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Theorem List for Intuitionistic Logic Explorer - 2001-2100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdfsb7 2001* An alternate definition of proper substitution df-sb 1773. 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 1897, first 𝑧 for 𝑥 then 𝑦 for 𝑧. Compare Definition 2.1'' of [Quine] p. 17. Theorem sb7f 2002 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)))
 
Theoremsb7f 2002* This version of dfsb7 2001 does not require that 𝜑 and 𝑧 be disjoint. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-17 1536, i.e., that does not have the concept of a variable not occurring in a formula. (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)))
 
Theoremsb7af 2003* An alternate definition of proper substitution df-sb 1773. Similar to dfsb7a 2004 but does not require that 𝜑 and 𝑧 be distinct. Similar to sb7f 2002 in that it involves a dummy variable 𝑧, but expressed in terms of rather than . (Contributed by Jim Kingdon, 5-Feb-2018.)
𝑧𝜑       ([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧𝜑)))
 
Theoremdfsb7a 2004* An alternate definition of proper substitution df-sb 1773. Similar to dfsb7 2001 in that it involves a dummy variable 𝑧, but expressed in terms of rather than . For a version which only requires 𝑧𝜑 rather than 𝑧 and 𝜑 being distinct, see sb7af 2003. (Contributed by Jim Kingdon, 5-Feb-2018.)
([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧𝜑)))
 
Theoremsb10f 2005* 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.)
(𝜑 → ∀𝑥𝜑)       ([𝑦 / 𝑧]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑧]𝜑))
 
Theoremsbid2v 2006* 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]𝜑𝜑)
 
Theoremsbelx 2007* Elimination of substitution. (Contributed by NM, 5-Aug-1993.)
(𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑))
 
Theoremsbel2x 2008* Elimination of double substitution. (Contributed by NM, 5-Aug-1993.)
(𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑))
 
Theoremsbalyz 2009* Move universal quantifier in and out of substitution. Identical to sbal 2010 except that it has an additional distinct variable constraint on 𝑦 and 𝑧. (Contributed by Jim Kingdon, 29-Dec-2017.)
([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
 
Theoremsbal 2010* Move universal quantifier in and out of substitution. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.)
([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
 
Theoremsbal1yz 2011* Lemma for proving sbal1 2012. Same as sbal1 2012 but with an additional disjoint variable condition on 𝑦, 𝑧. (Contributed by Jim Kingdon, 23-Feb-2018.)
(¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
 
Theoremsbal1 2012* A theorem used in elimination of disjoint variable conditions on 𝑥, 𝑦 by replacing it with a distinctor ¬ ∀𝑥𝑥 = 𝑧. (Contributed by NM, 5-Aug-1993.) (Proof rewitten by Jim Kingdon, 24-Feb-2018.)
(¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
 
Theoremsbexyz 2013* Move existential quantifier in and out of substitution. Identical to sbex 2014 except that it has an additional disjoint variable condition on 𝑦, 𝑧. (Contributed by Jim Kingdon, 29-Dec-2017.)
([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
 
Theoremsbex 2014* Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.)
([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
 
Theoremsbalv 2015* Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)
([𝑦 / 𝑥]𝜑𝜓)       ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓)
 
Theoremsbco4lem 2016* Lemma for sbco4 2017. 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 2017* 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)
 
Theoremexsb 2018* An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.)
(∃𝑥𝜑 ↔ ∃𝑦𝑥(𝑥 = 𝑦𝜑))
 
Theorem2exsb 2019* An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.)
(∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
 
TheoremdvelimALT 2020* Version of dvelim 2027 that doesn't use ax-10 1515. Because it has different distinct variable constraints than dvelim 2027 and is used in important proofs, it would be better if it had a name which does not end in ALT (ideally more close to set.mm naming). (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
 
Theoremdvelimfv 2021* Like dvelimf 2025 but with a distinct variable constraint on 𝑥 and 𝑧. (Contributed by Jim Kingdon, 6-Mar-2018.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑧𝜓)    &   (𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
 
Theoremhbsb4 2022 A variable not free remains so after substitution with a distinct variable. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)
(𝜑 → ∀𝑧𝜑)       (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑))
 
Theoremhbsb4t 2023 A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 2022). (Contributed by NM, 7-Apr-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(∀𝑥𝑧(𝜑 → ∀𝑧𝜑) → (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)))
 
Theoremnfsb4t 2024 A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 2022). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof rewritten by Jim Kingdon, 9-May-2018.)
(∀𝑥𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
 
Theoremdvelimf 2025 Version of dvelim 2027 without any variable restrictions. (Contributed by NM, 1-Oct-2002.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑧𝜓)    &   (𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
 
Theoremdvelimdf 2026 Deduction form of dvelimf 2025. This version may be useful if we want to avoid ax-17 1536 and use ax-16 1824 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)
𝑥𝜑    &   𝑧𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑧𝜒)    &   (𝜑 → (𝑧 = 𝑦 → (𝜓𝜒)))       (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒))
 
Theoremdvelim 2027* This theorem can be used to eliminate a distinct variable restriction on 𝑥 and 𝑧 and replace it with the "distinctor" ¬ ∀𝑥𝑥 = 𝑦 as an antecedent. 𝜑 normally has 𝑧 free and can be read 𝜑(𝑧), and 𝜓 substitutes 𝑦 for 𝑧 and can be read 𝜑(𝑦). We don't require that 𝑥 and 𝑦 be distinct: if they aren't, the distinctor will become false (in multiple-element domains of discourse) and "protect" the consequent.

To obtain a closed-theorem form of this inference, prefix the hypotheses with 𝑥𝑧, conjoin them, and apply dvelimdf 2026.

Other variants of this theorem are dvelimf 2025 (with no distinct variable restrictions) and dvelimALT 2020 (that avoids ax-10 1515). (Contributed by NM, 23-Nov-1994.)

(𝜑 → ∀𝑥𝜑)    &   (𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
 
Theoremdvelimor 2028* Disjunctive distinct variable constraint elimination. A user of this theorem starts with a formula 𝜑 (containing 𝑧) and a distinct variable constraint between 𝑥 and 𝑧. The theorem makes it possible to replace the distinct variable constraint with the disjunct 𝑥𝑥 = 𝑦 (𝜓 is just a version of 𝜑 with 𝑦 substituted for 𝑧). (Contributed by Jim Kingdon, 11-May-2018.)
𝑥𝜑    &   (𝑧 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥𝜓)
 
Theoremdveeq1 2029* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) (Proof rewritten by Jim Kingdon, 19-Feb-2018.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
 
Theoremsbal2 2030* Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
 
Theoremnfsb4or 2031 A variable not free remains so after substitution with a distinct variable. (Contributed by Jim Kingdon, 11-May-2018.)
𝑧𝜑       (∀𝑧 𝑧 = 𝑦 ∨ Ⅎ𝑧[𝑦 / 𝑥]𝜑)
 
Theoremnfd2 2032 Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Wolf Lammen, 16-Sep-2021.)
(𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)
 
Theoremhbe1a 2033 Dual statement of hbe1 1505. (Contributed by Wolf Lammen, 15-Sep-2021.)
(∃𝑥𝑥𝜑 → ∀𝑥𝜑)
 
Theoremnf5-1 2034 One direction of nf5 . (Contributed by Wolf Lammen, 16-Sep-2021.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)
 
Theoremnf5d 2035 Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)
 
1.4.6  Existential uniqueness
 
Syntaxweu 2036 Extend wff definition to include existential uniqueness ("there exists a unique 𝑥 such that 𝜑").
wff ∃!𝑥𝜑
 
Syntaxwmo 2037 Extend wff definition to include uniqueness ("there exists at most one 𝑥 such that 𝜑").
wff ∃*𝑥𝜑
 
Theoremeujust 2038* A soundness justification theorem for df-eu 2039, showing that the definition is equivalent to itself with its dummy variable renamed. Note that 𝑦 and 𝑧 needn't be distinct variables. (Contributed by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
 
Definitiondf-eu 2039* 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 2061, eu2 2080, eu3 2082, and eu5 2083 (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 2129). (Contributed by NM, 5-Aug-1993.)
(∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
 
Definitiondf-mo 2040 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 2090. For another possible definition see mo4 2097. (Contributed by NM, 5-Aug-1993.)
(∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
 
Theoremeuf 2041* A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.)
(𝜑 → ∀𝑦𝜑)       (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
 
Theoremeubidh 2042 Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
 
Theoremeubid 2043 Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
 
Theoremeubidv 2044* Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
 
Theoremeubii 2045 Introduce unique existential quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)
(𝜑𝜓)       (∃!𝑥𝜑 ↔ ∃!𝑥𝜓)
 
Theoremhbeu1 2046 Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.)
(∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑)
 
Theoremnfeu1 2047 Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥∃!𝑥𝜑
 
Theoremnfmo1 2048 Bound-variable hypothesis builder for "at most one". (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥∃*𝑥𝜑
 
Theoremsb8eu 2049 Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑦𝜑       (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
 
Theoremsb8mo 2050 Variable substitution for "at most one". (Contributed by Alexander van der Vekens, 17-Jun-2017.)
𝑦𝜑       (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)
 
Theoremnfeudv 2051* Deduction version of nfeu 2055. Similar to nfeud 2052 but has the additional constraint that 𝑥 and 𝑦 must be distinct. (Contributed by Jim Kingdon, 25-May-2018.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
 
Theoremnfeud 2052 Deduction version of nfeu 2055. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim Kingdon, 25-May-2018.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
 
Theoremnfmod 2053 Bound-variable hypothesis builder for "at most one". (Contributed by Mario Carneiro, 14-Nov-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃*𝑦𝜓)
 
Theoremnfeuv 2054* Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 2055 but has the additional condition that 𝑥 and 𝑦 must be distinct. (Contributed by Jim Kingdon, 23-May-2018.)
𝑥𝜑       𝑥∃!𝑦𝜑
 
Theoremnfeu 2055 Bound-variable hypothesis builder for existential uniqueness. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim Kingdon, 23-May-2018.)
𝑥𝜑       𝑥∃!𝑦𝜑
 
Theoremnfmo 2056 Bound-variable hypothesis builder for "at most one". (Contributed by NM, 9-Mar-1995.)
𝑥𝜑       𝑥∃*𝑦𝜑
 
Theoremhbeu 2057 Bound-variable hypothesis builder for uniqueness. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Proof rewritten by Jim Kingdon, 24-May-2018.)
(𝜑 → ∀𝑥𝜑)       (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑)
 
Theoremhbeud 2058 Deduction version of hbeu 2057. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 25-May-2018.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → ∀𝑦𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → (∃!𝑦𝜓 → ∀𝑥∃!𝑦𝜓))
 
Theoremsb8euh 2059 Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Andrew Salmon, 9-Jul-2011.)
(𝜑 → ∀𝑦𝜑)       (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
 
Theoremcbveu 2060 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
 
Theoremeu1 2061* An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.)
(𝜑 → ∀𝑦𝜑)       (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
 
Theoremeuor 2062 Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 21-Oct-2005.)
(𝜑 → ∀𝑥𝜑)       ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))
 
Theoremeuorv 2063* Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 23-Mar-1995.)
((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))
 
Theoremmo2n 2064* There is at most one of something which does not exist. (Contributed by Jim Kingdon, 2-Jul-2018.)
𝑦𝜑       (¬ ∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
 
Theoremmon 2065 There is at most one of something which does not exist. (Contributed by Jim Kingdon, 5-Jul-2018.)
(¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
 
Theoremeuex 2066 Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(∃!𝑥𝜑 → ∃𝑥𝜑)
 
Theoremeumo0 2067* Existential uniqueness implies "at most one". (Contributed by NM, 8-Jul-1994.)
(𝜑 → ∀𝑦𝜑)       (∃!𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
 
Theoremeumo 2068 Existential uniqueness implies "at most one". (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.)
(∃!𝑥𝜑 → ∃*𝑥𝜑)
 
Theoremeumoi 2069 "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)
∃!𝑥𝜑       ∃*𝑥𝜑
 
Theoremmobidh 2070 Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
 
Theoremmobid 2071 Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
 
Theoremmobidv 2072* Formula-building rule for "at most one" quantifier (deduction form). (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
 
Theoremmobii 2073 Formula-building rule for "at most one" quantifier (inference form). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.)
(𝜓𝜒)       (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)
 
Theoremhbmo1 2074 Bound-variable hypothesis builder for "at most one". (Contributed by NM, 8-Mar-1995.)
(∃*𝑥𝜑 → ∀𝑥∃*𝑥𝜑)
 
Theoremhbmo 2075 Bound-variable hypothesis builder for "at most one". (Contributed by NM, 9-Mar-1995.)
(𝜑 → ∀𝑥𝜑)       (∃*𝑦𝜑 → ∀𝑥∃*𝑦𝜑)
 
Theoremcbvmo 2076 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
 
Theoremmo23 2077* An implication between two definitions of "there exists at most one." (Contributed by Jim Kingdon, 25-Jun-2018.)
𝑦𝜑       (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
 
Theoremmor 2078* Converse of mo23 2077 with an additional 𝑥𝜑 condition. (Contributed by Jim Kingdon, 25-Jun-2018.)
𝑦𝜑       (∃𝑥𝜑 → (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
 
Theoremmodc 2079* Equivalent definitions of "there exists at most one," given decidable existence. (Contributed by Jim Kingdon, 1-Jul-2018.)
𝑦𝜑       (DECID𝑥𝜑 → (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
 
Theoremeu2 2080* An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.)
𝑦𝜑       (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
 
Theoremeu3h 2081* An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.)
(𝜑 → ∀𝑦𝜑)       (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
 
Theoremeu3 2082* An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.)
𝑦𝜑       (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
 
Theoremeu5 2083 Uniqueness in terms of "at most one". (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.)
(∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
 
Theoremexmoeu2 2084 Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.)
(∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑))
 
Theoremmoabs 2085 Absorption of existence condition by "at most one". (Contributed by NM, 4-Nov-2002.)
(∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
 
Theoremexmodc 2086 If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.)
(DECID𝑥𝜑 → (∃𝑥𝜑 ∨ ∃*𝑥𝜑))
 
Theoremexmonim 2087 There is at most one of something which does not exist. Unlike exmodc 2086 there is no decidability condition. (Contributed by Jim Kingdon, 22-Sep-2018.)
(¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
 
Theoremmo2r 2088* A condition which implies "at most one". (Contributed by Jim Kingdon, 2-Jul-2018.)
𝑦𝜑       (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑)
 
Theoremmo3h 2089* 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.) (New usage is discouraged.)
(𝜑 → ∀𝑦𝜑)       (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
 
Theoremmo3 2090* 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.)
𝑦𝜑       (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
 
Theoremmo2dc 2091* Alternate definition of "at most one" where existence is decidable. (Contributed by Jim Kingdon, 2-Jul-2018.)
𝑦𝜑       (DECID𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
 
Theoremeuan 2092 Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(𝜑 → ∀𝑥𝜑)       (∃!𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))
 
Theoremeuanv 2093* Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 23-Mar-1995.)
(∃!𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))
 
Theoremeuor2 2094 Introduce or eliminate a disjunct in a unique existential quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(¬ ∃𝑥𝜑 → (∃!𝑥(𝜑𝜓) ↔ ∃!𝑥𝜓))
 
Theoremsbmo 2095* Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)
([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑)
 
Theoremmo4f 2096* "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))
 
Theoremmo4 2097* "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))
 
Theoremeu4 2098* Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦)))
 
Theoremexmoeudc 2099 Existence in terms of "at most one" and uniqueness. (Contributed by Jim Kingdon, 3-Jul-2018.)
(DECID𝑥𝜑 → (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑)))
 
Theoremmoim 2100 "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.)
(∀𝑥(𝜑𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))
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