P. 454 of Theorem List - Metamath Proof Explorer

Type

Label

Description

Statement

Theorem

mulvfv 45301

Scalar multiplication at a value. (Contributed by Andrew Salmon, 27-Jan-2012.)

⊢ ((𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ ℝ) → ((𝐴._𝑣𝐵)‘𝐶) = (𝐴 · (𝐵‘𝐶)))

Theorem

addrfn 45302

Vector addition produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.)

⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+_𝑟𝐵) Fn ℝ)

Theorem

subrfn 45303

Vector subtraction produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.)

⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴-_𝑟𝐵) Fn ℝ)

Theorem

mulvfn 45304

Scalar multiplication producees a function. (Contributed by Andrew Salmon, 27-Jan-2012.)

⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴._𝑣𝐵) Fn ℝ)

Theorem

addrcom 45305

Vector addition is commutative. (Contributed by Andrew Salmon, 28-Jan-2012.)

⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+_𝑟𝐵) = (𝐵+_𝑟𝐴))

Definition

df-ptdf 45306*

Define the predicate PtDf, which is a utility definition used to shorten definitions and simplify proofs. (Contributed by Andrew Salmon, 15-Jul-2012.)

⊢ PtDf(𝐴, 𝐵) = (𝑥 ∈ ℝ ↦ (((𝑥._𝑣(𝐵-_𝑟𝐴)) +_𝑣 𝐴) “ {1, 2, 3}))

Definition

df-rr3 45307

Define the set of all points RR3. We define each point 𝐴 as a function to allow the use of vector addition and subtraction as well as scalar multiplication in our proofs. (Contributed by Andrew Salmon, 15-Jul-2012.)

⊢ RR3 = (ℝ ↑_m {1, 2, 3})

Definition

df-line3 45308*

Define the set of all lines. A line is an infinite subset of RR3 that satisfies a PtDf property. (Contributed by Andrew Salmon, 15-Jul-2012.)

⊢ line3 = {𝑥 ∈ 𝒫 RR3 ∣ (2_o ≼ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑧 ≠ 𝑦 → ran PtDf(𝑦, 𝑧) = 𝑥))}

21.43  Mathbox for Alan Sare

We are sad to report the passing of long-time contributor Alan Sare (Nov. 9, 1954 - Mar. 23, 2019).

Alan's first contribution to Metamath was a shorter proof for tfrlem8 8377 in 2008.

He developed a tool called "completeusersproof" that assists developing proofs using his "virtual deduction" method: https://us.metamath.org/other.html#completeusersproof 8377. His virtual deduction method is explained in the comment for wvd1 45400.

Below are some excerpts from his first emails to NM in 2007:

...I have been interested in proving set theory theorems for many years for mental exercise. I enjoy it. I have used a book by Martin Zuckerman. It is informal. I am interested in completely and perfectly proving theorems. Mr. Zuckerman leaves out most of the steps of a proof, of course, like most authors do, as you have noted. A complete proof for higher theorems would require a volume of writing similar to the Metamath documents. So I am frustrated when I am not capable of constructing a proof and Zuckerman leaves out steps I do not understand. I could search for the steps in other texts, but I don't do that too much. Metamath may be the answer for me....

...If we go beyond mathematics, I believe that it is possible to write down all human knowledge in a way similar to the way you have explicated large areas of mathematics. Of course, that would be a much, much more difficult job. For example, it is possible to take a hard science like physics, construct axioms based on experimental results, and to cast all of physics into a collection of axioms and theorems. Maybe this has already been attempted, although I am not familiar with it. When one then moves on to the soft sciences such as social science, this job gets much more difficult. The key is: All human thought consists of logical operations on abstract objects. Usually, these logical operations are done informally. There is no reason why one cannot take any subject and explicate it and take it down to the indivisible postulates in a formal rigorous way....

...When I read a math book or an engineering book I come across something I don't understand and I am compelled to understand it. But, often it is hopeless. I don't have the time. Or, I would have to read the same thing by multiple authors in the hope that different authors would give parts of the working proof that others have omitted. It is very inefficient. Because I have always been inclined to "get to the bottom" for a 100% fully understood proof....

21.43.1  Auxiliary theorems for the Virtual Deduction tool

Theorem

idiALT 45309

Placeholder for idi 1. Though unnecessary, this theorem is sometimes used in proofs in this mathbox for pedagogical purposes. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ 𝜑    ⇒   ⊢ 𝜑

Theorem

exbir 45310

Exportation implication also converting the consequent from a biconditional to an implication. Derived automatically from exbirVD 45683. (Contributed by Alan Sare, 31-Dec-2011.)

⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → (𝜑 → (𝜓 → (𝜃 → 𝜒))))

Theorem

3impexpbicom 45311

Version of 3impexp 1377 where in addition the consequent is commuted. (Contributed by Alan Sare, 31-Dec-2011.)

⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃)))))

Theorem

3impexpbicomi 45312

Inference associated with 3impexpbicom 45311. Derived automatically from 3impexpbicomiVD 45688. (Contributed by Alan Sare, 31-Dec-2011.)

⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃))))

21.43.2  Supplementary unification deductions

Theorem

bi1imp 45313

Importation inference similar to imp 412, except the outermost implication of the hypothesis is a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)

⊢ (𝜑 ↔ (𝜓 → 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Theorem

bi2imp 45314

Importation inference similar to imp 412, except both implications of the hypothesis are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

⊢ (𝜑 ↔ (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Theorem

bi3impb 45315

Similar to 3impb 1132 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.)

⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Theorem

bi3impa 45316

Similar to 3impa 1127 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.)

⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Theorem

bi23impib 45317

3impib 1134 with the inner implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)

⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ 𝜃))    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Theorem

bi13impib 45318

3impib 1134 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)

⊢ (𝜑 ↔ ((𝜓 ∧ 𝜒) → 𝜃))    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Theorem

bi123impib 45319

3impib 1134 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

⊢ (𝜑 ↔ ((𝜓 ∧ 𝜒) ↔ 𝜃))    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Theorem

bi13impia 45320

3impia 1135 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)

⊢ ((𝜑 ∧ 𝜓) ↔ (𝜒 → 𝜃))    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Theorem

bi123impia 45321

3impia 1135 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

⊢ ((𝜑 ∧ 𝜓) ↔ (𝜒 ↔ 𝜃))    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Theorem

bi33imp12 45322

3imp 1128 with innermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)

⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Theorem

bi13imp23 45323

3imp 1128 with outermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)

⊢ (𝜑 ↔ (𝜓 → (𝜒 → 𝜃)))    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Theorem

bi13imp2 45324

Similar to 3imp 1128 except the outermost and innermost implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

⊢ (𝜑 ↔ (𝜓 → (𝜒 ↔ 𝜃)))    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Theorem

bi12imp3 45325

Similar to 3imp 1128 except all but innermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

⊢ (𝜑 ↔ (𝜓 ↔ (𝜒 → 𝜃)))    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Theorem

bi23imp1 45326

Similar to 3imp 1128 except all but outermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

⊢ (𝜑 → (𝜓 ↔ (𝜒 ↔ 𝜃)))    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Theorem

bi123imp0 45327

Similar to 3imp 1128 except all implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

⊢ (𝜑 ↔ (𝜓 ↔ (𝜒 ↔ 𝜃)))    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Theorem

4animp1 45328

A single hypothesis unification deduction with an assertion which is an implication with a 4-right-nested conjunction antecedent. (Contributed by Alan Sare, 30-May-2018.)

⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜏 ↔ 𝜃))    ⇒   ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theorem

4an31 45329

A rearrangement of conjuncts for a 4-right-nested conjunction. (Contributed by Alan Sare, 30-May-2018.)

⊢ ((((𝜒 ∧ 𝜓) ∧ 𝜑) ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theorem

4an4132 45330

A rearrangement of conjuncts for a 4-right-nested conjunction. (Contributed by Alan Sare, 30-May-2018.)

⊢ ((((𝜃 ∧ 𝜒) ∧ 𝜓) ∧ 𝜑) → 𝜏)    ⇒   ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theorem

expcomdg 45331

Biconditional form of expcomd 422. (Contributed by Alan Sare, 22-Jul-2012.) (New usage is discouraged.)

⊢ ((𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) ↔ (𝜑 → (𝜒 → (𝜓 → 𝜃))))

21.43.3  Conventional Metamath proofs, some derived from VD proofs

Theorem

iidn3 45332

idn3 45446 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 → 𝜒)))

Theorem

ee222 45333

e222 45467 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 7-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜓 → 𝜃))    &   ⊢ (𝜑 → (𝜓 → 𝜏))    &   ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂)))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜂))

Theorem

ee3bir 45334

Right-biconditional form of e3 45567 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 22-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜏 ↔ 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))

Theorem

ee13 45335

e13 45578 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 28-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → (𝜒 → (𝜃 → 𝜏)))    &   ⊢ (𝜓 → (𝜏 → 𝜂))    ⇒   ⊢ (𝜑 → (𝜒 → (𝜃 → 𝜂)))

Theorem

ee121 45336

e121 45487 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → (𝜒 → 𝜃))    &   ⊢ (𝜑 → 𝜏)    &   ⊢ (𝜓 → (𝜃 → (𝜏 → 𝜂)))    ⇒   ⊢ (𝜑 → (𝜒 → 𝜂))

Theorem

ee122 45337

e122 45484 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → (𝜒 → 𝜃))    &   ⊢ (𝜑 → (𝜒 → 𝜏))    &   ⊢ (𝜓 → (𝜃 → (𝜏 → 𝜂)))    ⇒   ⊢ (𝜑 → (𝜒 → 𝜂))

Theorem

ee333 45338

e333 45563 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))    &   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))    &   ⊢ (𝜃 → (𝜏 → (𝜂 → 𝜁)))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜁)))

Theorem

ee323 45339

e323 45596 without virtual deductions. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜑 → (𝜓 → 𝜏))    &   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))    &   ⊢ (𝜃 → (𝜏 → (𝜂 → 𝜁)))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜁)))

Theorem

3ornot23 45340

If the second and third disjuncts of a true triple disjunction are false, then the first disjunct is true. Automatically derived from 3ornot23VD 45677. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒 ∨ 𝜑 ∨ 𝜓) → 𝜒))

Theorem

orbi1r 45341

orbi1 931 with order of disjuncts reversed. Derived from orbi1rVD 45678. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓)))

Theorem

3orbi123 45342

pm4.39 992 with a 3-conjunct antecedent. This proof is 3orbi123VD 45680 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂)))

Theorem

syl5imp 45343

Closed form of syl5 35. Derived automatically from syl5impVD 45693. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜓) → (𝜑 → (𝜃 → 𝜒))))

Theorem

impexpd 45344

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

1::	⊢ (((𝜓 ∧ 𝜒) → 𝜃) ↔ (𝜓 → (𝜒 → 𝜃)))
qed:1:	⊢ ((𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃))))

⊢ ((𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃))))

Theorem

com3rgbi 45345

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

1::	⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → (𝜑 → (𝜒 → (𝜓 → 𝜃))))
2::	⊢ ((𝜑 → (𝜒 → (𝜓 → 𝜃))) → (𝜒 → (𝜑 → (𝜓 → 𝜃))))
3:1,2:	⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → (𝜒 → (𝜑 → (𝜓 → 𝜃))))
4::	⊢ ((𝜒 → (𝜑 → (𝜓 → 𝜃))) → (𝜑 → (𝜒 → (𝜓 → 𝜃))))
5::	⊢ ((𝜑 → (𝜒 → (𝜓 → 𝜃))) → (𝜑 → (𝜓 → (𝜒 → 𝜃))))
6:4,5:	⊢ ((𝜒 → (𝜑 → (𝜓 → 𝜃))) → (𝜑 → (𝜓 → (𝜒 → 𝜃))))
qed:3,6:	⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) ↔ (𝜒 → (𝜑 → (𝜓 → 𝜃))))

⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) ↔ (𝜒 → (𝜑 → (𝜓 → 𝜃))))

Theorem

impexpdcom 45346

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

1::	⊢ ((𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃))))
2::	⊢ ((𝜓 → (𝜒 → (𝜑 → 𝜃))) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃))))
qed:1,2:	⊢ ((𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) ↔ (𝜓 → (𝜒 → (𝜑 → 𝜃))))

⊢ ((𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) ↔ (𝜓 → (𝜒 → (𝜑 → 𝜃))))

Theorem

ee1111 45347

Non-virtual deduction form of e1111 45506. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.

h1::	⊢ (𝜑 → 𝜓)
h2::	⊢ (𝜑 → 𝜒)
h3::	⊢ (𝜑 → 𝜃)
h4::	⊢ (𝜑 → 𝜏)
h5::	⊢ (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))
6:1,5:	⊢ (𝜑 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))
7:6:	⊢ (𝜒 → (𝜑 → (𝜃 → (𝜏 → 𝜂))))
8:2,7:	⊢ (𝜑 → (𝜑 → (𝜃 → (𝜏 → 𝜂))))
9:8:	⊢ (𝜑 → (𝜃 → (𝜏 → 𝜂)))
10:9:	⊢ (𝜃 → (𝜑 → (𝜏 → 𝜂)))
11:3,10:	⊢ (𝜑 → (𝜑 → (𝜏 → 𝜂)))
12:11:	⊢ (𝜑 → (𝜏 → 𝜂))
13:12:	⊢ (𝜏 → (𝜑 → 𝜂))
14:4,13:	⊢ (𝜑 → (𝜑 → 𝜂))
qed:14:	⊢ (𝜑 → 𝜂)

⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → 𝜃)    &   ⊢ (𝜑 → 𝜏)    &   ⊢ (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))    ⇒   ⊢ (𝜑 → 𝜂)

Theorem

pm2.43bgbi 45348

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

1::	⊢ ((𝜑 → (𝜓 → (𝜑 → 𝜒))) → (𝜑 → (𝜑 → (𝜓 → 𝜒))))
2::	⊢ ((𝜑 → (𝜑 → (𝜓 → 𝜒))) → (𝜑 → (𝜓 → 𝜒)))
3:1,2:	⊢ ((𝜑 → (𝜓 → (𝜑 → 𝜒))) → (𝜑 → (𝜓 → 𝜒)))
4::	⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))
5:3,4:	⊢ ((𝜑 → (𝜓 → (𝜑 → 𝜒))) → (𝜓 → (𝜑 → 𝜒)))
6::	⊢ ((𝜓 → (𝜑 → 𝜒)) → (𝜑 → (𝜓 → (𝜑 → 𝜒))))
qed:5,6:	⊢ ((𝜑 → (𝜓 → (𝜑 → 𝜒))) ↔ (𝜓 → (𝜑 → 𝜒)))

⊢ ((𝜑 → (𝜓 → (𝜑 → 𝜒))) ↔ (𝜓 → (𝜑 → 𝜒)))

Theorem

pm2.43cbi 45349

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

1::	⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜑 → 𝜃))) ) → (𝜑 → (𝜓 → (𝜑 → (𝜒 → 𝜃)))))
2::	⊢ ((𝜑 → (𝜓 → (𝜑 → (𝜒 → 𝜃))) ) → (𝜓 → (𝜑 → (𝜒 → 𝜃))))
3:1,2:	⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜑 → 𝜃))) ) → (𝜓 → (𝜑 → (𝜒 → 𝜃))))
4::	⊢ ((𝜓 → (𝜑 → (𝜒 → 𝜃))) → (𝜓 → (𝜒 → (𝜑 → 𝜃))))
5:3,4:	⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜑 → 𝜃))) ) → (𝜓 → (𝜒 → (𝜑 → 𝜃))))
6::	⊢ ((𝜓 → (𝜒 → (𝜑 → 𝜃))) → (𝜑 → (𝜓 → (𝜒 → (𝜑 → 𝜃)))))
qed:5,6:	⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜑 → 𝜃))) ) ↔ (𝜓 → (𝜒 → (𝜑 → 𝜃))))

⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜑 → 𝜃)))) ↔ (𝜓 → (𝜒 → (𝜑 → 𝜃))))

Theorem

ee233 45350

Non-virtual deduction form of e233 45595. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.

h1::	⊢ (𝜑 → (𝜓 → 𝜒))
h2::	⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏)))
h3::	⊢ (𝜑 → (𝜓 → (𝜃 → 𝜂)))
h4::	⊢ (𝜒 → (𝜏 → (𝜂 → 𝜁)))
5:1,4:	⊢ (𝜑 → (𝜓 → (𝜏 → (𝜂 → 𝜁))) )
6:5:	⊢ (𝜏 → (𝜑 → (𝜓 → (𝜂 → 𝜁))) )
7:2,6:	⊢ (𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁))))))
8:7:	⊢ (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁)))))
9:8:	⊢ (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁))) )
10:9:	⊢ (𝜑 → (𝜓 → (𝜃 → (𝜂 → 𝜁))) )
11:10:	⊢ (𝜂 → (𝜑 → (𝜓 → (𝜃 → 𝜁))) )
12:3,11:	⊢ (𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁))))))
13:12:	⊢ (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁)))))
14:13:	⊢ (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁))) )
qed:14:	⊢ (𝜑 → (𝜓 → (𝜃 → 𝜁)))

⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏)))    &   ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜂)))    &   ⊢ (𝜒 → (𝜏 → (𝜂 → 𝜁)))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜁)))

Theorem

imbi13 45351

Join three logical equivalences to form equivalence of implications. imbi13 45351 is imbi13VD 45704 without virtual deductions and was automatically derived from imbi13VD 45704 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.)

⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂))))))

Theorem

ee33 45352

Non-virtual deduction form of e33 45564. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.

h1::	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
h2::	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))
h3::	⊢ (𝜃 → (𝜏 → 𝜂))
4:1,3:	⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 → 𝜂))))
5:4:	⊢ (𝜏 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))
6:2,5:	⊢ (𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))))
7:6:	⊢ (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))
8:7:	⊢ (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))
qed:8:	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))

⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))    &   ⊢ (𝜃 → (𝜏 → 𝜂))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))

Theorem

con5 45353

Biconditional contraposition variation. This proof is con5VD 45730 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 → 𝜓))

Theorem

con5i 45354

Inference form of con5 45353. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 ↔ ¬ 𝜓)    ⇒   ⊢ (¬ 𝜑 → 𝜓)

Theorem

exlimexi 45355

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

⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (∃𝑥𝜑 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥𝜑 → 𝜓)

Theorem

sb5ALT 45356*

Equivalence for substitution. Alternate proof of sb5 2311. This proof is sb5ALTVD 45743 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 45357

exinst01 45456 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 45358

exinst11 45457 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 45359

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 45359 is vk15.4jVD 45744 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 45360

Converse of double negation. Alternate proof of notnotr 131. This proof is notnotrALTVD 45745 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (¬ ¬ 𝜑 → 𝜑)

Theorem

con3ALT2 45361

Contraposition. Alternate proof of con3 154. This proof is con3ALTVD 45746 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 45362*

Quantification restricted to a subclass for two quantifiers. ssralv 4003 for two quantifiers. The proof of ssralv2 45362 was automatically generated by minimizing the automatically translated proof of ssralv2VD 45696. 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 45363

sbcor 3792 with a 3-disjuncts. This proof is sbc3orgVD 45681 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 45364*

Closed form of alrimi 2251 with 2 additional conjuncts having no occurrences of the quantifying variable. This proof is 19.21a3con13vVD 45682 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 45365*

rspsbc 3829 with two quantifying variables. This proof is rspsbc2VD 45685 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 45366*

Substitution of a setvar variable for another setvar variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 45689. (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 45367*

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 45691. (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 45368

An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 6383 using the Axiom of Regularity indirectly through dford2 9603. 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 45368 is ordelordALTVD 45697 without virtual deductions and was automatically derived from ordelordALTVD 45697 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 45369

Distribution of class substitution over a left-nested implication. Similar to sbcimg 3790. sbcim2g 45369 is sbcim2gVD 45705 without virtual deductions and was automatically derived from sbcim2gVD 45705 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 45370

Implication form of sbcbii 3798. sbcbi 45370 is sbcbiVD 45706 without virtual deductions and was automatically derived from sbcbiVD 45706 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 45371*

Formula-building inference rule for class substitution, substituting a class variable for the setvar variable of the transitivity predicate. trsbc 45371 is trsbcVD 45707 without virtual deductions and was automatically derived from trsbcVD 45707 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 45372*

The union of a class of transitive sets is transitive. Alternate proof of truni 5232. truniALT 45372 is truniALTVD 45708 without virtual deductions and was automatically derived from truniALTVD 45708 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 45373*

Lemma for onfrALT 45380. (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 45374*

Lemma for onfrALT 45380. (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 45375*

Lemma for onfrALT 45380. (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 45376*

gen31 45452 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 45377*

Lemma for onfrALT 45380. (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 45378*

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 45379*

Lemma for onfrALT 45380. (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 45380

The membership relation is foundational on the class of ordinal numbers. onfrALT 45380 is an alternate proof of onfr 6401. onfrALTVD 45721 is the Virtual Deduction proof from which onfrALT 45380 is derived. The Virtual Deduction proof mirrors the working proof of onfr 6401 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 45721. 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 45381

Closed form of right-to-left implication of 19.41 2273, Theorem 19.41 of [Margaris] p. 90. Derived from 19.41rgVD 45732. (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 45382*

Ordered pair membership in a class abstraction of ordered pairs. Compare to elopab 5509. (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 45383

pm13.193 45243 for two variables. pm13.193 45243 is Theorem *13.193 in [WhiteheadRussell] p. 179. Derived from 2pm13.193VD 45733. (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 45384

A closed form of hbn 2330. hbnt 2329 is another closed form of hbn 2330. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑))

Theorem

hbimpg 45385

A closed form of hbim 2334. Derived from hbimpgVD 45734. (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 45386

Closed form of hbal 2204. Derived from hbalgVD 45735. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))

Theorem

hbexg 45387

Closed form of nfex 2356. Derived from hbexgVD 45736. (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 45388*

Alternate form of ax6e 2414 for non-distinct 𝑥, 𝑦 and 𝑢 = 𝑣. ax6e2eq 45388 is derived from ax6e2eqVD 45737. (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 45389*

If at least two sets exist (dtru 5416), then the same is true expressed in an alternate form similar to the form of ax6e 2414. ax6e2nd 45389 is derived from ax6e2ndVD 45738. (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 45390*

"At least two sets exist" expressed in the form of dtru 5416 is logically equivalent to the same expressed in a form similar to ax6e 2414 if dtru 5416 is false implies 𝑢 = 𝑣. ax6e2ndeq 45390 is derived from ax6e2ndeqVD 45739. (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 45391*

Equivalence for double substitution 2sb5 2313 without distinct 𝑥, 𝑦 requirement. 2sb5nd 45391 is derived from 2sb5ndVD 45740. (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 45392*

Distribute the unabbreviated form of proper substitution in and out of a conjunction. 2uasbanh 45392 is derived from 2uasbanhVD 45741. (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 45393*

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 45394

Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind 45394 is derived from e2ebindVD 45742. (Contributed by Alan Sare, 27-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑))

Theorem

elpwgded 45395

elpwgdedVD 45747 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 45396

Deduction form of trel 5224. 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 45397

Deduction form of jao 975. 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 45398

A substitution into a theorem remains true. sbt 2103 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 45399

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 45399 using completeusersproof, which is verified by the Metamath program. https://us.metamath.org/other/completeusersproof/not12an2impnot1ro.html 45399 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)

21.43.4  What is Virtual Deduction?

Syntax

wvd1 45400

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 it 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 originally 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 his (Gentzen's) Hauptsatz or normal form theorem 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 sequences 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)

if ⊢ → 𝜑 in G1 then ⊢ 𝜑 in H

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

if ⊢ 𝜑 → 𝜓 in G1 then ⊢ (   𝜑   ▶   𝜓   ) in H

if ⊢ 𝜑   ,   𝜓 → 𝜒 in G1 then ⊢ (   (   𝜑   ,   𝜓   )   ▶   𝜒   ) in H

if ⊢ 𝜑   ,   𝜓   ,   𝜒 → 𝜃 in G1 then ⊢ (   (   𝜑   ,   𝜓   ,   𝜒   )   ▶   𝜃   ) in H

   ▶    denotes the same connective denoted by →. " , " , in the context of Virtual Deduction, denotes the same connective denoted by ∧. This Virtual Deduction notation is specified by the following set.mm definitions:

df-vd1 45401	⊢ ((   𝜑   ▶   𝜓   ) ↔ (𝜑 → 𝜓))
dfvd2an 45413	⊢ ((   (   𝜑   ,   𝜓   )   ▶   𝜒   ) ↔ ((𝜑 ∧ 𝜓) → 𝜒))
dfvd3an 45425	⊢ ((   (   𝜑   ,   𝜓   ,   𝜒   )   ▶   𝜃   ) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃))

   ▶    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)

if ⊢ 𝜑 in H then ⊢ → 𝜑 in G1

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

if ⊢ (   𝜑   ▶   𝜓   ) in H then ⊢ 𝜑 → 𝜓 in G1

if ⊢ (   (   𝜑   ,   𝜓   )   ▶   𝜒   ) in H then ⊢ 𝜑   ,   𝜓 → 𝜒 in G1

if ⊢ (   (   𝜑   ,   𝜓   ,   𝜒   )   ▶   𝜃   ) in H then ⊢ 𝜑   ,   𝜓   ,   𝜒 → 𝜃 in G1

To prove in H, the User simply proves in G1 and translates each G1-proof step into a H-proof step. The translation is trivial and immediate. The proof in H is in Virtual Deduction notation. It is a working proof in the sense that, if it has no errors, each theorem and deduction of the proof is true, but may or may not, after being translated into conventional notation, unify with any theorem or deduction 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 postulates. 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 45425, https://us.metamath.org/other/completeusersproof/sineq0altvd.html 45425, https://us.metamath.org/other/completeusersproof/iunconlem2vd.html 45425, https://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html 45425, and https://us.metamath.org/other/completeusersproof/chordthmaltvd.html 45425 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   )