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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | sbid2 2401 | An identity law for substitution. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑) | ||
Theorem | sbidm 2402 | An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Jul-2019.) |
⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | ||
Theorem | sbco2 2403 | A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Sep-2018.) |
⊢ Ⅎ𝑧𝜑 ⇒ ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | ||
Theorem | sbco2d 2404 | A composition law for substitution. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑧𝜑 & ⊢ (𝜑 → Ⅎ𝑧𝜓) ⇒ ⊢ (𝜑 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓)) | ||
Theorem | sbco3 2405 | A composition law for substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Wolf Lammen, 18-Sep-2018.) |
⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑) | ||
Theorem | sbcom 2406 | A commutativity law for substitution. (Contributed by NM, 27-May-1997.) (Proof shortened by Wolf Lammen, 20-Sep-2018.) |
⊢ ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑) | ||
Theorem | sbt 2407 | A substitution into a theorem yields a theorem. (See chvar 2250 and chvarv 2251 for versions using implicit substitution.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 20-Jul-2018.) |
⊢ 𝜑 ⇒ ⊢ [𝑦 / 𝑥]𝜑 | ||
Theorem | sbtrt 2408 | Partially closed form of sbtr 2409. (Contributed by BJ, 4-Jun-2019.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 → 𝜑) | ||
Theorem | sbtr 2409 | A partial converse to sbt 2407. If the substitution of a variable for a non-free one in a wff gives a theorem, then the original wff is a theorem. (Contributed by BJ, 15-Sep-2018.) |
⊢ Ⅎ𝑦𝜑 & ⊢ [𝑦 / 𝑥]𝜑 ⇒ ⊢ 𝜑 | ||
Theorem | sb5rf 2410 | Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 20-Sep-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (𝜑 ↔ ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑)) | ||
Theorem | sb6rf 2411 | Reversed substitution. (Contributed by NM, 1-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)) | ||
Theorem | sb8 2412 | Substitution of variable in universal quantifier. (Contributed by NM, 16-May-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | sb8e 2413 | Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | sb9 2414 | Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) Allow a shortening of sb9i 2415. (Revised by Wolf Lammen, 15-Jun-2019.) |
⊢ (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | sb9i 2415 | Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 15-Jun-2019.) |
⊢ (∀𝑥[𝑥 / 𝑦]𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | ax12vALT 2416* | Alternate proof of ax12v2 2036, shorter, but depending on more axioms. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | sb6 2417* | Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. The implication "to the left" is sb2 2340 and does not require any dv condition. Theorem sb6f 2373 replaces the dv condition with a non-freeness hypothesis. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) |
⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | sb5 2418* | Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. The implication "to the right" is sb1 1870 and does not require any dv condition. Theorem sb5f 2374 replaces the dv condition with a non-freeness hypothesis. (Contributed by NM, 18-Aug-1993.) |
⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | ||
Theorem | equsb3lem 2419* | Lemma for equsb3 2420. (Contributed by Raph Levien and FL, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
⊢ ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧) | ||
Theorem | equsb3 2420* | Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.) Remove dependency on ax-11 2021. (Revised by Wolf Lammen, 21-Sep-2018.) |
⊢ ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧) | ||
Theorem | equsb3ALT 2421* | Alternate proof of equsb3 2420, shorter but requiring ax-11 2021. (Contributed by Raph Levien and FL, 4-Dec-2005.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧) | ||
Theorem | elsb3 2422* | Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
⊢ ([𝑥 / 𝑦]𝑦 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧) | ||
Theorem | elsb4 2423* | Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
⊢ ([𝑥 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑥) | ||
Theorem | hbs1 2424* | The setvar 𝑥 is not free in [𝑦 / 𝑥]𝜑 when 𝑥 and 𝑦 are distinct. (Contributed by NM, 26-May-1993.) |
⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | nfs1v 2425* | The setvar 𝑥 is not free in [𝑦 / 𝑥]𝜑 when 𝑥 and 𝑦 are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | ||
Theorem | sbhb 2426* | Two ways of expressing "𝑥 is (effectively) not free in 𝜑." (Contributed by NM, 29-May-2009.) |
⊢ ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)) | ||
Theorem | sbnf2 2427* | Two ways of expressing "𝑥 is (effectively) not free in 𝜑." (Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Sep-2018.) |
⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑦∀𝑧([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)) | ||
Theorem | nfsb 2428* | If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑧𝜑 ⇒ ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 | ||
Theorem | hbsb 2429* | If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. (Contributed by NM, 12-Aug-1993.) |
⊢ (𝜑 → ∀𝑧𝜑) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑) | ||
Theorem | nfsbd 2430* | Deduction version of nfsb 2428. (Contributed by NM, 15-Feb-2013.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑧𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓) | ||
Theorem | 2sb5 2431* | Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.) |
⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑)) | ||
Theorem | 2sb6 2432* | Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.) |
⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑)) | ||
Theorem | sbcom2 2433* | Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.) (Proof shortened by Wolf Lammen, 24-Sep-2018.) |
⊢ ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑) | ||
Theorem | sbcom4 2434* | Commutativity law for substitution. This theorem was incorrectly used as our previous version of pm11.07 2435 but may still be useful. (Contributed by Andrew Salmon, 17-Jun-2011.) (Proof shortened by Jim Kingdon, 22-Jan-2018.) |
⊢ ([𝑤 / 𝑥][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑) | ||
Theorem | pm11.07 2435 | Axiom *11.07 in [WhiteheadRussell] p. 159. The original reads: *11.07 "Whatever possible argument 𝑥 may be, 𝜑(𝑥, 𝑦) is true whatever possible argument 𝑦 may be" implies the corresponding statement with 𝑥 and 𝑦 interchanged except in "𝜑(𝑥, 𝑦)". Under our formalism this appears to correspond to idi 2 and not to sbcom4 2434 as earlier thought. See https://groups.google.com/d/msg/metamath/iS0fOvSemC8/M1zTH8wxCAAJ. (Contributed by BJ, 16-Sep-2018.) (New usage is discouraged.) |
⊢ 𝜑 ⇒ ⊢ 𝜑 | ||
Theorem | sb6a 2436* | Equivalence for substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Wolf Lammen, 23-Sep-2018.) |
⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑)) | ||
Theorem | 2ax6elem 2437 | We can always find values matching 𝑥 and 𝑦, as long as they are represented by distinct variables. This theorem merges two ax6e 2238 instances ∃𝑧𝑧 = 𝑥 and ∃𝑤𝑤 = 𝑦 into a common expression. Alan Sare contributed a variant of this theorem with distinct variable conditions before, see ax6e2nd 37795. (Contributed by Wolf Lammen, 27-Sep-2018.) |
⊢ (¬ ∀𝑤 𝑤 = 𝑧 → ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦)) | ||
Theorem | 2ax6e 2438* | We can always find values matching 𝑥 and 𝑦, as long as they are represented by distinct variables. Version of 2ax6elem 2437 with a distinct variable constraint. (Contributed by Wolf Lammen, 28-Sep-2018.) |
⊢ ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦) | ||
Theorem | 2sb5rf 2439* | Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) Remove distinct variable constraints. (Revised by Wolf Lammen, 28-Sep-2018.) |
⊢ Ⅎ𝑧𝜑 & ⊢ Ⅎ𝑤𝜑 ⇒ ⊢ (𝜑 ↔ ∃𝑧∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)) | ||
Theorem | 2sb6rf 2440* | Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) Remove variable constraints. (Revised by Wolf Lammen, 28-Sep-2018.) |
⊢ Ⅎ𝑧𝜑 & ⊢ Ⅎ𝑤𝜑 ⇒ ⊢ (𝜑 ↔ ∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)) | ||
Theorem | sb7f 2441* | This version of dfsb7 2443 does not require that 𝜑 and 𝑧 be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-5 1827 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1868 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Revised by Mario Carneiro, 6-Oct-2016.) |
⊢ Ⅎ𝑧𝜑 ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑))) | ||
Theorem | sb7h 2442* | This version of dfsb7 2443 does not require that 𝜑 and 𝑧 be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-5 1827 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1868 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (𝜑 → ∀𝑧𝜑) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑))) | ||
Theorem | dfsb7 2443* | An alternate definition of proper substitution df-sb 1868. By introducing a dummy variable 𝑧 in the definiens, we are able to eliminate any distinct variable restrictions among the variables 𝑥, 𝑦, and 𝜑 of the definiendum. No distinct variable conflicts arise because 𝑧 effectively insulates 𝑥 from 𝑦. To achieve this, we use a chain of two substitutions in the form of sb5 2418, first 𝑧 for 𝑥 then 𝑦 for 𝑧. Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 2597. Theorem sb7h 2442 provides a version where 𝜑 and 𝑧 don't have to be distinct. (Contributed by NM, 28-Jan-2004.) |
⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑))) | ||
Theorem | sb10f 2444* | Hao Wang's identity axiom P6 in Irving Copi, Symbolic Logic (5th ed., 1979), p. 328. In traditional predicate calculus, this is a sole axiom for identity from which the usual ones can be derived. (Contributed by NM, 9-May-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ ([𝑦 / 𝑧]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑧]𝜑)) | ||
Theorem | sbid2v 2445* | An identity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.) |
⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑) | ||
Theorem | sbelx 2446* | Elimination of substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑)) | ||
Theorem | sbel2x 2447* | Elimination of double substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 29-Sep-2018.) |
⊢ (𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑)) | ||
Theorem | sbal1 2448* | A theorem used in elimination of disjoint variable restriction on 𝑥 and 𝑦 by replacing it with a distinctor ¬ ∀𝑥𝑥 = 𝑧. (Contributed by NM, 15-May-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2018.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)) | ||
Theorem | sbal2 2449* | Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.) Remove a distinct variable constraint. (Revised by Wolf Lammen, 3-Oct-2018.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)) | ||
Theorem | sbal 2450* | Move universal quantifier in and out of substitution. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 29-Sep-2018.) |
⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑) | ||
Theorem | sbex 2451* | Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.) |
⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑) | ||
Theorem | sbalv 2452* | Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.) |
⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) ⇒ ⊢ ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓) | ||
Theorem | sbco4lem 2453* | Lemma for sbco4 2454. It replaces the temporary variable 𝑣 with another temporary variable 𝑤. (Contributed by Jim Kingdon, 26-Sep-2018.) |
⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑) | ||
Theorem | sbco4 2454* | Two ways of exchanging two variables. Both sides of the biconditional exchange 𝑥 and 𝑦, either via two temporary variables 𝑢 and 𝑣, or a single temporary 𝑤. (Contributed by Jim Kingdon, 25-Sep-2018.) |
⊢ ([𝑦 / 𝑢][𝑥 / 𝑣][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑) | ||
Theorem | 2sb8e 2455* | An equivalent expression for double existence. (Contributed by Wolf Lammen, 2-Nov-2019.) |
⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑) | ||
Theorem | exsb 2456* | An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.) |
⊢ (∃𝑥𝜑 ↔ ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | 2exsb 2457* | An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Wolf Lammen, 30-Sep-2018.) |
⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑)) | ||
Syntax | weu 2458 | Extend wff definition to include existential uniqueness ("there exists a unique 𝑥 such that 𝜑"). |
wff ∃!𝑥𝜑 | ||
Syntax | wmo 2459 | Extend wff definition to include uniqueness ("there exists at most one 𝑥 such that 𝜑"). |
wff ∃*𝑥𝜑 | ||
Theorem | eujust 2460* | A soundness justification theorem for df-eu 2462, showing that the definition is equivalent to itself with its dummy variable renamed. Note that 𝑦 and 𝑧 needn't be distinct variables. See eujustALT 2461 for a proof that provides an example of how it can be achieved through the use of dvelim 2325. (Contributed by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | ||
Theorem | eujustALT 2461* | Alternate proof of eujust 2460 illustrating the use of dvelim 2325. (Contributed by NM, 11-Mar-2010.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | ||
Definition | df-eu 2462* | Define existential uniqueness, i.e. "there exists exactly one 𝑥 such that 𝜑." Definition 10.1 of [BellMachover] p. 97; also Definition *14.02 of [WhiteheadRussell] p. 175. Other possible definitions are given by eu1 2498, eu2 2497, eu3v 2486, and eu5 2484 (which in some cases we show with a hypothesis 𝜑 → ∀𝑦𝜑 in place of a distinct variable condition on 𝑦 and 𝜑). Double uniqueness is tricky: ∃!𝑥∃!𝑦𝜑 does not mean "exactly one 𝑥 and one 𝑦 " (see 2eu4 2544). (Contributed by NM, 12-Aug-1993.) |
⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | ||
Definition | df-mo 2463 | Define "there exists at most one 𝑥 such that 𝜑." Here we define it in terms of existential uniqueness. Notation of [BellMachover] p. 460, whose definition we show as mo3 2495. For other possible definitions see mo2 2467 and mo4 2505. (Contributed by NM, 8-Mar-1995.) |
⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | ||
Theorem | euequ1 2464* | Equality has existential uniqueness. Special case of eueq1 3346 proved using only predicate calculus. The proof needs 𝑦 = 𝑧 be free of 𝑥. This is ensured by having 𝑥 and 𝑦 be distinct. Alternatively, a distinctor ¬ ∀𝑥𝑥 = 𝑦 could have been used instead. (Contributed by Stefan Allan, 4-Dec-2008.) (Proof shortened by Wolf Lammen, 8-Sep-2019.) |
⊢ ∃!𝑥 𝑥 = 𝑦 | ||
Theorem | mo2v 2465* | Alternate definition of "at most one." Unlike mo2 2467, which is slightly more general, it does not depend on ax-11 2021 and ax-13 2234, whence it is preferable within predicate logic. Elsewhere, most theorems depend on these axioms anyway, so this advantage is no longer important. (Contributed by Wolf Lammen, 27-May-2019.) (Proof shortened by Wolf Lammen, 10-Nov-2019.) |
⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | ||
Theorem | euf 2466* | A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Oct-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | ||
Theorem | mo2 2467* | Alternate definition of "at most one." (Contributed by NM, 8-Mar-1995.) Restrict dummy variable z. (Revised by Wolf Lammen, 28-May-2019.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | ||
Theorem | nfeu1 2468 | Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎ𝑥∃!𝑥𝜑 | ||
Theorem | nfmo1 2469 | Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎ𝑥∃*𝑥𝜑 | ||
Theorem | nfeud2 2470 | Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.) (Proof shortened by Wolf Lammen, 4-Oct-2018.) |
⊢ Ⅎ𝑦𝜑 & ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) | ||
Theorem | nfmod2 2471 | Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ Ⅎ𝑦𝜑 & ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓) | ||
Theorem | nfeud 2472 | Deduction version of nfeu 2474. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) | ||
Theorem | nfmod 2473 | Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓) | ||
Theorem | nfeu 2474 | Bound-variable hypothesis builder for uniqueness. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃!𝑦𝜑 | ||
Theorem | nfmo 2475 | Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃*𝑦𝜑 | ||
Theorem | eubid 2476 | Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒)) | ||
Theorem | mobid 2477 | Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) | ||
Theorem | eubidv 2478* | Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒)) | ||
Theorem | mobidv 2479* | Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by Mario Carneiro, 7-Oct-2016.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) | ||
Theorem | eubii 2480 | Introduce uniqueness quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥𝜓) | ||
Theorem | mobii 2481 | Formula-building rule for "at most one" quantifier (inference rule). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.) |
⊢ (𝜓 ↔ 𝜒) ⇒ ⊢ (∃*𝑥𝜓 ↔ ∃*𝑥𝜒) | ||
Theorem | euex 2482 | Existential uniqueness implies existence. For a shorter proof using more axioms, see euexALT 2499. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2018.) |
⊢ (∃!𝑥𝜑 → ∃𝑥𝜑) | ||
Theorem | exmo 2483 | Something exists or at most one exists. (Contributed by NM, 8-Mar-1995.) |
⊢ (∃𝑥𝜑 ∨ ∃*𝑥𝜑) | ||
Theorem | eu5 2484 | Uniqueness in terms of "at most one." Revised to reduce dependencies on axioms. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 25-May-2019.) |
⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑)) | ||
Theorem | exmoeu2 2485 | Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.) |
⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑)) | ||
Theorem | eu3v 2486* | An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) Add a distinct variable condition on 𝜑. (Revised by Wolf Lammen, 29-May-2019.) |
⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) | ||
Theorem | eumo 2487 | Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.) |
⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑) | ||
Theorem | eumoi 2488 | "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.) |
⊢ ∃!𝑥𝜑 ⇒ ⊢ ∃*𝑥𝜑 | ||
Theorem | moabs 2489 | Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.) |
⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑)) | ||
Theorem | exmoeu 2490 | Existence in terms of "at most one" and uniqueness. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Wolf Lammen, 5-Dec-2018.) |
⊢ (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑)) | ||
Theorem | sb8eu 2491 | Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Aug-2019.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | sb8mo 2492 | Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | cbveu 2493 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) | ||
Theorem | cbvmo 2494 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) | ||
Theorem | mo3 2495* | Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that 𝑦 not occur in 𝜑 in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) (Proof shortened by Wolf Lammen, 18-Aug-2019.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) | ||
Theorem | mo 2496* | Equivalent definitions of "there exists at most one." (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Dec-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) | ||
Theorem | eu2 2497* | An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.) (Proof shortened by Wolf Lammen, 2-Dec-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))) | ||
Theorem | eu1 2498* | An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 29-Oct-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦))) | ||
Theorem | euexALT 2499 | Alternate proof of euex 2482. Shorter but uses more axioms. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∃!𝑥𝜑 → ∃𝑥𝜑) | ||
Theorem | euor 2500 | Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∨ 𝜓)) |
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