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Type | Label | Description | ||||||||||||||
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Statement | ||||||||||||||||
Theorem | sb5ALT 41001* | Equivalence for substitution. Alternate proof of sb5 2276. This proof is sb5ALTVD 41389 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | ||||||||||||||||
Theorem | eexinst01 41002 | exinst01 41101 without virtual deductions. (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 | eexinst11 41003 | exinst11 41102 without virtual deductions. (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 | vk15.4j 41004 | 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 41004 is vk15.4jVD 41390 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ¬ (∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒)) & ⊢ (∀𝑥𝜒 → ¬ ∃𝑥(𝜃 ∧ 𝜏)) & ⊢ ¬ ∀𝑥(𝜏 → 𝜑) ⇒ ⊢ (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜓) | ||||||||||||||||
Theorem | notnotrALT 41005 | Converse of double negation. Alternate proof of notnotr 132. This proof is notnotrALTVD 41391 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (¬ ¬ 𝜑 → 𝜑) | ||||||||||||||||
Theorem | con3ALT2 41006 | Contraposition. Alternate proof of con3 156. This proof is con3ALTVD 41392 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑)) | ||||||||||||||||
Theorem | ssralv2 41007* | Quantification restricted to a subclass for two quantifiers. ssralv 4009 for two quantifiers. The proof of ssralv2 41007 was automatically generated by minimizing the automatically translated proof of ssralv2VD 41342. The automatic translation is by the tools program translate_without_overwriting.cmd. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝜑)) | ||||||||||||||||
Theorem | sbc3or 41008 | sbcor 3800 with a 3-disjuncts. This proof is sbc3orgVD 41327 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 41009* | Closed form of alrimi 2213 with 2 additional conjuncts having no occurrences of the quantifying variable. This proof is 19.21a3con13vVD 41328 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 41010* | rspsbc 3839 with two quantifying variables. This proof is rspsbc2VD 41331 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 41011* | Substitution of a setvar variable for another setvar variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 41335. (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 41012* | 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 41337. (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 41013 | An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 6187 using the Axiom of Regularity indirectly through dford2 9059. 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 41013 is ordelordALTVD 41343 without virtual deductions and was automatically derived from ordelordALTVD 41343 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 41014 | Distribution of class substitution over a left-nested implication. Similar to sbcimg 3798. sbcim2g 41014 is sbcim2gVD 41351 without virtual deductions and was automatically derived from sbcim2gVD 41351 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 41015 | Implication form of sbcbii 3807. sbcbi 41015 is sbcbiVD 41352 without virtual deductions and was automatically derived from sbcbiVD 41352 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 41016* | Formula-building inference rule for class substitution, substituting a class variable for the setvar variable of the transitivity predicate. trsbc 41016 is trsbcVD 41353 without virtual deductions and was automatically derived from trsbcVD 41353 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 41017* | The union of a class of transitive sets is transitive. Alternate proof of truni 5160. truniALT 41017 is truniALTVD 41354 without virtual deductions and was automatically derived from truniALTVD 41354 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 41018* | Lemma for onfrALT 41025. (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 41019* | Lemma for onfrALT 41025. (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 41020* | Lemma for onfrALT 41025. (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 41021* | gen31 41097 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 41022* | Lemma for onfrALT 41025. (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 41023* | 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 41024* | Lemma for onfrALT 41025. (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 41025 | The membership relation is foundational on the class of ordinal numbers. onfrALT 41025 is an alternate proof of onfr 6204. onfrALTVD 41367 is the Virtual Deduction proof from which onfrALT 41025 is derived. The Virtual Deduction proof mirrors the working proof of onfr 6204 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 41367. 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 41026 | Closed form of right-to-left implication of 19.41 2237, Theorem 19.41 of [Margaris] p. 90. Derived from 19.41rgVD 41378. (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 41027* | Ordered pair membership in a class abstraction of pairs. Compare to elopab 5388. (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 41028 | pm13.193 40885 for two variables. pm13.193 40885 is Theorem *13.193 in [WhiteheadRussell] p. 179. Derived from 2pm13.193VD 41379. (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 41029 | A closed form of hbn 2303. hbnt 2302 is another closed form of hbn 2303. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑)) | ||||||||||||||||
Theorem | hbimpg 41030 | A closed form of hbim 2307. Derived from hbimpgVD 41380. (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 41031 | Closed form of hbal 2174. Derived from hbalgVD 41381. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)) | ||||||||||||||||
Theorem | hbexg 41032 | Closed form of nfex 2343. Derived from hbexgVD 41382. (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 41033* | Alternate form of ax6e 2401 for non-distinct 𝑥, 𝑦 and 𝑢 = 𝑣. ax6e2eq 41033 is derived from ax6e2eqVD 41383. (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 41034* | If at least two sets exist (dtru 5245) , then the same is true expressed in an alternate form similar to the form of ax6e 2401. ax6e2nd 41034 is derived from ax6e2ndVD 41384. (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 41035* | "At least two sets exist" expressed in the form of dtru 5245 is logically equivalent to the same expressed in a form similar to ax6e 2401 if dtru 5245 is false implies 𝑢 = 𝑣. ax6e2ndeq 41035 is derived from ax6e2ndeqVD 41385. (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 41036* | Equivalence for double substitution 2sb5 2282 without distinct 𝑥, 𝑦 requirement. 2sb5nd 41036 is derived from 2sb5ndVD 41386. (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 41037* | Distribute the unabbreviated form of proper substitution in and out of a conjunction. 2uasbanh 41037 is derived from 2uasbanhVD 41387. (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 41038* | 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 41039 | Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind 41039 is derived from e2ebindVD 41388. (Contributed by Alan Sare, 27-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑)) | ||||||||||||||||
Theorem | elpwgded 41040 | elpwgdedVD 41393 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 41041 | Deduction form of trel 5153. 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 41042 | Deduction form of jao 957. 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 41043 | A substitution into a theorem remains true. sbt 2071 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 41044 | 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 41044 using completeusersproof, which is verified by the Metamath program. https://us.metamath.org/other/completeusersproof/not12an2impnot1ro.html 41044 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 41045 |
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 41070, https://us.metamath.org/other/completeusersproof/sineq0altvd.html 41070, https://us.metamath.org/other/completeusersproof/iunconlem2vd.html 41070, https://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html 41070, and https://us.metamath.org/other/completeusersproof/chordthmaltvd.html 41070 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 41046 | Definition of virtual deduction. (Contributed by Alan Sare, 21-Apr-2011.) (New usage is discouraged.) | ||||||||||||||
⊢ (( 𝜑 ▶ 𝜓 ) ↔ (𝜑 → 𝜓)) | ||||||||||||||||
Theorem | in1 41047 | Inference form of df-vd1 41046. 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 41048 | in1 41047 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||||||||||||||||
Theorem | dfvd1ir 41049 | Inference form of df-vd1 41046 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 41050 | 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 41051 | 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 41052 | 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 41053 | Syntax for a 2-hypothesis virtual deduction. (New usage is discouraged.) | ||||||||||||||
wff ( 𝜑 , 𝜓 ▶ 𝜒 ) | ||||||||||||||||
Definition | df-vd2 41054 | 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 41055 | 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 41056 | 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 41057 | 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 41058 | 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 41059 | Inference form of dfvd2an 41058. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( ( 𝜑 , 𝜓 ) ▶ 𝜒 ) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||||||||||||||||
Theorem | dfvd2anir 41060 | Right-to-left inference form of dfvd2an 41058. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ ( ( 𝜑 , 𝜓 ) ▶ 𝜒 ) | ||||||||||||||||
Theorem | dfvd2i 41061 | Inference form of dfvd2 41055. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) ⇒ ⊢ (𝜑 → (𝜓 → 𝜒)) | ||||||||||||||||
Theorem | dfvd2ir 41062 | Right-to-left inference form of dfvd2 41055. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) | ||||||||||||||||
Syntax | wvd3 41063 | Syntax for a 3-hypothesis virtual deduction. (New usage is discouraged.) | ||||||||||||||
wff ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) | ||||||||||||||||
Syntax | wvhc3 41064 | 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 41065 | 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 41066 | 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 41067 | 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 41068 | Inference form of dfvd3 41067. (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 41069 | Right-to-left inference form of dfvd3 41067. (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 41070 | 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 41071 | Inference form of dfvd3an 41070. (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 41072 | Right-to-left inference form of dfvd3an 41070. (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 41073 | 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 41074 | 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 41075 | 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 41076 | 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 41077 | 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 41078 | 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 41079 | 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 41080 | 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 41081 | 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 41082 | The virtual deduction introduction rule of converting the end virtual hypothesis of 2 virtual hypotheses into an antecedent. Conventional form of int2 41082 is ex 415. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( ( 𝜑 , 𝜓 ) ▶ 𝜒 ) ⇒ ⊢ ( 𝜑 ▶ (𝜓 → 𝜒) ) | ||||||||||||||||
Theorem | iin2 41083 | in2 41081 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 41084 | The virtual deduction introduction rule converting the second conjunct of the second virtual hypothesis into the antecedent of the conclusion. expd 418 is the non-virtual deduction form of in2an 41084. (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 41085 | 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 41086 | in3 41085 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 41087 | The virtual deduction introduction rule converting the second conjunct of the third virtual hypothesis into the antecedent of the conclusion. exp4a 434 is the non-virtual deduction form of in3an 41087. (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 41088 | The virtual deduction introduction rule of converting the end virtual hypothesis of 3 virtual hypotheses into an antecedent. Conventional form of int3 41088 is 3expia 1117. (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 41089 | 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 41090 | Virtual deduction identity rule. simpr 487 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 41091 | 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 41092* | Virtual deduction generalizing rule for one quantifying variable and one virtual hypothesis. alrimiv 1928 is gen11 41092 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ ( 𝜑 ▶ 𝜓 ) ⇒ ⊢ ( 𝜑 ▶ ∀𝑥𝜓 ) | ||||||||||||||||
Theorem | gen11nv 41093 | Virtual deduction generalizing rule for one quantifying variable and one virtual hypothesis without distinct variables. alrimih 1824 is gen11nv 41093 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 41094* | Virtual deduction generalizing rule for two quantifying variables and one virtual hypothesis. gen12 41094 is alrimivv 1929 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 41095* | Virtual deduction generalizing rule for one quantifying variables and two virtual hypothesis. gen21 41095 is alrimdv 1930 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 41096 | Virtual deduction form of alrimdh 1864. (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 41097* | Virtual deduction generalizing rule for one quantifying variable and three virtual hypothesis. gen31 41097 is ggen31 41021 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 41098* | 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 41099* | gen22 41098 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 41100 | Existential Instantiation. Virtual deduction form of exlimexi 41000. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||
⊢ (𝜓 → ∀𝑥𝜓) & ⊢ ( ∃𝑥𝜑 , 𝜑 ▶ 𝜓 ) ⇒ ⊢ (∃𝑥𝜑 → 𝜓) |
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