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Theorem List for Intuitionistic Logic Explorer - 1901-2000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem2sb5 1901* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ 𝜑))
 
Theorem2sb6 1902* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
 
Theoremsbcom2v 1903* Lemma for proving sbcom2 1905. It is the same as sbcom2 1905 but with additional distinct variable constraints on 𝑥 and 𝑦, and on 𝑤 and 𝑧. (Contributed by Jim Kingdon, 19-Feb-2018.)
([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
 
Theoremsbcom2v2 1904* Lemma for proving sbcom2 1905. It is the same as sbcom2v 1903 but removes the distinct variable constraint on 𝑥 and 𝑦. (Contributed by Jim Kingdon, 19-Feb-2018.)
([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
 
Theoremsbcom2 1905* 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 modified to be intuitionistic by Jim Kingdon, 19-Feb-2018.)
([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
 
Theoremsb6a 1906* Equivalence for substitution. (Contributed by NM, 5-Aug-1993.)
([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑))
 
Theorem2sb5rf 1907* Reversed double substitution. (Contributed by NM, 3-Feb-2005.)
(𝜑 → ∀𝑧𝜑)    &   (𝜑 → ∀𝑤𝜑)       (𝜑 ↔ ∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
 
Theorem2sb6rf 1908* Reversed double substitution. (Contributed by NM, 3-Feb-2005.)
(𝜑 → ∀𝑧𝜑)    &   (𝜑 → ∀𝑤𝜑)       (𝜑 ↔ ∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
 
Theoremdfsb7 1909* An alternate definition of proper substitution df-sb 1687. 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 1809, first 𝑧 for 𝑥 then 𝑦 for 𝑧. Compare Definition 2.1'' of [Quine] p. 17. Theorem sb7f 1910 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 1910* This version of dfsb7 1909 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-17 1460 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1687 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)))
 
Theoremsb7af 1911* An alternate definition of proper substitution df-sb 1687. Similar to dfsb7a 1912 but does not require that 𝜑 and 𝑧 be distinct. Similar to sb7f 1910 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 1912* An alternate definition of proper substitution df-sb 1687. Similar to dfsb7 1909 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 1911. (Contributed by Jim Kingdon, 5-Feb-2018.)
([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧𝜑)))
 
Theoremsb10f 1913* 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 1914* 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 1915* Elimination of substitution. (Contributed by NM, 5-Aug-1993.)
(𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑))
 
Theoremsbel2x 1916* Elimination of double substitution. (Contributed by NM, 5-Aug-1993.)
(𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑))
 
Theoremsbalyz 1917* Move universal quantifier in and out of substitution. Identical to sbal 1918 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 1918* 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 1919* Lemma for proving sbal1 1920. Same as sbal1 1920 but with an additional distinct variable constraint on 𝑦 and 𝑧. (Contributed by Jim Kingdon, 23-Feb-2018.)
(¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
 
Theoremsbal1 1920* A theorem used in elimination of disjoint variable restriction on 𝑥 and 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 1921* Move existential quantifier in and out of substitution. Identical to sbex 1922 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)
 
Theoremsbex 1922* 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 1923* Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)
([𝑦 / 𝑥]𝜑𝜓)       ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓)
 
Theoremsbco4lem 1924* Lemma for sbco4 1925. 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 1925* 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 1926* An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.)
(∃𝑥𝜑 ↔ ∃𝑦𝑥(𝑥 = 𝑦𝜑))
 
Theorem2exsb 1927* An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.)
(∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
 
TheoremdvelimALT 1928* Version of dvelim 1935 that doesn't use ax-10 1437. Because it has different distinct variable constraints than dvelim 1935 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 1929* Like dvelimf 1933 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 1930 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 1931 A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1930). (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 1932 A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1930). (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 1933 Version of dvelim 1935 without any variable restrictions. (Contributed by NM, 1-Oct-2002.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑧𝜓)    &   (𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
 
Theoremdvelimdf 1934 Deduction form of dvelimf 1933. This version may be useful if we want to avoid ax-17 1460 and use ax-16 1736 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 1935* 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 1934.

Other variants of this theorem are dvelimf 1933 (with no distinct variable restrictions) and dvelimALT 1928 (that avoids ax-10 1437). (Contributed by NM, 23-Nov-1994.)

(𝜑 → ∀𝑥𝜑)    &   (𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
 
Theoremdvelimor 1936* 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 1937* 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))
 
Theoremdveel1 1938* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦𝑧 → ∀𝑥 𝑦𝑧))
 
Theoremdveel2 1939* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑧𝑦 → ∀𝑥 𝑧𝑦))
 
Theoremsbal2 1940* Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
 
Theoremnfsb4or 1941 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)
 
1.4.6  Existential uniqueness
 
Syntaxweu 1942 Extend wff definition to include existential uniqueness ("there exists a unique 𝑥 such that 𝜑").
wff ∃!𝑥𝜑
 
Syntaxwmo 1943 Extend wff definition to include uniqueness ("there exists at most one 𝑥 such that 𝜑").
wff ∃*𝑥𝜑
 
Theoremeujust 1944* A soundness justification theorem for df-eu 1945, 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 1945* 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 1967, eu2 1986, eu3 1988, and eu5 1989 (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 2035). (Contributed by NM, 5-Aug-1993.)
(∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
 
Definitiondf-mo 1946 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 1996. For another possible definition see mo4 2003. (Contributed by NM, 5-Aug-1993.)
(∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
 
Theoremeuf 1947* 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 1948 Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
 
Theoremeubid 1949 Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
 
Theoremeubidv 1950* Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
 
Theoremeubii 1951 Introduce uniqueness quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)
(𝜑𝜓)       (∃!𝑥𝜑 ↔ ∃!𝑥𝜓)
 
Theoremhbeu1 1952 Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.)
(∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑)
 
Theoremnfeu1 1953 Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥∃!𝑥𝜑
 
Theoremnfmo1 1954 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥∃*𝑥𝜑
 
Theoremsb8eu 1955 Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑦𝜑       (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
 
Theoremsb8mo 1956 Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)
𝑦𝜑       (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)
 
Theoremnfeudv 1957* Deduction version of nfeu 1961. Similar to nfeud 1958 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 1958 Deduction version of nfeu 1961. (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 1959 Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃*𝑦𝜓)
 
Theoremnfeuv 1960* Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 1961 but has the additional constraint that 𝑥 and 𝑦 must be distinct. (Contributed by Jim Kingdon, 23-May-2018.)
𝑥𝜑       𝑥∃!𝑦𝜑
 
Theoremnfeu 1961 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 1962 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)
𝑥𝜑       𝑥∃*𝑦𝜑
 
Theoremhbeu 1963 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 1964 Deduction version of hbeu 1963. (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 1965 Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Andrew Salmon, 9-Jul-2011.)
(𝜑 → ∀𝑦𝜑)       (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
 
Theoremcbveu 1966 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 1967* 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 1968 Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.)
(𝜑 → ∀𝑥𝜑)       ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))
 
Theoremeuorv 1969* Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)
((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))
 
Theoremmo2n 1970* There is at most one of something which does not exist. (Contributed by Jim Kingdon, 2-Jul-2018.)
𝑦𝜑       (¬ ∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
 
Theoremmon 1971 There is at most one of something which does not exist. (Contributed by Jim Kingdon, 5-Jul-2018.)
(¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
 
Theoremeuex 1972 Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(∃!𝑥𝜑 → ∃𝑥𝜑)
 
Theoremeumo0 1973* Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.)
(𝜑 → ∀𝑦𝜑)       (∃!𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
 
Theoremeumo 1974 Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.)
(∃!𝑥𝜑 → ∃*𝑥𝜑)
 
Theoremeumoi 1975 "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)
∃!𝑥𝜑       ∃*𝑥𝜑
 
Theoremmobidh 1976 Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
 
Theoremmobid 1977 Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
 
Theoremmobidv 1978* Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
 
Theoremmobii 1979 Formula-building rule for "at most one" quantifier (inference rule). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.)
(𝜓𝜒)       (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)
 
Theoremhbmo1 1980 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.)
(∃*𝑥𝜑 → ∀𝑥∃*𝑥𝜑)
 
Theoremhbmo 1981 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)
(𝜑 → ∀𝑥𝜑)       (∃*𝑦𝜑 → ∀𝑥∃*𝑦𝜑)
 
Theoremcbvmo 1982 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 1983* 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 1984* Converse of mo23 1983 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 1985* 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 1986* 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 1987* 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 1988* An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.)
𝑦𝜑       (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
 
Theoremeu5 1989 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 1990 Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.)
(∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑))
 
Theoremmoabs 1991 Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.)
(∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
 
Theoremexmodc 1992 If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.)
(DECID𝑥𝜑 → (∃𝑥𝜑 ∨ ∃*𝑥𝜑))
 
Theoremexmonim 1993 There is at most one of something which does not exist. Unlike exmodc 1992 there is no decidability condition. (Contributed by Jim Kingdon, 22-Sep-2018.)
(¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
 
Theoremmo2r 1994* A condition which implies "at most one." (Contributed by Jim Kingdon, 2-Jul-2018.)
𝑦𝜑       (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑)
 
Theoremmo3h 1995* 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 1996* 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 1997* Alternate definition of "at most one" where existence is decidable. (Contributed by Jim Kingdon, 2-Jul-2018.)
𝑦𝜑       (DECID𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
 
Theoremeuan 1998 Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(𝜑 → ∀𝑥𝜑)       (∃!𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))
 
Theoremeuanv 1999* Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)
(∃!𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))
 
Theoremeuor2 2000 Introduce or eliminate a disjunct in a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(¬ ∃𝑥𝜑 → (∃!𝑥(𝜑𝜓) ↔ ∃!𝑥𝜓))
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