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Theorem List for Metamath Proof Explorer - 38501-38600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmulvfv 38501 Scalar multiplication at a value. (Contributed by Andrew Salmon, 27-Jan-2012.)
((𝐴𝐸𝐵𝐷𝐶 ∈ ℝ) → ((𝐴.𝑣𝐵)‘𝐶) = (𝐴 · (𝐵𝐶)))

Theoremaddrfn 38502 Vector addition produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.)
((𝐴𝐶𝐵𝐷) → (𝐴+𝑟𝐵) Fn ℝ)

Theoremsubrfn 38503 Vector subtraction produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.)
((𝐴𝐶𝐵𝐷) → (𝐴-𝑟𝐵) Fn ℝ)

Theoremmulvfn 38504 Scalar multiplication producees a function. (Contributed by Andrew Salmon, 27-Jan-2012.)
((𝐴𝐶𝐵𝐷) → (𝐴.𝑣𝐵) Fn ℝ)

((𝐴𝐶𝐵𝐷) → (𝐴+𝑟𝐵) = (𝐵+𝑟𝐴))

Definitiondf-ptdf 38506* 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}))

Definitiondf-rr3 38507 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})

Definitiondf-line3 38508* 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 ∣ (2𝑜𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑧𝑦 → ran PtDf(𝑦, 𝑧) = 𝑥))}

20.31  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 7477 in 2008.

He developed a tool called "completeusersproof" that assists developing proofs using his "virtual deduction" method: http://us.metamath.org/other.html#completeusersproof. His virtual deduction method is explained in the comment for wvd1 38611.

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....

20.31.1  Auxiliary theorems for the Virtual Deduction tool

TheoremidiALT 38509 Placeholder for idi 2. 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

Theoremexbir 38510 Exportation implication also converting the consequent from a biconditional to an implication. Derived automatically from exbirVD 38914. (Contributed by Alan Sare, 31-Dec-2011.)
(((𝜑𝜓) → (𝜒𝜃)) → (𝜑 → (𝜓 → (𝜃𝜒))))

Theorem3impexpbicom 38511 Version of 3impexp 1288 where in addition the consequent is commuted. (Contributed by Alan Sare, 31-Dec-2011.)
(((𝜑𝜓𝜒) → (𝜃𝜏)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃)))))

Theorem3impexpbicomi 38512 Inference associated with 3impexpbicom 38511. Derived automatically from 3impexpbicomiVD 38919. (Contributed by Alan Sare, 31-Dec-2011.)
((𝜑𝜓𝜒) → (𝜃𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃))))

20.31.2  Supplementary unification deductions

Theorembi1imp 38513 Importation inference similar to imp 445, except the outermost implication of the hypothesis is a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 ↔ (𝜓𝜒))       ((𝜑𝜓) → 𝜒)

Theorembi2imp 38514 Importation inference similar to imp 445, except both implications of the hypothesis are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 ↔ (𝜓𝜒))       ((𝜑𝜓) → 𝜒)

Theorembi3impb 38515 Similar to 3impb 1259 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
((𝜑 ∧ (𝜓𝜒)) ↔ 𝜃)       ((𝜑𝜓𝜒) → 𝜃)

Theorembi3impa 38516 Similar to 3impa 1258 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
(((𝜑𝜓) ∧ 𝜒) ↔ 𝜃)       ((𝜑𝜓𝜒) → 𝜃)

Theorembi23impib 38517 3impib 1261 with the inner implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 → ((𝜓𝜒) ↔ 𝜃))       ((𝜑𝜓𝜒) → 𝜃)

Theorembi13impib 38518 3impib 1261 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 ↔ ((𝜓𝜒) → 𝜃))       ((𝜑𝜓𝜒) → 𝜃)

Theorembi123impib 38519 3impib 1261 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 ↔ ((𝜓𝜒) ↔ 𝜃))       ((𝜑𝜓𝜒) → 𝜃)

Theorembi13impia 38520 3impia 1260 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
((𝜑𝜓) ↔ (𝜒𝜃))       ((𝜑𝜓𝜒) → 𝜃)

Theorembi123impia 38521 3impia 1260 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
((𝜑𝜓) ↔ (𝜒𝜃))       ((𝜑𝜓𝜒) → 𝜃)

Theorembi33imp12 38522 3imp 1255 with innermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 → (𝜓 → (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)

Theorembi23imp13 38523 3imp 1255 with middle implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)

Theorembi13imp23 38524 3imp 1255 with outermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 ↔ (𝜓 → (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)

Theorembi13imp2 38525 Similar to 3imp 1255 except the outermost and innermost implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 ↔ (𝜓 → (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)

Theorembi12imp3 38526 Similar to 3imp 1255 except all but innermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 ↔ (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)

Theorembi23imp1 38527 Similar to 3imp 1255 except all but outermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)

Theorembi123imp0 38528 Similar to 3imp 1255 except all implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 ↔ (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)

Theorem4animp1 38529 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.)
((𝜑𝜓𝜒) → (𝜏𝜃))       ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theorem4an31 38530 A rearrangement of conjuncts for a 4-right-nested conjunction. (Contributed by Alan Sare, 30-May-2018.)
((((𝜒𝜓) ∧ 𝜑) ∧ 𝜃) → 𝜏)       ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theorem4an4132 38531 A rearrangement of conjuncts for a 4-right-nested conjunction. (Contributed by Alan Sare, 30-May-2018.)
((((𝜃𝜒) ∧ 𝜓) ∧ 𝜑) → 𝜏)       ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theoremexpcomdg 38532 Biconditional form of expcomd 454. (Contributed by Alan Sare, 22-Jul-2012.) (New usage is discouraged.)
((𝜑 → ((𝜓𝜒) → 𝜃)) ↔ (𝜑 → (𝜒 → (𝜓𝜃))))

20.31.3  Conventional Metamath proofs, some derived from VD proofs

Theoremiidn3 38533 idn3 38666 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 → (𝜒𝜒)))

Theoremee222 38534 e222 38687 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 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜓𝜏))    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜓𝜂))

Theoremee3bir 38535 Right-biconditional form of e3 38790 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 → (𝜒𝜃)))    &   (𝜏𝜃)       (𝜑 → (𝜓 → (𝜒𝜏)))

Theoremee13 38536 e13 38801 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.)
(𝜑𝜓)    &   (𝜑 → (𝜒 → (𝜃𝜏)))    &   (𝜓 → (𝜏𝜂))       (𝜑 → (𝜒 → (𝜃𝜂)))

Theoremee121 38537 e121 38707 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜑 → (𝜒𝜃))    &   (𝜑𝜏)    &   (𝜓 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜒𝜂))

Theoremee122 38538 e122 38704 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜑 → (𝜒𝜃))    &   (𝜑 → (𝜒𝜏))    &   (𝜓 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜒𝜂))

Theoremee333 38539 e333 38786 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓 → (𝜒𝜏)))    &   (𝜑 → (𝜓 → (𝜒𝜂)))    &   (𝜃 → (𝜏 → (𝜂𝜁)))       (𝜑 → (𝜓 → (𝜒𝜁)))

Theoremee323 38540 e323 38819 without virtual deductions. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓𝜏))    &   (𝜑 → (𝜓 → (𝜒𝜂)))    &   (𝜃 → (𝜏 → (𝜂𝜁)))       (𝜑 → (𝜓 → (𝜒𝜁)))

Theorem3ornot23 38541 If the second and third disjuncts of a true triple disjunction are false, then the first disjunct is true. Automatically derived from 3ornot23VD 38908. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒𝜑𝜓) → 𝜒))

Theoremorbi1r 38542 orbi1 742 with order of disjuncts reversed. Derived from orbi1rVD 38909. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))

Theorem3orbi123 38543 pm4.39 915 with a 3-conjunct antecedent. This proof is 3orbi123VD 38911 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂)))

Theoremsyl5imp 38544 Closed form of syl5 34. Derived automatically from syl5impVD 38925. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜃𝜓) → (𝜑 → (𝜃𝜒))))

Theoremimpexpd 38545 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 → ((𝜓𝜒) → 𝜃)) ↔ (𝜑 → (𝜓 → (𝜒𝜃))))

Theoremcom3rgbi 38546 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 → (𝜒𝜃))) ↔ (𝜒 → (𝜑 → (𝜓𝜃))))

Theoremimpexpdcom 38547 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 → ((𝜓𝜒) → 𝜃)) ↔ (𝜓 → (𝜒 → (𝜑𝜃))))

Theoremee1111 38548 Non-virtual deduction form of e1111 38726. (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)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂))))       (𝜑𝜂)

Theorempm2.43bgbi 38549 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 → (𝜑𝜒))) ↔ (𝜓 → (𝜑𝜒)))

Theorempm2.43cbi 38550 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 → (𝜑𝜃)))) ↔ (𝜓 → (𝜒 → (𝜑𝜃))))

Theoremee233 38551 Non-virtual deduction form of e233 38818. (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 → (𝜓𝜒))    &   (𝜑 → (𝜓 → (𝜃𝜏)))    &   (𝜑 → (𝜓 → (𝜃𝜂)))    &   (𝜒 → (𝜏 → (𝜂𝜁)))       (𝜑 → (𝜓 → (𝜃𝜁)))

Theoremimbi13 38552 Join three logical equivalences to form equivalence of implications. imbi13 38552 is imbi13VD 38936 without virtual deductions and was automatically derived from imbi13VD 38936 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.)
((𝜑𝜓) → ((𝜒𝜃) → ((𝜏𝜂) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂))))))

Theoremee33 38553 Non-virtual deduction form of e33 38787. (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 → (𝜒𝜃)))    &   (𝜑 → (𝜓 → (𝜒𝜏)))    &   (𝜃 → (𝜏𝜂))       (𝜑 → (𝜓 → (𝜒𝜂)))

Theoremcon5 38554 Biconditional contraposition variation. This proof is con5VD 38962 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑𝜓))

Theoremcon5i 38555 Inference form of con5 38554. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ↔ ¬ 𝜓)       𝜑𝜓)

Theoremexlimexi 38556 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 → (𝜑𝜓))       (∃𝑥𝜑𝜓)

Theoremsb5ALT 38557* Equivalence for substitution. Alternate proof of sb5 2429. This proof is sb5ALTVD 38975 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))

Theoremeexinst01 38558 exinst01 38676 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜓    &   (𝜑 → (𝜓𝜒))    &   (𝜑 → ∀𝑥𝜑)    &   (𝜒 → ∀𝑥𝜒)       (𝜑𝜒)

Theoremeexinst11 38559 exinst11 38677 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝜓)    &   (𝜑 → (𝜓𝜒))    &   (𝜑 → ∀𝑥𝜑)    &   (𝜒 → ∀𝑥𝜒)       (𝜑𝜒)

Theoremvk15.4j 38560 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 38560 is vk15.4jVD 38976 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(𝜃𝜏))    &    ¬ ∀𝑥(𝜏𝜑)       (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜓)

TheoremnotnotrALT 38561 Converse of double negation. Alternate proof of notnotr 125. This proof is notnotrALTVD 38977 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ¬ 𝜑𝜑)

Theoremcon3ALT2 38562 Contraposition. Alternate proof of con3 149. This proof is con3ALTVD 38978 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))

Theoremssralv2 38563* Quantification restricted to a subclass for two quantifiers. ssralv 3664 for two quantifiers. The proof of ssralv2 38563 was automatically generated by minimizing the automatically translated proof of ssralv2VD 38928. The automatic translation is by the tools program translatewithout_overwriting.cmd. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐶𝐷) → (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥𝐴𝑦𝐶 𝜑))

Theoremsbc3or 38564 sbcor 3477 with a 3-disjuncts. This proof is sbc3orgVD 38912 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](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))

TheoremsbcangOLD 38565 Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.) Obsolete as of 17-Aug-2018. Use sbcan 3476 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))

TheoremsbcorgOLD 38566 Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.) Obsolete as of 17-Aug-2018. Use sbcor 3477 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))

TheoremsbcbiiOLD 38567 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.) Obsolete as of 17-Aug-2018. Use sbcbii 3489 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝜓)       (𝐴𝑉 → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))

Theoremsbc3orgOLD 38568 sbcorgOLD 38566 with a 3-disjuncts. This proof is sbc3orgVD 38912 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))

Theoremalrim3con13v 38569* Closed form of alrimi 2081 with 2 additional conjuncts having no occurrences of the quantifying variable. This proof is 19.21a3con13vVD 38913 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → ∀𝑥𝜑) → ((𝜓𝜑𝜒) → ∀𝑥(𝜓𝜑𝜒)))

Theoremrspsbc2 38570* rspsbc 3516 with two quantifying variables. This proof is rspsbc2VD 38916 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → (𝐶𝐷 → (∀𝑥𝐵𝑦𝐷 𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑)))

Theoremsbcoreleleq 38571* Substitution of a setvar variable for another setvar variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 38921. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐴𝐴𝑥𝑥 = 𝐴)))

Theoremtratrb 38572* 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 38923. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐵)

TheoremordelordALT 38573 An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 5743 using the Axiom of Regularity indirectly through dford2 8514. 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 38573 is ordelordALTVD 38929 without virtual deductions and was automatically derived from ordelordALTVD 38929 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 𝐴𝐵𝐴) → Ord 𝐵)

Theoremsbcim2g 38574 Distribution of class substitution over a left-nested implication. Similar to sbcimg 3475. sbcim2g 38574 is sbcim2gVD 38937 without virtual deductions and was automatically derived from sbcim2gVD 38937 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.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))))

Theoremsbcbi 38575 Implication form of sbcbiiOLD 38567. sbcbi 38575 is sbcbiVD 38938 without virtual deductions and was automatically derived from sbcbiVD 38938 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.)
(𝐴𝑉 → (∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))

Theoremtrsbc 38576* Formula-building inference rule for class substitution, substituting a class variable for the setvar variable of the transitivity predicate. trsbc 38576 is trsbcVD 38939 without virtual deductions and was automatically derived from trsbcVD 38939 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.)
(𝐴𝑉 → ([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴))

TheoremtruniALT 38577* The union of a class of transitive sets is transitive. Alternate proof of truni 4765. truniALT 38577 is truniALTVD 38940 without virtual deductions and was automatically derived from truniALTVD 38940 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.)
(∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)

TheoremsbcalgOLD 38578* Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.) Obsolete as of 17-Aug-2018. Use sbcal 3483 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑦]𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑))

TheoremsbcexgOLD 38579* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) Obsolete as of 17-Aug-2018. Use sbcex 3443 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑦]𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))

Theoremsbcel12gOLD 38580 Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) Obsolete as of 18-Aug-2018. Use sbcel12 3981 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

Theoremsbcel2gOLD 38581* Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.) Obsolete as of 18-Aug-2018. Use sbcel2 3987 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐵𝐴 / 𝑥𝐶))

TheoremsbcssOLD 38582 Distribute proper substitution through a subclass relation. This theorem was automatically derived from sbcssgVD 38945. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → ([𝐴 / 𝑥]𝐶𝐷𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))

TheoremonfrALTlem5 38583* Lemma for onfrALT 38590. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
([(𝑎𝑥) / 𝑏]((𝑏 ⊆ (𝑎𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦𝑏 (𝑏𝑦) = ∅) ↔ (((𝑎𝑥) ⊆ (𝑎𝑥) ∧ (𝑎𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅))

TheoremonfrALTlem4 38584* Lemma for onfrALT 38590. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))

TheoremonfrALTlem3 38585* Lemma for onfrALT 38590. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅))

Theoremggen31 38586* gen31 38672 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → (𝜓 → (𝜒 → ∀𝑥𝜃)))

TheoremonfrALTlem2 38587* Lemma for onfrALT 38590. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))

Theoremcbvexsv 38588* 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)

TheoremonfrALTlem1 38589* Lemma for onfrALT 38590. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ (𝑎𝑥) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))

TheoremonfrALT 38590 The epsilon relation is foundational on the class of ordinal numbers. onfrALT 38590 is an alternate proof of onfr 5761. onfrALTVD 38953 is the Virtual Deduction proof from which onfrALT 38590 is derived. The Virtual Deduction proof mirrors the working proof of onfr 5761 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 38953. 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

Theoremcsbeq2gOLD 38591 Formula-building implication rule for class substitution. Closed form of csbeq2i 3991. csbeq2gOLD 38591 is derived from the virtual deduction proof csbeq2gVD 38954. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete version of csbeq2 3535 as of 11-Oct-2018. (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → (∀𝑥 𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))

Theorem19.41rg 38592 Closed form of right-to-left implication of 19.41 2102, Theorem 19.41 of [Margaris] p. 90. Derived from 19.41rgVD 38964. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓)))

Theoremopelopab4 38593* Ordered pair membership in a class abstraction of pairs. Compare to elopab 4981. (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} ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))

Theorem2pm13.193 38594 pm13.193 38438 for two variables. pm13.193 38438 is Theorem *13.193 in [WhiteheadRussell] p. 179. Derived from 2pm13.193VD 38965. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))

Theoremhbntal 38595 A closed form of hbn 2145. hbnt 2143 is another closed form of hbn 2145. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))

Theoremhbimpg 38596 A closed form of hbim 2126. Derived from hbimpgVD 38966. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓)) → ∀𝑥((𝜑𝜓) → ∀𝑥(𝜑𝜓)))

Theoremhbalg 38597 Closed form of hbal 2035. Derived from hbalgVD 38967. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥𝑦𝜑))

Theoremhbexg 38598 Closed form of nfex 2153. Derived from hbexgVD 38968. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 12-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑))

Theoremax6e2eq 38599* Alternate form of ax6e 2249 for non-distinct 𝑥, 𝑦 and 𝑢 = 𝑣. ax6e2eq 38599 is derived from ax6e2eqVD 38969. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))

Theoremax6e2nd 38600* If at least two sets exist (dtru 4855) , then the same is true expressed in an alternate form similar to the form of ax6e 2249. ax6e2nd 38600 is derived from ax6e2ndVD 38970. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))

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