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Type | Label | Description | ||||||||||||||||||||||||||||||
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Statement | ||||||||||||||||||||||||||||||||
Theorem | addrval 39501* | 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 39502* | 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 39503* | 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 39504 | 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 39505 | 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 39506 | 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 39507 | Vector addition produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.) | ||||||||||||||||||||||||||||||
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+𝑟𝐵) Fn ℝ) | ||||||||||||||||||||||||||||||||
Theorem | subrfn 39508 | Vector subtraction produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.) | ||||||||||||||||||||||||||||||
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴-𝑟𝐵) Fn ℝ) | ||||||||||||||||||||||||||||||||
Theorem | mulvfn 39509 | Scalar multiplication producees a function. (Contributed by Andrew Salmon, 27-Jan-2012.) | ||||||||||||||||||||||||||||||
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴.𝑣𝐵) Fn ℝ) | ||||||||||||||||||||||||||||||||
Theorem | addrcom 39510 | Vector addition is commutative. (Contributed by Andrew Salmon, 28-Jan-2012.) | ||||||||||||||||||||||||||||||
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+𝑟𝐵) = (𝐵+𝑟𝐴)) | ||||||||||||||||||||||||||||||||
Definition | df-ptdf 39511* | 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 39512 | 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 39513* | 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 ∣ (2o ≼ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑧 ≠ 𝑦 → ran PtDf(𝑦, 𝑧) = 𝑥))} | ||||||||||||||||||||||||||||||||
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 7746 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 39606. 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.... | ||||||||||||||||||||||||||||||||
Theorem | idiALT 39514 | 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 | ||||||||||||||||||||||||||||||||
Theorem | exbir 39515 | Exportation implication also converting the consequent from a biconditional to an implication. Derived automatically from exbirVD 39900. (Contributed by Alan Sare, 31-Dec-2011.) | ||||||||||||||||||||||||||||||
⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → (𝜑 → (𝜓 → (𝜃 → 𝜒)))) | ||||||||||||||||||||||||||||||||
Theorem | 3impexpbicom 39516 | Version of 3impexp 1471 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 39517 | Inference associated with 3impexpbicom 39516. Derived automatically from 3impexpbicomiVD 39905. (Contributed by Alan Sare, 31-Dec-2011.) | ||||||||||||||||||||||||||||||
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃)))) | ||||||||||||||||||||||||||||||||
Theorem | bi1imp 39518 | Importation inference similar to imp 397, 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 39519 | Importation inference similar to imp 397, except both implications of the hypothesis are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.) | ||||||||||||||||||||||||||||||
⊢ (𝜑 ↔ (𝜓 ↔ 𝜒)) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||||||||||||||||||||||||||||||||
Theorem | bi3impb 39520 | Similar to 3impb 1147 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.) | ||||||||||||||||||||||||||||||
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||||||||||||||||||||||||||||||||
Theorem | bi3impa 39521 | Similar to 3impa 1140 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.) | ||||||||||||||||||||||||||||||
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||||||||||||||||||||||||||||||||
Theorem | bi23impib 39522 | 3impib 1148 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 39523 | 3impib 1148 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 39524 | 3impib 1148 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.) | ||||||||||||||||||||||||||||||
⊢ (𝜑 ↔ ((𝜓 ∧ 𝜒) ↔ 𝜃)) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||||||||||||||||||||||||||||||||
Theorem | bi13impia 39525 | 3impia 1149 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 39526 | 3impia 1149 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.) | ||||||||||||||||||||||||||||||
⊢ ((𝜑 ∧ 𝜓) ↔ (𝜒 ↔ 𝜃)) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||||||||||||||||||||||||||||||||
Theorem | bi33imp12 39527 | 3imp 1141 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 39528 | 3imp 1141 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 39529 | 3imp 1141 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 39530 | Similar to 3imp 1141 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 39531 | Similar to 3imp 1141 except all but innermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.) | ||||||||||||||||||||||||||||||
⊢ (𝜑 ↔ (𝜓 ↔ (𝜒 → 𝜃))) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||||||||||||||||||||||||||||||||
Theorem | bi23imp1 39532 | Similar to 3imp 1141 except all but outermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.) | ||||||||||||||||||||||||||||||
⊢ (𝜑 → (𝜓 ↔ (𝜒 ↔ 𝜃))) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||||||||||||||||||||||||||||||||
Theorem | bi123imp0 39533 | Similar to 3imp 1141 except all implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.) | ||||||||||||||||||||||||||||||
⊢ (𝜑 ↔ (𝜓 ↔ (𝜒 ↔ 𝜃))) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||||||||||||||||||||||||||||||||
Theorem | 4animp1 39534 | 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 39535 | 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 39536 | 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 39537 | Biconditional form of expcomd 408. (Contributed by Alan Sare, 22-Jul-2012.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
⊢ ((𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) ↔ (𝜑 → (𝜒 → (𝜓 → 𝜃)))) | ||||||||||||||||||||||||||||||||
Theorem | iidn3 39538 | idn3 39661 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 39539 | e222 39682 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 39540 | Right-biconditional form of e3 39784 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 39541 | e13 39795 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 39542 | e121 39702 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 39543 | e122 39699 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 39544 | e333 39780 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 39545 | e323 39813 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 39546 | If the second and third disjuncts of a true triple disjunction are false, then the first disjunct is true. Automatically derived from 3ornot23VD 39894. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒 ∨ 𝜑 ∨ 𝜓) → 𝜒)) | ||||||||||||||||||||||||||||||||
Theorem | orbi1r 39547 | orbi1 946 with order of disjuncts reversed. Derived from orbi1rVD 39895. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓))) | ||||||||||||||||||||||||||||||||
Theorem | 3orbi123 39548 | pm4.39 1004 with a 3-conjunct antecedent. This proof is 3orbi123VD 39897 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 39549 | Closed form of syl5 34. Derived automatically from syl5impVD 39910. (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 39550 |
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.)
| ||||||||||||||||||||||||||||||
⊢ ((𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃)))) | ||||||||||||||||||||||||||||||||
Theorem | com3rgbi 39551 |
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.)
| ||||||||||||||||||||||||||||||
⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) ↔ (𝜒 → (𝜑 → (𝜓 → 𝜃)))) | ||||||||||||||||||||||||||||||||
Theorem | impexpdcom 39552 |
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.)
| ||||||||||||||||||||||||||||||
⊢ ((𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) ↔ (𝜓 → (𝜒 → (𝜑 → 𝜃)))) | ||||||||||||||||||||||||||||||||
Theorem | ee1111 39553 |
Non-virtual deduction form of e1111 39721. (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.
| ||||||||||||||||||||||||||||||
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → 𝜒) & ⊢ (𝜑 → 𝜃) & ⊢ (𝜑 → 𝜏) & ⊢ (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂)))) ⇒ ⊢ (𝜑 → 𝜂) | ||||||||||||||||||||||||||||||||
Theorem | pm2.43bgbi 39554 |
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.
| ||||||||||||||||||||||||||||||
⊢ ((𝜑 → (𝜓 → (𝜑 → 𝜒))) ↔ (𝜓 → (𝜑 → 𝜒))) | ||||||||||||||||||||||||||||||||
Theorem | pm2.43cbi 39555 |
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.
| ||||||||||||||||||||||||||||||
⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜑 → 𝜃)))) ↔ (𝜓 → (𝜒 → (𝜑 → 𝜃)))) | ||||||||||||||||||||||||||||||||
Theorem | ee233 39556 |
Non-virtual deduction form of e233 39812. (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.
| ||||||||||||||||||||||||||||||
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏))) & ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜂))) & ⊢ (𝜒 → (𝜏 → (𝜂 → 𝜁))) ⇒ ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜁))) | ||||||||||||||||||||||||||||||||
Theorem | imbi13 39557 | Join three logical equivalences to form equivalence of implications. imbi13 39557 is imbi13VD 39921 without virtual deductions and was automatically derived from imbi13VD 39921 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 39558 |
Non-virtual deduction form of e33 39781. (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.
| ||||||||||||||||||||||||||||||
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) & ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏))) & ⊢ (𝜃 → (𝜏 → 𝜂)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂))) | ||||||||||||||||||||||||||||||||
Theorem | con5 39559 | Biconditional contraposition variation. This proof is con5VD 39947 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 39560 | Inference form of con5 39559. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
⊢ (𝜑 ↔ ¬ 𝜓) ⇒ ⊢ (¬ 𝜑 → 𝜓) | ||||||||||||||||||||||||||||||||
Theorem | exlimexi 39561 | 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 39562* | Equivalence for substitution. Alternate proof of sb5 2308. This proof is sb5ALTVD 39960 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 39563 | exinst01 39671 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 39564 | exinst11 39672 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 39565 | 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 39565 is vk15.4jVD 39961 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 39566 | Converse of double negation. Alternate proof of notnotr 128. This proof is notnotrALTVD 39962 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
⊢ (¬ ¬ 𝜑 → 𝜑) | ||||||||||||||||||||||||||||||||
Theorem | con3ALT2 39567 | Contraposition. Alternate proof of con3 151. This proof is con3ALTVD 39963 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 39568* | Quantification restricted to a subclass for two quantifiers. ssralv 3891 for two quantifiers. The proof of ssralv2 39568 was automatically generated by minimizing the automatically translated proof of ssralv2VD 39913. 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.) | ||||||||||||||||||||||||||||||
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝜑)) | ||||||||||||||||||||||||||||||||
Theorem | sbc3or 39569 | sbcor 3706 with a 3-disjuncts. This proof is sbc3orgVD 39898 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 39570* | Closed form of alrimi 2256 with 2 additional conjuncts having no occurrences of the quantifying variable. This proof is 19.21a3con13vVD 39899 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 39571* | rspsbc 3742 with two quantifying variables. This proof is rspsbc2VD 39902 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 39572* | Substitution of a setvar variable for another setvar variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 39906. (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 39573* | 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 39908. (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 39574 | An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 5985 using the Axiom of Regularity indirectly through dford2 8794. 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 39574 is ordelordALTVD 39914 without virtual deductions and was automatically derived from ordelordALTVD 39914 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 39575 | Distribution of class substitution over a left-nested implication. Similar to sbcimg 3704. sbcim2g 39575 is sbcim2gVD 39922 without virtual deductions and was automatically derived from sbcim2gVD 39922 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 39576 | Implication form of sbcbii 3718. sbcbi 39576 is sbcbiVD 39923 without virtual deductions and was automatically derived from sbcbiVD 39923 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 39577* | Formula-building inference rule for class substitution, substituting a class variable for the setvar variable of the transitivity predicate. trsbc 39577 is trsbcVD 39924 without virtual deductions and was automatically derived from trsbcVD 39924 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 39578* | The union of a class of transitive sets is transitive. Alternate proof of truni 4989. truniALT 39578 is truniALTVD 39925 without virtual deductions and was automatically derived from truniALTVD 39925 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 39579* | Lemma for onfrALT 39586. (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 39580* | Lemma for onfrALT 39586. (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 39581* | Lemma for onfrALT 39586. (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 39582* | gen31 39667 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 39583* | Lemma for onfrALT 39586. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)) | ||||||||||||||||||||||||||||||||
Theorem | cbvexsv 39584* | A theorem pertaining to the substitution for an existentially quantified variable when the substituted variable does not occur in the quantified wff. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) | ||||||||||||||||||||||||||||||||
Theorem | onfrALTlem1 39585* | Lemma for onfrALT 39586. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)) | ||||||||||||||||||||||||||||||||
Theorem | onfrALT 39586 | The epsilon relation is foundational on the class of ordinal numbers. onfrALT 39586 is an alternate proof of onfr 6002. onfrALTVD 39938 is the Virtual Deduction proof from which onfrALT 39586 is derived. The Virtual Deduction proof mirrors the working proof of onfr 6002 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 39938. This theorem does not rely on the Axiom of Regularity. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
⊢ E Fr On | ||||||||||||||||||||||||||||||||
Theorem | 19.41rg 39587 | Closed form of right-to-left implication of 19.41 2278, Theorem 19.41 of [Margaris] p. 90. Derived from 19.41rgVD 39949. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓))) | ||||||||||||||||||||||||||||||||
Theorem | opelopab4 39588* | Ordered pair membership in a class abstraction of pairs. Compare to elopab 5209. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 6-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
⊢ (〈𝑢, 𝑣〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)) | ||||||||||||||||||||||||||||||||
Theorem | 2pm13.193 39589 | pm13.193 39444 for two variables. pm13.193 39444 is Theorem *13.193 in [WhiteheadRussell] p. 179. Derived from 2pm13.193VD 39950. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)) | ||||||||||||||||||||||||||||||||
Theorem | hbntal 39590 | A closed form of hbn 2327. hbnt 2326 is another closed form of hbn 2327. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑)) | ||||||||||||||||||||||||||||||||
Theorem | hbimpg 39591 | A closed form of hbim 2331. Derived from hbimpgVD 39951. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
⊢ ((∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓)) → ∀𝑥((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))) | ||||||||||||||||||||||||||||||||
Theorem | hbalg 39592 | Closed form of hbal 2217. Derived from hbalgVD 39952. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)) | ||||||||||||||||||||||||||||||||
Theorem | hbexg 39593 | Closed form of nfex 2356. Derived from hbexgVD 39953. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 12-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)) | ||||||||||||||||||||||||||||||||
Theorem | ax6e2eq 39594* | Alternate form of ax6e 2402 for non-distinct 𝑥, 𝑦 and 𝑢 = 𝑣. ax6e2eq 39594 is derived from ax6e2eqVD 39954. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
⊢ (∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))) | ||||||||||||||||||||||||||||||||
Theorem | ax6e2nd 39595* | If at least two sets exist (dtru 5070) , then the same is true expressed in an alternate form similar to the form of ax6e 2402. ax6e2nd 39595 is derived from ax6e2ndVD 39955. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) | ||||||||||||||||||||||||||||||||
Theorem | ax6e2ndeq 39596* | "At least two sets exist" expressed in the form of dtru 5070 is logically equivalent to the same expressed in a form similar to ax6e 2402 if dtru 5070 is false implies 𝑢 = 𝑣. ax6e2ndeq 39596 is derived from ax6e2ndeqVD 39956. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) ↔ ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) | ||||||||||||||||||||||||||||||||
Theorem | 2sb5nd 39597* | Equivalence for double substitution 2sb5 2309 without distinct 𝑥, 𝑦 requirement. 2sb5nd 39597 is derived from 2sb5ndVD 39957. (Contributed by Alan Sare, 30-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))) | ||||||||||||||||||||||||||||||||
Theorem | 2uasbanh 39598* | Distribute the unabbreviated form of proper substitution in and out of a conjunction. 2uasbanh 39598 is derived from 2uasbanhVD 39958. (Contributed by Alan Sare, 31-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
⊢ (𝜒 ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))) ⇒ ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)) ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))) | ||||||||||||||||||||||||||||||||
Theorem | 2uasban 39599* | Distribute the unabbreviated form of proper substitution in and out of a conjunction. (Contributed by Alan Sare, 31-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)) ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))) | ||||||||||||||||||||||||||||||||
Theorem | e2ebind 39600 | Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind 39600 is derived from e2ebindVD 39959. (Contributed by Alan Sare, 27-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑)) |
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