P. 445 of Theorem List - Metamath Proof Explorer

Type

Label

Description

Statement

Theorem

pm14.18 44401

Theorem *14.18 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.)

⊢ (∃!𝑥𝜑 → (∀𝑥𝜓 → [(℩𝑥𝜑) / 𝑥]𝜓))

Theorem

iotaequ 44402*

Theorem *14.2 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.)

⊢ (℩𝑥𝑥 = 𝑦) = 𝑦

Theorem

iotavalb 44403*

Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 6460. (Contributed by Andrew Salmon, 11-Jul-2011.)

⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))

Theorem

iotasbc5 44404*

Theorem *14.205 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 11-Jul-2011.)

⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ 𝜓)))

Theorem

pm14.24 44405*

Theorem *14.24 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)

⊢ (∃!𝑥𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = (℩𝑥𝜑)))

Theorem

iotavalsb 44406*

Theorem *14.242 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)

⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜓 ↔ [(℩𝑥𝜑) / 𝑧]𝜓))

Theorem

sbiota1 44407

Theorem *14.25 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.)

⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓))

Theorem

sbaniota 44408

Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.)

⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓))

Theorem

eubiOLD 44409

Obsolete proof of eubi 2577 as of 7-Oct-2022. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑥𝜓))

Theorem

iotasbcq 44410

Theorem *14.272 in [WhiteheadRussell] p. 193. (Contributed by Andrew Salmon, 11-Jul-2011.)

⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([(℩𝑥𝜑) / 𝑦]𝜒 ↔ [(℩𝑥𝜓) / 𝑦]𝜒))

21.40.5  Set Theory

Theorem

elnev 44411*

Any set that contains one element less than the universe is not equal to it. (Contributed by Andrew Salmon, 16-Jun-2011.)

⊢ (𝐴 ∈ V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V)

Theorem

rusbcALT 44412

A version of Russell's paradox which is proven using proper substitution. (Contributed by Andrew Salmon, 18-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V

Theorem

compeq 44413*

Equality between two ways of saying "the complement of 𝐴". (Contributed by Andrew Salmon, 15-Jul-2011.)

⊢ (V ∖ 𝐴) = {𝑥 ∣ ¬ 𝑥 ∈ 𝐴}

Theorem

compne 44414

The complement of 𝐴 is not equal to 𝐴. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by BJ, 11-Nov-2021.)

⊢ (V ∖ 𝐴) ≠ 𝐴

Theorem

compab 44415

Two ways of saying "the complement of a class abstraction". (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

⊢ (V ∖ {𝑧 ∣ 𝜑}) = {𝑧 ∣ ¬ 𝜑}

Theorem

conss2 44416

Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)

⊢ (𝐴 ⊆ (V ∖ 𝐵) ↔ 𝐵 ⊆ (V ∖ 𝐴))

Theorem

conss1 44417

Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)

⊢ ((V ∖ 𝐴) ⊆ 𝐵 ↔ (V ∖ 𝐵) ⊆ 𝐴)

Theorem

ralbidar 44418

More general form of ralbida 3240. (Contributed by Andrew Salmon, 25-Jul-2011.)

⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜑)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))

Theorem

rexbidar 44419

More general form of rexbida 3241. (Contributed by Andrew Salmon, 25-Jul-2011.)

⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜑)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))

Theorem

dropab1 44420

Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.)

⊢ (∀𝑥 𝑥 = 𝑦 → {⟨𝑥, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ 𝜑})

Theorem

dropab2 44421

Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.)

⊢ (∀𝑥 𝑥 = 𝑦 → {⟨𝑧, 𝑥⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜑})

Theorem

ipo0 44422

If the identity relation partially orders any class, then that class is the null class. (Contributed by Andrew Salmon, 25-Jul-2011.)

⊢ ( I Po 𝐴 ↔ 𝐴 = ∅)

Theorem

ifr0 44423

A class that is founded by the identity relation is null. (Contributed by Andrew Salmon, 25-Jul-2011.)

⊢ ( I Fr 𝐴 ↔ 𝐴 = ∅)

Theorem

ordpss 44424

ordelpss 6339 with an antecedent removed. (Contributed by Andrew Salmon, 25-Jul-2011.)

⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → 𝐴 ⊊ 𝐵))

Theorem

fvsb 44425*

Explicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.)

⊢ (∃!𝑦 𝐴𝐹𝑦 → ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑)))

Theorem

fveqsb 44426*

Implicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.)

⊢ (𝑥 = (𝐹‘𝐴) → (𝜑 ↔ 𝜓))    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝜓 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑)))

Theorem

xpexb 44427

A Cartesian product exists iff its converse does. Corollary 6.9(1) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.)

⊢ ((𝐴 × 𝐵) ∈ V ↔ (𝐵 × 𝐴) ∈ V)

Theorem

trelpss 44428

An element of a transitive set is a proper subset of it. Theorem 7.2 in [TakeutiZaring] p. 35. Unlike tz7.2 5606, ax-reg 9503 is required for its proof. (Contributed by Andrew Salmon, 13-Nov-2011.)

⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊊ 𝐴)

21.40.6  Arithmetic

Theorem

addcomgi 44429

Generalization of commutative law for addition. Simplifies proofs dealing with vectors. However, it is dependent on our particular definition of ordered pair. (Contributed by Andrew Salmon, 28-Jan-2012.) (Revised by Mario Carneiro, 6-May-2015.)

⊢ (𝐴 + 𝐵) = (𝐵 + 𝐴)

21.40.7  Geometry

Syntax

cplusr 44430

Introduce the operation of vector addition.

class +_𝑟

Syntax

cminusr 44431

Introduce the operation of vector subtraction.

class -_𝑟

Syntax

ctimesr 44432

Introduce the operation of scalar multiplication.

class ._𝑣

Syntax

cptdfc 44433

PtDf is a predicate that is crucial for the definition of lines as well as proving a number of important theorems.

class PtDf(𝐴, 𝐵)

Syntax

crr3c 44434

RR3 is a class.

class RR3

Syntax

cline3 44435

line3 is a class.

class line3

Definition

df-addr 44436*

Define the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.)

⊢ +_𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) + (𝑦‘𝑣))))

Definition

df-subr 44437*

Define the operation of vector subtraction. (Contributed by Andrew Salmon, 27-Jan-2012.)

⊢ -_𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) − (𝑦‘𝑣))))

Definition

df-mulv 44438*

Define the operation of scalar multiplication. (Contributed by Andrew Salmon, 27-Jan-2012.)

⊢ ._𝑣 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ (𝑥 · (𝑦‘𝑣))))

Theorem

addrval 44439*

Value of the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.)

⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+_𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) + (𝐵‘𝑣))))

Theorem

subrval 44440*

Value of the operation of vector subtraction. (Contributed by Andrew Salmon, 27-Jan-2012.)

⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴-_𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) − (𝐵‘𝑣))))

Theorem

mulvval 44441*

Value of the operation of scalar multiplication. (Contributed by Andrew Salmon, 27-Jan-2012.)

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

Theorem

addrfv 44442

Vector addition at a value. The operation takes each vector 𝐴 and 𝐵 and forms a new vector whose values are the sum of each of the values of 𝐴 and 𝐵. (Contributed by Andrew Salmon, 27-Jan-2012.)

⊢ ((𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ ℝ) → ((𝐴+_𝑟𝐵)‘𝐶) = ((𝐴‘𝐶) + (𝐵‘𝐶)))

Theorem

subrfv 44443

Vector subtraction at a value. (Contributed by Andrew Salmon, 27-Jan-2012.)

⊢ ((𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ ℝ) → ((𝐴-_𝑟𝐵)‘𝐶) = ((𝐴‘𝐶) − (𝐵‘𝐶)))

Theorem

mulvfv 44444

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 44445

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

⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+_𝑟𝐵) Fn ℝ)

Theorem

subrfn 44446

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

⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴-_𝑟𝐵) Fn ℝ)

Theorem

mulvfn 44447

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

⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴._𝑣𝐵) Fn ℝ)

Theorem

addrcom 44448

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

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

Definition

df-ptdf 44449*

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 44450

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 44451*

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))}

21.41  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 8313 in 2008.

He developed a tool called "completeusersproof" that assists developing proofs using his "virtual deduction" method: https://us.metamath.org/other.html#completeusersproof 8313. His virtual deduction method is explained in the comment for wvd1 44543.

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....

21.41.1  Auxiliary theorems for the Virtual Deduction tool

Theorem

idiALT 44452

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 44453

Exportation implication also converting the consequent from a biconditional to an implication. Derived automatically from exbirVD 44826. (Contributed by Alan Sare, 31-Dec-2011.)

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

Theorem

3impexpbicom 44454

Version of 3impexp 1359 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 44455

Inference associated with 3impexpbicom 44454. Derived automatically from 3impexpbicomiVD 44831. (Contributed by Alan Sare, 31-Dec-2011.)

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

21.41.2  Supplementary unification deductions

Theorem

bi1imp 44456

Importation inference similar to imp 406, 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 44457

Importation inference similar to imp 406, except both implications of the hypothesis are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

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

Theorem

bi3impb 44458

Similar to 3impb 1114 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.)

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

Theorem

bi3impa 44459

Similar to 3impa 1109 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.)

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

Theorem

bi23impib 44460

3impib 1116 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 44461

3impib 1116 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 44462

3impib 1116 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

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

Theorem

bi13impia 44463

3impia 1117 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 44464

3impia 1117 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

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

Theorem

bi33imp12 44465

3imp 1110 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 44466

3imp 1110 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 44467

Similar to 3imp 1110 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 44468

Similar to 3imp 1110 except all but innermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

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

Theorem

bi23imp1 44469

Similar to 3imp 1110 except all but outermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

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

Theorem

bi123imp0 44470

Similar to 3imp 1110 except all implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

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

Theorem

4animp1 44471

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 44472

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 44473

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 44474

Biconditional form of expcomd 416. (Contributed by Alan Sare, 22-Jul-2012.) (New usage is discouraged.)

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

21.41.3  Conventional Metamath proofs, some derived from VD proofs

Theorem

iidn3 44475

idn3 44589 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 44476

e222 44610 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 44477

Right-biconditional form of e3 44710 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 44478

e13 44721 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 44479

e121 44630 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 44480

e122 44627 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 44481

e333 44706 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 44482

e323 44739 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 44483

If the second and third disjuncts of a true triple disjunction are false, then the first disjunct is true. Automatically derived from 3ornot23VD 44820. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒 ∨ 𝜑 ∨ 𝜓) → 𝜒))

Theorem

orbi1r 44484

orbi1 917 with order of disjuncts reversed. Derived from orbi1rVD 44821. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓)))

Theorem

3orbi123 44485

pm4.39 978 with a 3-conjunct antecedent. This proof is 3orbi123VD 44823 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 44486

Closed form of syl5 34. Derived automatically from syl5impVD 44836. (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 44487

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 44488

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 44489

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 44490

Non-virtual deduction form of e1111 44649. (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 44491

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 44492

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 44493

Non-virtual deduction form of e233 44738. (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 44494

Join three logical equivalences to form equivalence of implications. imbi13 44494 is imbi13VD 44847 without virtual deductions and was automatically derived from imbi13VD 44847 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 44495

Non-virtual deduction form of e33 44707. (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 44496

Biconditional contraposition variation. This proof is con5VD 44873 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 44497

Inference form of con5 44496. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 ↔ ¬ 𝜓)    ⇒   ⊢ (¬ 𝜑 → 𝜓)

Theorem

exlimexi 44498

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 44499*

Equivalence for substitution. Alternate proof of sb5 2276. This proof is sb5ALTVD 44886 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 44500

exinst01 44599 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)