference form of dfvd2 41648. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) ⇒ ⊢ (𝜑 → (𝜓 → 𝜒)) | ||||||||||||||||
Theorem | dfvd2ir 41655 | Right-to-left inference form of dfvd2 41648. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) | ||||||||||||||||
Syntax | wvd3 41656 | Syntax for a 3-hypothesis virtual deduction. (New usage is discouraged.) | ||||||||||||||
wff ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) | ||||||||||||||||
Syntax | wvhc3 41657 | Syntax for a 3-element virtual hypotheses collection. (Contributed by Alan Sare, 13-Jun-2015.) (New usage is discouraged.) | ||||||||||||||
wff ( 𝜑 , 𝜓 , 𝜒 ) | ||||||||||||||||
Definition | df-vhc3 41658 | Definition of a 3-element virtual hypotheses collection. (Contributed by Alan Sare, 13-Jun-2015.) (New usage is discouraged.) | ||||||||||||||
⊢ (( 𝜑 , 𝜓 , 𝜒 ) ↔ (𝜑 ∧ 𝜓 ∧ 𝜒)) | ||||||||||||||||
Definition | df-vd3 41659 | Definition of a 3-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (New usage is discouraged.) | ||||||||||||||
⊢ (( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)) | ||||||||||||||||
Theorem | dfvd3 41660 | Definition of a 3-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃)))) | ||||||||||||||||
Theorem | dfvd3i 41661 | Inference form of dfvd3 41660. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) | ||||||||||||||||
Theorem | dfvd3ir 41662 | Right-to-left inference form of dfvd3 41660. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) ⇒ ⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) | ||||||||||||||||
Theorem | dfvd3an 41663 | 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.) | ||||||||||||||
⊢ (( ( 𝜑 , 𝜓 , 𝜒 ) ▶ 𝜃 ) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)) | ||||||||||||||||
Theorem | dfvd3ani 41664 | Inference form of dfvd3an 41663. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( ( 𝜑 , 𝜓 , 𝜒 ) ▶ 𝜃 ) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||||||||||||||||
Theorem | dfvd3anir 41665 | Right-to-left inference form of dfvd3an 41663. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ( ( 𝜑 , 𝜓 , 𝜒 ) ▶ 𝜃 ) | ||||||||||||||||
Theorem | vd01 41666 | A virtual hypothesis virtually infers a theorem. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ 𝜑 ⇒ ⊢ ( 𝜓 ▶ 𝜑 ) | ||||||||||||||||
Theorem | vd02 41667 | Two virtual hypotheses virtually infer a theorem. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ 𝜑 ⇒ ⊢ ( 𝜓 , 𝜒 ▶ 𝜑 ) | ||||||||||||||||
Theorem | vd03 41668 | 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.) | ||||||||||||||
⊢ 𝜑 ⇒ ⊢ ( 𝜓 , 𝜒 , 𝜃 ▶ 𝜑 ) | ||||||||||||||||
Theorem | vd12 41669 | 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.) | ||||||||||||||
⊢ ( 𝜑 ▶ 𝜓 ) ⇒ ⊢ ( 𝜑 , 𝜒 ▶ 𝜓 ) | ||||||||||||||||
Theorem | vd13 41670 | 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.) | ||||||||||||||
⊢ ( 𝜑 ▶ 𝜓 ) ⇒ ⊢ ( 𝜑 , 𝜒 , 𝜃 ▶ 𝜓 ) | ||||||||||||||||
Theorem | vd23 41671 | 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.) | ||||||||||||||
⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) ⇒ ⊢ ( 𝜑 , 𝜓 , 𝜃 ▶ 𝜒 ) | ||||||||||||||||
Theorem | dfvd2imp 41672 | 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.) | ||||||||||||||
⊢ (( 𝜑 , 𝜓 ▶ 𝜒 ) → (𝜑 → (𝜓 → 𝜒))) | ||||||||||||||||
Theorem | dfvd2impr 41673 | 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.) | ||||||||||||||
⊢ ((𝜑 → (𝜓 → 𝜒)) → ( 𝜑 , 𝜓 ▶ 𝜒 )) | ||||||||||||||||
Theorem | in2 41674 | 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.) | ||||||||||||||
⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) ⇒ ⊢ ( 𝜑 ▶ (𝜓 → 𝜒) ) | ||||||||||||||||
Theorem | int2 41675 | The virtual deduction introduction rule of converting the end virtual hypothesis of 2 virtual hypotheses into an antecedent. Conventional form of int2 41675 is ex 417. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( ( 𝜑 , 𝜓 ) ▶ 𝜒 ) ⇒ ⊢ ( 𝜑 ▶ (𝜓 → 𝜒) ) | ||||||||||||||||
Theorem | iin2 41676 | in2 41674 without virtual deductions. (Contributed by Alan Sare, 20-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → 𝜒)) | ||||||||||||||||
Theorem | in2an 41677 | The virtual deduction introduction rule converting the second conjunct of the second virtual hypothesis into the antecedent of the conclusion. expd 420 is the non-virtual deduction form of in2an 41677. (Contributed by Alan Sare, 30-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( 𝜑 , (𝜓 ∧ 𝜒) ▶ 𝜃 ) ⇒ ⊢ ( 𝜑 , 𝜓 ▶ (𝜒 → 𝜃) ) | ||||||||||||||||
Theorem | in3 41678 | 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.) | ||||||||||||||
⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) ⇒ ⊢ ( 𝜑 , 𝜓 ▶ (𝜒 → 𝜃) ) | ||||||||||||||||
Theorem | iin3 41679 | in3 41678 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) | ||||||||||||||||
Theorem | in3an 41680 | The virtual deduction introduction rule converting the second conjunct of the third virtual hypothesis into the antecedent of the conclusion. exp4a 436 is the non-virtual deduction form of in3an 41680. (Contributed by Alan Sare, 25-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( 𝜑 , 𝜓 , (𝜒 ∧ 𝜃) ▶ 𝜏 ) ⇒ ⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ (𝜃 → 𝜏) ) | ||||||||||||||||
Theorem | int3 41681 | The virtual deduction introduction rule of converting the end virtual hypothesis of 3 virtual hypotheses into an antecedent. Conventional form of int3 41681 is 3expia 1119. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( ( 𝜑 , 𝜓 , 𝜒 ) ▶ 𝜃 ) ⇒ ⊢ ( ( 𝜑 , 𝜓 ) ▶ (𝜒 → 𝜃) ) | ||||||||||||||||
Theorem | idn2 41682 | Virtual deduction identity rule which is idd 24 with virtual deduction symbols. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( 𝜑 , 𝜓 ▶ 𝜓 ) | ||||||||||||||||
Theorem | iden2 41683 | Virtual deduction identity rule. simpr 489 in conjunction form Virtual Deduction notation. (Contributed by Alan Sare, 5-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( ( 𝜑 , 𝜓 ) ▶ 𝜓 ) | ||||||||||||||||
Theorem | idn3 41684 | Virtual deduction identity rule for three virtual hypotheses. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜒 ) | ||||||||||||||||
Theorem | gen11 41685* | Virtual deduction generalizing rule for one quantifying variable and one virtual hypothesis. alrimiv 1929 is gen11 41685 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( 𝜑 ▶ 𝜓 ) ⇒ ⊢ ( 𝜑 ▶ ∀𝑥𝜓 ) | ||||||||||||||||
Theorem | gen11nv 41686 | Virtual deduction generalizing rule for one quantifying variable and one virtual hypothesis without distinct variables. alrimih 1826 is gen11nv 41686 without virtual deductions. (Contributed by Alan Sare, 12-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ ( 𝜑 ▶ 𝜓 ) ⇒ ⊢ ( 𝜑 ▶ ∀𝑥𝜓 ) | ||||||||||||||||
Theorem | gen12 41687* | Virtual deduction generalizing rule for two quantifying variables and one virtual hypothesis. gen12 41687 is alrimivv 1930 with virtual deductions. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( 𝜑 ▶ 𝜓 ) ⇒ ⊢ ( 𝜑 ▶ ∀𝑥∀𝑦𝜓 ) | ||||||||||||||||
Theorem | gen21 41688* | Virtual deduction generalizing rule for one quantifying variables and two virtual hypothesis. gen21 41688 is alrimdv 1931 with virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) ⇒ ⊢ ( 𝜑 , 𝜓 ▶ ∀𝑥𝜒 ) | ||||||||||||||||
Theorem | gen21nv 41689 | Virtual deduction form of alrimdh 1865. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜓 → ∀𝑥𝜓) & ⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) ⇒ ⊢ ( 𝜑 , 𝜓 ▶ ∀𝑥𝜒 ) | ||||||||||||||||
Theorem | gen31 41690* | Virtual deduction generalizing rule for one quantifying variable and three virtual hypothesis. gen31 41690 is ggen31 41614 with virtual deductions. (Contributed by Alan Sare, 22-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) ⇒ ⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ ∀𝑥𝜃 ) | ||||||||||||||||
Theorem | gen22 41691* | Virtual deduction generalizing rule for two quantifying variables and two virtual hypothesis. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) ⇒ ⊢ ( 𝜑 , 𝜓 ▶ ∀𝑥∀𝑦𝜒 ) | ||||||||||||||||
Theorem | ggen22 41692* | gen22 41691 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥∀𝑦𝜒)) | ||||||||||||||||
Theorem | exinst 41693 | Existential Instantiation. Virtual deduction form of exlimexi 41593. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (𝜓 → ∀𝑥𝜓) & ⊢ ( ∃𝑥𝜑 , 𝜑 ▶ 𝜓 ) ⇒ ⊢ (∃𝑥𝜑 → 𝜓) | ||||||||||||||||
Theorem | exinst01 41694 | 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.) | ||||||||||||||
⊢ ∃𝑥𝜓 & ⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) & ⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜒 → ∀𝑥𝜒) ⇒ ⊢ ( 𝜑 ▶ 𝜒 ) | ||||||||||||||||
Theorem | exinst11 41695 | 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.) | ||||||||||||||
⊢ ( 𝜑 ▶ ∃𝑥𝜓 ) & ⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) & ⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜒 → ∀𝑥𝜒) ⇒ ⊢ ( 𝜑 ▶ 𝜒 ) | ||||||||||||||||
Theorem | e1a 41696 | A Virtual deduction elimination rule. syl 17 is e1a 41696 without virtual deductions. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( 𝜑 ▶ 𝜓 ) & ⊢ (𝜓 → 𝜒) ⇒ ⊢ ( 𝜑 ▶ 𝜒 ) | ||||||||||||||||
Theorem | el1 41697 | A Virtual deduction elimination rule. syl 17 is el1 41697 without virtual deductions. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( 𝜑 ▶ 𝜓 ) & ⊢ (𝜓 → 𝜒) ⇒ ⊢ ( 𝜑 ▶ 𝜒 ) | ||||||||||||||||
Theorem | e1bi 41698 | Biconditional form of e1a 41696. sylib 221 is e1bi 41698 without virtual deductions. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( 𝜑 ▶ 𝜓 ) & ⊢ (𝜓 ↔ 𝜒) ⇒ ⊢ ( 𝜑 ▶ 𝜒 ) | ||||||||||||||||
Theorem | e1bir 41699 | Right biconditional form of e1a 41696. sylibr 237 is e1bir 41699 without virtual deductions. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( 𝜑 ▶ 𝜓 ) & ⊢ (𝜒 ↔ 𝜓) ⇒ ⊢ ( 𝜑 ▶ 𝜒 ) | ||||||||||||||||
Theorem | e2 41700 | A virtual deduction elimination rule. syl6 35 is e2 41700 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) & ⊢ (𝜒 → 𝜃) ⇒ ⊢ ( 𝜑 , 𝜓 ▶ 𝜃 ) |
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