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Theorem List for Metamath Proof Explorer - 2501-2600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsb8 2501 Substitution of variable in universal quantifier. For a version requiring disjoint variables, but fewer axioms, see sb8v 2321. (Contributed by NM, 16-May-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
𝑦𝜑       (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Theoremsb8e 2502 Substitution of variable in existential quantifier. For a version requiring disjoint variables, but fewer axioms, see sb8ev 2322. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
𝑦𝜑       (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

Theoremsb9 2503 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) Allow a shortening of sb9i 2504. (Revised by Wolf Lammen, 15-Jun-2019.)
(∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Theoremsb9i 2504 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)

Theoremax12vALT 2505* Alternate proof of ax12v2 2165, 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 = 𝑦𝜑)))

Theoremsb6OLD 2506* Obsolete proof of sb6 2250 as of 28-Jul-2022. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theoremhbs1OLD 2507* Obsolete version of hbs1 2255 as of 28-Jul-2022. (Contributed by NM, 26-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Theoremnfs1vOLD 2508* Obsolete version of nfs1v 2254 as of 28-Jul-2022. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥[𝑦 / 𝑥]𝜑

Theoremequsb3vOLD 2509* Obsolete version of equsb3v 2291 as of 19-Jan-2023. (Contributed by Raph Levien and FL, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)

Theoremequsb3 2510* Substitution in an equality. For a version requiring disjoint variables, but fewer axioms, see equsb3v 2291. (Contributed by Raph Levien and FL, 4-Dec-2005.) Remove dependency on ax-11 2150. (Revised by Wolf Lammen, 21-Sep-2018.)
([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)

Theoremequsb3ALT 2511* Alternate proof of equsb3 2510, shorter but requiring ax-11 2150. (Contributed by Raph Levien and FL, 4-Dec-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)

Theoremelsb3v 2512* Specialization of elsb3 2513 with an extra distinct variable condition, but no dependency on ax-13 2334. Adapted from equsb3v 2291. (Contributed by Wolf Lammen, 27-Jul-2022.)
([𝑥 / 𝑦]𝑦𝑧𝑥𝑧)

Theoremelsb3 2513* Substitution applied to an atomic membership wff. For a version requiring more disjoint variables, but fewer axioms, see elsb3v 2512. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) Remove dependency on ax-11 2150. (Revised by Wolf Lammen, 27-Jul-2022.)
([𝑥 / 𝑦]𝑦𝑧𝑥𝑧)

Theoremelsb3OLD 2514* Obsolete version of elsb3 2513 as of 27-Jul-2022. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑥 / 𝑦]𝑦𝑧𝑥𝑧)

Theoremelsb4v 2515* Specialization of elsb4 2516 with an extra distinct variable condition, but no dependency on ax-13 2334. Adapted from equsb3v 2291. (Contributed by Wolf Lammen, 28-Jul-2022.)
([𝑥 / 𝑦]𝑧𝑦𝑧𝑥)

Theoremelsb4 2516* Substitution applied to an atomic membership wff. For a version requiring more disjoint variables, but fewer axioms, see elsb4v 2515. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) Remove dependency on ax-11 2150. (Revised by Wolf Lammen, 28-Jul-2022.)
([𝑥 / 𝑦]𝑧𝑦𝑧𝑥)

Theoremelsb4OLD 2517* Obsolete version of elsb3 2513 as of 27-Jul-2022. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑥 / 𝑦]𝑧𝑦𝑧𝑥)

Theoremsbhb 2518* Two ways of expressing "𝑥 is (effectively) not free in 𝜑". (Contributed by NM, 29-May-2009.)
((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))

Theoremsbnf2OLD 2519* Obsolete version of sbnf2 2326 as of 30-Jan-2023. (Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(Ⅎ𝑥𝜑 ↔ ∀𝑦𝑧([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))

Theoremnfsb 2520* If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. For a version requiring more disjoint variables, but fewer axioms, see nfsbv 2306. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑧𝜑       𝑧[𝑦 / 𝑥]𝜑

Theoremhbsb 2521* 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)

Theoremnfsbd 2522* Deduction version of nfsb 2520. (Contributed by NM, 15-Feb-2013.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑧𝜓)       (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)

Theoremsbcom2 2523* 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, 23-Dec-2022.)
([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)

Theoremsbcom2OLD 2524* Obsolete version of sbcom2 2523 as of 23-Dec-2022. (Contributed by NM, 27-May-1997.) (Proof shortened by Wolf Lammen, 24-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)

Theoremsbcom4 2525* Commutativity law for substitution. This theorem was incorrectly used as our previous version of pm11.07 2526 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)

Theorempm11.07 2526 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 2525 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

Theoremsb6a 2527* 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))

Theorem2ax6elem 2528 We can always find values matching 𝑥 and 𝑦, as long as they are represented by distinct variables. This theorem merges two ax6e 2347 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 39728. (Contributed by Wolf Lammen, 27-Sep-2018.)
(¬ ∀𝑤 𝑤 = 𝑧 → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦))

Theorem2ax6e 2529* We can always find values matching 𝑥 and 𝑦, as long as they are represented by distinct variables. Version of 2ax6elem 2528 with a distinct variable constraint. (Contributed by Wolf Lammen, 28-Sep-2018.)
𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦)

Theorem2sb5rf 2530* 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.)
𝑧𝜑    &   𝑤𝜑       (𝜑 ↔ ∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))

Theorem2sb6rf 2531* 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.) (Proof shortened by Wolf Lammen, 13-Apr-2023.)
𝑧𝜑    &   𝑤𝜑       (𝜑 ↔ ∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))

Theorem2sb6rfOLD 2532* Obsolete version of 2sb6rf 2531 as of 13-Apr-2023. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) Remove variable constraints. (Revised by Wolf Lammen, 28-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑧𝜑    &   𝑤𝜑       (𝜑 ↔ ∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))

Theoremsb7f 2533* This version of dfsb7 2535 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 1953 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 2012 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.)
𝑧𝜑       ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧𝜑)))

Theoremsb7h 2534* This version of dfsb7 2535 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 1953 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 2012 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 = 𝑧𝜑)))

Theoremdfsb7 2535* An alternate definition of proper substitution df-sb 2012. 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 2251, 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 2764. Theorem sb7h 2534 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 = 𝑧𝜑)))

Theoremsb10f 2536* 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.)
𝑥𝜑       ([𝑦 / 𝑧]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑧]𝜑))

Theoremsbelx 2537* Elimination of substitution. (Contributed by NM, 5-Aug-1993.)
(𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑))

Theoremsbel2x 2538* Elimination of double substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 29-Sep-2018.)
(𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑))

Theoremsbal1 2539* 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))

Theoremsbal2 2540* Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.) Remove a distinct variable constraint. (Revised by Wolf Lammen, 24-Dec-2022.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))

Theoremsbal2OLD 2541* Obsolete version of sbal2 2540 as of 24-Dec-2022. (Contributed by NM, 2-Jan-2002.) Remove a distinct variable constraint. (Revised by Wolf Lammen, 3-Oct-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))

Theoremsbal 2542* 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)

Theoremsbex 2543* Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.)
([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)

Theoremsbalv 2544* Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)
([𝑦 / 𝑥]𝜑𝜓)       ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓)

Theoremsbco4lem 2545* Lemma for sbco4 2546. 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)

Theoremsbco4 2546* 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)

Theorem2sb8e 2547* An equivalent expression for double existence. For a version requiring more disjoint variables, but fewer axioms, see 2sb8ev 2323. (Contributed by Wolf Lammen, 2-Nov-2019.)
(∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)

TheoremexsbOLD 2548* Obsolete version of exsb 2327 as of 16-Oct-2022. (Contributed by NM, 2-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥𝜑 ↔ ∃𝑦𝑥(𝑥 = 𝑦𝜑))

1.6  Uniqueness and unique existence

1.6.1  Uniqueness: the at-most-one quantifier

Syntaxwmo 2549 Extend wff definition to include the at-most-one quantifier ("there exists at most one 𝑥 such that 𝜑").
wff ∃*𝑥𝜑

Theoremmojust 2550* Soundness justification theorem for df-mo 2551 (note that 𝑦 and 𝑧 need not be disjoint, although the weaker theorem with that disjoint variable condition added would be enough to justify the soundness of the definition). (Contributed by NM, 11-Mar-2010.) Added this theorem by adapting the proof of eujust 2589. (Revised by BJ, 30-Sep-2022.)
(∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))

Definitiondf-mo 2551* Define the at-most-one quantifier. The expression ∃*𝑥𝜑 is read "there exists at most one 𝑥 such that 𝜑". This is also called the "uniqueness quantifier" but that expression is also used for the unique existential quantifier df-eu 2587, therefore we avoid that ambiguous name.

Notation of [BellMachover] p. 460, whose definition we show as mo3 2580. For other possible definitions see moeu 2603 and mo4 2585. (Contributed by Wolf Lammen, 27-May-2019.) Make this the definition (which used to be moeu 2603, while this definition was then proved as dfmo 2615). (Revised by BJ, 30-Sep-2022.)

(∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))

Theoremnexmo 2552 Nonexistence implies uniqueness. (Contributed by BJ, 30-Sep-2022.) Avoid ax-11 2150. (Revised by Wolf Lammen, 16-Oct-2022.)
(¬ ∃𝑥𝜑 → ∃*𝑥𝜑)

TheoremnexmoOLD 2553 Obsolete version of nexmo 2552 as of 16-Oct-2022. (Contributed by BJ, 30-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∃𝑥𝜑 → ∃*𝑥𝜑)

Theoremexmo 2554 Any proposition holds for some 𝑥 or holds for at most one 𝑥. (Contributed by NM, 8-Mar-1995.) Shorten proof and avoid df-eu 2587. (Revised by BJ, 14-Oct-2022.)
(∃𝑥𝜑 ∨ ∃*𝑥𝜑)

Theoremmoabs 2555 Absorption of existence condition by uniqueness. (Contributed by NM, 4-Nov-2002.) Shorten proof and avoid df-eu 2587. (Revised by BJ, 14-Oct-2022.)
(∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))

Theoremmoim 2556 The at-most-one quantifier reverses implication. (Contributed by NM, 22-Apr-1995.)
(∀𝑥(𝜑𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))

Theoremmoimi 2557 The at-most-one quantifier reverses implication. (Contributed by NM, 15-Feb-2006.) Remove use of ax-5 1953. (Revised by Steven Nguyen, 9-May-2023.)
(𝜑𝜓)       (∃*𝑥𝜓 → ∃*𝑥𝜑)

TheoremmoimiOLD 2558 Obsolete version of moimi 2557 as of 9-May-2023. The at-most-one quantifier reverses implication. (Contributed by NM, 15-Feb-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       (∃*𝑥𝜓 → ∃*𝑥𝜑)

Theoremmoimdv 2559* The at-most-one quantifier reverses implication, deduction form. (Contributed by Thierry Arnoux, 25-Feb-2017.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝜒 → ∃*𝑥𝜓))

Theoremmobi 2560 Equivalence theorem for the at-most-one quantifier. (Contributed by BJ, 7-Oct-2022.) (Proof shortened by Wolf Lammen, 18-Feb-2023.)
(∀𝑥(𝜑𝜓) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓))

TheoremmobiOLD 2561 Obsolete version of mobi 2560 as of 18-Feb-2023. (Contributed by BJ, 7-Oct-2022.) (Proof shortened by Wolf Lammen, 16-Oct-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥(𝜑𝜓) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓))

TheoremmobiOLDOLD 2562 Obsolete proof of mobi 2560 as of 15-Oct-2022. (Contributed by BJ, 7-Oct-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥(𝜑𝜓) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓))

Theoremmobii 2563 Formula-building rule for the at-most-one quantifier (inference form). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.)
(𝜓𝜒)       (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)

Theoremmobidv 2564* Formula-building rule for the at-most-one quantifier (deduction form). (Contributed by Mario Carneiro, 7-Oct-2016.) Reduce axiom dependencies and shorten proof. (Revised by BJ, 7-Oct-2022.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

Theoremmobid 2565 Formula-building rule for the at-most-one quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) Remove dependency on ax-10 2135, ax-11 2150, ax-13 2334. (Revised by BJ, 14-Oct-2022.) (Proof shortened by Wolf Lammen, 18-Feb-2023.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

TheoremmobidOLD 2566 Obsolete version of mobid 2565 as of 18-Feb-2023. (Contributed by NM, 8-Mar-1995.) Remove dependency on ax-10 2135, ax-11 2150, ax-13 2334. (Revised by BJ, 14-Oct-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

Theoremmoa1 2567 If an implication holds for at most one value, then its consequent holds for at most one value. See also ala1 1857 and exa1 1881. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Wolf Lammen, 22-Dec-2018.) (Revised by BJ, 29-Mar-2021.)
(∃*𝑥(𝜑𝜓) → ∃*𝑥𝜓)

Theoremmoan 2568 "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.)
(∃*𝑥𝜑 → ∃*𝑥(𝜓𝜑))

Theoremmoani 2569 "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.)
∃*𝑥𝜑       ∃*𝑥(𝜓𝜑)

Theoremmoor 2570 "At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.)
(∃*𝑥(𝜑𝜓) → ∃*𝑥𝜑)

Theoremmooran1 2571 "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
((∃*𝑥𝜑 ∨ ∃*𝑥𝜓) → ∃*𝑥(𝜑𝜓))

Theoremmooran2 2572 "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(∃*𝑥(𝜑𝜓) → (∃*𝑥𝜑 ∧ ∃*𝑥𝜓))

Theoremnfmo1 2573 Bound-variable hypothesis builder for the at-most-one quantifier. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) Adapt to new definition. (Revised by BJ, 1-Oct-2022.)
𝑥∃*𝑥𝜑

Theoremnfmod2 2574 Bound-variable hypothesis builder for the at-most-one quantifier. (Contributed by Mario Carneiro, 14-Nov-2016.) Avoid df-eu 2587. (Revised by BJ, 14-Oct-2022.)
𝑦𝜑    &   ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Theoremnfmodv 2575* Bound-variable hypothesis builder for the at-most-one quantifier. See nfmod 2576 for a version without disjoint variable conditions but requiring ax-13 2334. (Contributed by BJ, 28-Jan-2023.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Theoremnfmod 2576 Bound-variable hypothesis builder for the at-most-one quantifier. Deduction version of nfmo 2577. (Contributed by Mario Carneiro, 14-Nov-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Theoremnfmo 2577 Bound-variable hypothesis builder for the at-most-one quantifier. Note that 𝑥 and 𝑦 need not be disjoint. (Contributed by NM, 9-Mar-1995.)
𝑥𝜑       𝑥∃*𝑦𝜑

Theoremmof 2578* Version of df-mo 2551 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by NM, 8-Mar-1995.) Extract dfmo 2615 from this proof, and prove mof 2578 from it (as of 30-Sep-2022, directly from df-mo 2551). (Revised by Wolf Lammen, 28-May-2019.) Avoid ax-13 2334. (Revised by Wolf Lammen, 16-Oct-2022.)
𝑦𝜑       (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))

TheoremmofOLD 2579* Obsolete version of mof 2578 as of 16-Oct-2022. (Contributed by NM, 8-Mar-1995.) Extract dfmo 2615 from this proof, and prove mof 2578 from it (as of 30-Sep-2022, directly from df-mo 2551). (Revised by Wolf Lammen, 28-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦𝜑       (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))

Theoremmo3 2580* Alternate definition of the at-most-one quantifier. 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.) Remove dependency on ax-13 2334. (Revised by BJ and WL, 29-Jan-2023.)
𝑦𝜑       (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))

Theoremmo3OLD 2581* Obsolete version of mo3 2580 as of 29-Jan-2023. (Contributed by NM, 8-Mar-1995.) (Proof shortened by Wolf Lammen, 18-Aug-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦𝜑       (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))

Theoremmo 2582* 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.)
𝑦𝜑       (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))

Theoremmo4f 2583* At-most-one quantifier expressed using implicit substitution. Note that the disjoint variable condition on 𝑦, 𝜑 can be replaced by the nonfreeness hypothesis 𝑦𝜑 with essentially the same proof. (Contributed by NM, 10-Apr-2004.) Remove dependency on ax-13 2334. (Revised by Wolf Lammen, 19-Jan-2023.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))

Theoremmo4fOLD 2584* Obsolete version of mo4f 2583 as of 19-Jan-2023. (Contributed by NM, 10-Apr-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))

Theoremmo4 2585* At-most-one quantifier expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))

1.6.2  Unique existence: the unique existential quantifier

Syntaxweu 2586 Extend wff definition to include the unique existential quantifier ("there exists a unique 𝑥 such that 𝜑").
wff ∃!𝑥𝜑

Definitiondf-eu 2587 Define the existential uniqueness quantifier. This expresses unique existence, or existential uniqueness, which is the conjunction of existence (df-ex 1824) and uniqueness (df-mo 2551). The expression ∃!𝑥𝜑 is read "there exists exactly one 𝑥 such that 𝜑 " or "there exists a unique 𝑥 such that 𝜑". This is also called the "uniqueness quantifier" but that expression is also used for the at-most-one quantifier df-mo 2551, therefore we avoid that ambiguous name.

Definition 10.1 of [BellMachover] p. 97; also Definition *14.02 of [WhiteheadRussell] p. 175. Other possible definitions are given by eu1 2642, eu2 2641, eu3v 2588, and eu6 2592. As for double unique existence, beware that the expression ∃!𝑥∃!𝑦𝜑 means "there exists a unique 𝑥 such that there exists a unique 𝑦 such that 𝜑 " which is a weaker property than "there exists exactly one 𝑥 and one 𝑦 such that 𝜑 " (see 2eu4 2685). (Contributed by NM, 12-Aug-1993.) Make this the definition (which used to be eu6 2592, while this definition was then proved as dfeu 2614). (Revised by BJ, 30-Sep-2022.)

(∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))

Theoremeu3v 2588* An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) Replace a nonfreeness hypothesis with a disjoint variable condition on 𝜑, 𝑦 to reduce axiom usage. (Revised by Wolf Lammen, 29-May-2019.)
(∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))

Theoremeujust 2589* Soundness justification theorem for eu6 2592 when this was the definition of the unique existential quantifier (note that 𝑦 and 𝑧 need not be disjoint, although the weaker theorem with that disjoint variable condition added would be enough to justify the soundness of the definition). See eujustALT 2590 for a proof that provides an example of how it can be achieved through the use of dvelim 2417. (Contributed by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))

TheoremeujustALT 2590* Alternate proof of eujust 2589 illustrating the use of dvelim 2417. (Contributed by NM, 11-Mar-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))

Theoremeu6lem 2591* Proof lines shared by eu6 2592 and eu6im 2593. (Contributed by NM, 12-Aug-1993.) This used to be the definition of the unique existential quantifier, while df-eu 2587 was then proved as dfeu 2614. (Revised by BJ, 30-Sep-2022.) (Proof shortened by Wolf Lammen, 3-Jan-2023.) Extract common proof lines. (Revised by Wolf Lammen, 3-Mar-2023.)
(∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ (∃𝑦𝑥(𝑥 = 𝑦𝜑) ∧ ∃𝑧𝑥(𝜑𝑥 = 𝑧)))

Theoremeu6 2592* Alternate definition of the unique existential quantifier df-eu 2587 not using the at-most-one quantifier. (Contributed by NM, 12-Aug-1993.) This used to be the definition of the unique existential quantifier, while df-eu 2587 was then proved as dfeu 2614. (Revised by BJ, 30-Sep-2022.) (Proof shortened by Wolf Lammen, 3-Jan-2023.)
(∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))

Theoremeu6im 2593* One direction of eu6 2592 needs fewer axioms. (Contributed by Wolf Lammen, 2-Mar-2023.)
(∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃!𝑥𝜑)

Theoremeu6OLD 2594* Obsolete version of eu6 2592 as of 28-Dec-2022. (Contributed by NM, 12-Aug-1993.) This used to be the definition of the unique existential quantifier, while df-eu 2587 was then proved as dfeu 2614. (Revised by BJ, 30-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))

Theoremeuf 2595* Version of eu6 2592 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Oct-2018.) Avoid ax-13 2334. (Revised by Wolf Lammen, 16-Oct-2022.)
𝑦𝜑       (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))

TheoremeufOLD 2596* Obsolete version of euf 2595 as of 16-Oct-2022. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Oct-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦𝜑       (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))

Theoremeuex 2597 Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2018.) (Proof shortened by BJ, 7-Oct-2022.)
(∃!𝑥𝜑 → ∃𝑥𝜑)

Theoremeumo 2598 Existential uniqueness implies uniqueness. (Contributed by NM, 23-Mar-1995.)
(∃!𝑥𝜑 → ∃*𝑥𝜑)

Theoremeumoi 2599 Uniqueness inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)
∃!𝑥𝜑       ∃*𝑥𝜑

Theoremexmoeub 2600 Existence implies that uniqueness is equivalent to unique existence. (Contributed by NM, 5-Apr-2004.)
(∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑))

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