P. 409 of Theorem List - Metamath Proof Explorer

Type

Label

Description

Statement

Theorem

ipo0 40801

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 40802

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

⊢ ( I Fr 𝐴 ↔ 𝐴 = ∅)

Theorem

ordpss 40803

ordelpss 6219 with an antecedent removed. (Contributed by Andrew Salmon, 25-Jul-2011.)

⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → 𝐴 ⊊ 𝐵))

Theorem

fvsb 40804*

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 40805*

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 40806

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 40807

An element of a transitive set is a proper subset of it. Theorem 7.2 in [TakeutiZaring] p. 35. Unlike tz7.2 5539, ax-reg 9056 is required for its proof. (Contributed by Andrew Salmon, 13-Nov-2011.)

⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊊ 𝐴)

20.35.6  Arithmetic

Theorem

addcomgi 40808

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)

20.35.7  Geometry

Syntax

cplusr 40809

Introduce the operation of vector addition.

class +_𝑟

Syntax

cminusr 40810

Introduce the operation of vector subtraction.

class -_𝑟

Syntax

ctimesr 40811

Introduce the operation of scalar multiplication.

class ._𝑣

Syntax

cptdfc 40812

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 40813

RR3 is a class.

class RR3

Syntax

cline3 40814

line3 is a class.

class line3

Definition

df-addr 40815*

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 40816*

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 40817*

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 40818*

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 40819*

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 40820*

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 40821

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 40822

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 40823

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 40824

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

⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+_𝑟𝐵) Fn ℝ)

Theorem

subrfn 40825

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

⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴-_𝑟𝐵) Fn ℝ)

Theorem

mulvfn 40826

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

⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴._𝑣𝐵) Fn ℝ)

Theorem

addrcom 40827

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

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

Definition

df-ptdf 40828*

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 40829

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 40830*

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))}

20.36  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 8020 in 2008.

He developed a tool called "completeusersproof" that assists developing proofs using his "virtual deduction" method: https://us.metamath.org/other.html#completeusersproof 8020. His virtual deduction method is explained in the comment for wvd1 40923.

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....

20.36.1  Auxiliary theorems for the Virtual Deduction tool

Theorem

idiALT 40831

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 40832

Exportation implication also converting the consequent from a biconditional to an implication. Derived automatically from exbirVD 41207. (Contributed by Alan Sare, 31-Dec-2011.)

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

Theorem

3impexpbicom 40833

Version of 3impexp 1354 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 40834

Inference associated with 3impexpbicom 40833. Derived automatically from 3impexpbicomiVD 41212. (Contributed by Alan Sare, 31-Dec-2011.)

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

20.36.2  Supplementary unification deductions

Theorem

bi1imp 40835

Importation inference similar to imp 409, 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 40836

Importation inference similar to imp 409, except both implications of the hypothesis are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

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

Theorem

bi3impb 40837

Similar to 3impb 1111 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.)

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

Theorem

bi3impa 40838

Similar to 3impa 1106 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.)

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

Theorem

bi23impib 40839

3impib 1112 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 40840

3impib 1112 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 40841

3impib 1112 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

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

Theorem

bi13impia 40842

3impia 1113 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 40843

3impia 1113 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

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

Theorem

bi33imp12 40844

3imp 1107 with innermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)

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

Theorem

bi23imp13 40845

3imp 1107 with middle implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)

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

Theorem

bi13imp23 40846

3imp 1107 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 40847

Similar to 3imp 1107 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 40848

Similar to 3imp 1107 except all but innermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

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

Theorem

bi23imp1 40849

Similar to 3imp 1107 except all but outermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

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

Theorem

bi123imp0 40850

Similar to 3imp 1107 except all implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

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

Theorem

4animp1 40851

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 40852

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 40853

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 40854

Biconditional form of expcomd 419. (Contributed by Alan Sare, 22-Jul-2012.) (New usage is discouraged.)

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

20.36.3  Conventional Metamath proofs, some derived from VD proofs

Theorem

iidn3 40855

idn3 40969 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 40856

e222 40990 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 40857

Right-biconditional form of e3 41091 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 40858

e13 41102 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 40859

e121 41010 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 40860

e122 41007 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 40861

e333 41087 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 40862

e323 41120 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 40863

If the second and third disjuncts of a true triple disjunction are false, then the first disjunct is true. Automatically derived from 3ornot23VD 41201. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒 ∨ 𝜑 ∨ 𝜓) → 𝜒))

Theorem

orbi1r 40864

orbi1 914 with order of disjuncts reversed. Derived from orbi1rVD 41202. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓)))

Theorem

3orbi123 40865

pm4.39 973 with a 3-conjunct antecedent. This proof is 3orbi123VD 41204 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 40866

Closed form of syl5 34. Derived automatically from syl5impVD 41217. (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 40867

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 40868

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 40869

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 40870

Non-virtual deduction form of e1111 41029. (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 40871

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 40872

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 40873

Non-virtual deduction form of e233 41119. (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 40874

Join three logical equivalences to form equivalence of implications. imbi13 40874 is imbi13VD 41228 without virtual deductions and was automatically derived from imbi13VD 41228 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 40875

Non-virtual deduction form of e33 41088. (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 40876

Biconditional contraposition variation. This proof is con5VD 41254 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 40877

Inference form of con5 40876. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 ↔ ¬ 𝜓)    ⇒   ⊢ (¬ 𝜑 → 𝜓)

Theorem

exlimexi 40878

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 40879*

Equivalence for substitution. Alternate proof of sb5 2276. This proof is sb5ALTVD 41267 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 40880

exinst01 40979 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 40881

exinst11 40980 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 40882

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 40882 is vk15.4jVD 41268 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 40883

Converse of double negation. Alternate proof of notnotr 132. This proof is notnotrALTVD 41269 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (¬ ¬ 𝜑 → 𝜑)

Theorem

con3ALT2 40884

Contraposition. Alternate proof of con3 156. This proof is con3ALTVD 41270 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 40885*

Quantification restricted to a subclass for two quantifiers. ssralv 4033 for two quantifiers. The proof of ssralv2 40885 was automatically generated by minimizing the automatically translated proof of ssralv2VD 41220. 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 40886

sbcor 3822 with a 3-disjuncts. This proof is sbc3orgVD 41205 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 40887*

Closed form of alrimi 2213 with 2 additional conjuncts having no occurrences of the quantifying variable. This proof is 19.21a3con13vVD 41206 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 40888*

rspsbc 3862 with two quantifying variables. This proof is rspsbc2VD 41209 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 40889*

Substitution of a setvar variable for another setvar variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 41213. (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 40890*

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 41215. (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 40891

An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 6213 using the Axiom of Regularity indirectly through dford2 9083. 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 40891 is ordelordALTVD 41221 without virtual deductions and was automatically derived from ordelordALTVD 41221 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 40892

Distribution of class substitution over a left-nested implication. Similar to sbcimg 3820. sbcim2g 40892 is sbcim2gVD 41229 without virtual deductions and was automatically derived from sbcim2gVD 41229 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 40893

Implication form of sbcbii 3829. sbcbi 40893 is sbcbiVD 41230 without virtual deductions and was automatically derived from sbcbiVD 41230 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 40894*

Formula-building inference rule for class substitution, substituting a class variable for the setvar variable of the transitivity predicate. trsbc 40894 is trsbcVD 41231 without virtual deductions and was automatically derived from trsbcVD 41231 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 40895*

The union of a class of transitive sets is transitive. Alternate proof of truni 5186. truniALT 40895 is truniALTVD 41232 without virtual deductions and was automatically derived from truniALTVD 41232 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 40896*

Lemma for onfrALT 40903. (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 40897*

Lemma for onfrALT 40903. (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 40898*

Lemma for onfrALT 40903. (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 40899*

gen31 40975 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 40900*

Lemma for onfrALT 40903. (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) = ∅))