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Theorem List for Intuitionistic Logic Explorer - 1901-2000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsb9i 1901 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)
(∀𝑥[𝑥 / 𝑦]𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)

Theoremsbnf2 1902* 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.)
(Ⅎ𝑥𝜑 ↔ ∀𝑦𝑧([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))

Theoremhbsbd 1903* Deduction version of hbsb 1868. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → ∀𝑧𝜑)    &   (𝜑 → (𝜓 → ∀𝑧𝜓))       (𝜑 → ([𝑦 / 𝑥]𝜓 → ∀𝑧[𝑦 / 𝑥]𝜓))

Theorem2sb5 1904* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ 𝜑))

Theorem2sb6 1905* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))

Theoremsbcom2v 1906* Lemma for proving sbcom2 1908. It is the same as sbcom2 1908 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 1907* Lemma for proving sbcom2 1908. It is the same as sbcom2v 1906 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 1908* 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 1909* Equivalence for substitution. (Contributed by NM, 5-Aug-1993.)
([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑))

Theorem2sb5rf 1910* Reversed double substitution. (Contributed by NM, 3-Feb-2005.)
(𝜑 → ∀𝑧𝜑)    &   (𝜑 → ∀𝑤𝜑)       (𝜑 ↔ ∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))

Theorem2sb6rf 1911* Reversed double substitution. (Contributed by NM, 3-Feb-2005.)
(𝜑 → ∀𝑧𝜑)    &   (𝜑 → ∀𝑤𝜑)       (𝜑 ↔ ∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))

Theoremdfsb7 1912* An alternate definition of proper substitution df-sb 1690. 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 1812, first 𝑧 for 𝑥 then 𝑦 for 𝑧. Compare Definition 2.1'' of [Quine] p. 17. Theorem sb7f 1913 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 1913* This version of dfsb7 1912 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 1462 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1690 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 1914* An alternate definition of proper substitution df-sb 1690. Similar to dfsb7a 1915 but does not require that 𝜑 and 𝑧 be distinct. Similar to sb7f 1913 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 1915* An alternate definition of proper substitution df-sb 1690. Similar to dfsb7 1912 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 1914. (Contributed by Jim Kingdon, 5-Feb-2018.)
([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧𝜑)))

Theoremsb10f 1916* 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 1917* 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 1918* Elimination of substitution. (Contributed by NM, 5-Aug-1993.)
(𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑))

Theoremsbel2x 1919* Elimination of double substitution. (Contributed by NM, 5-Aug-1993.)
(𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑))

Theoremsbalyz 1920* Move universal quantifier in and out of substitution. Identical to sbal 1921 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 1921* 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 1922* Lemma for proving sbal1 1923. Same as sbal1 1923 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 1923* 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 1924* Move existential quantifier in and out of substitution. Identical to sbex 1925 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 1925* 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 1926* Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)
([𝑦 / 𝑥]𝜑𝜓)       ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓)

Theoremsbco4lem 1927* Lemma for sbco4 1928. 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 1928* 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 1929* An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.)
(∃𝑥𝜑 ↔ ∃𝑦𝑥(𝑥 = 𝑦𝜑))

Theorem2exsb 1930* An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.)
(∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))

TheoremdvelimALT 1931* Version of dvelim 1938 that doesn't use ax-10 1439. Because it has different distinct variable constraints than dvelim 1938 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 1932* Like dvelimf 1936 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 1933 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 1934 A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1933). (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 1935 A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1933). (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 1936 Version of dvelim 1938 without any variable restrictions. (Contributed by NM, 1-Oct-2002.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑧𝜓)    &   (𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Theoremdvelimdf 1937 Deduction form of dvelimf 1936. This version may be useful if we want to avoid ax-17 1462 and use ax-16 1739 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 1938* 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 1937.

Other variants of this theorem are dvelimf 1936 (with no distinct variable restrictions) and dvelimALT 1931 (that avoids ax-10 1439). (Contributed by NM, 23-Nov-1994.)

(𝜑 → ∀𝑥𝜑)    &   (𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Theoremdvelimor 1939* 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 1940* 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 1941* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦𝑧 → ∀𝑥 𝑦𝑧))

Theoremdveel2 1942* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑧𝑦 → ∀𝑥 𝑧𝑦))

Theoremsbal2 1943* Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))

Theoremnfsb4or 1944 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 1945 Extend wff definition to include existential uniqueness ("there exists a unique 𝑥 such that 𝜑").
wff ∃!𝑥𝜑

Syntaxwmo 1946 Extend wff definition to include uniqueness ("there exists at most one 𝑥 such that 𝜑").
wff ∃*𝑥𝜑

Theoremeujust 1947* A soundness justification theorem for df-eu 1948, 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 1948* 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 1970, eu2 1989, eu3 1991, and eu5 1992 (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 2038). (Contributed by NM, 5-Aug-1993.)
(∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))

Definitiondf-mo 1949 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 1999. For another possible definition see mo4 2006. (Contributed by NM, 5-Aug-1993.)
(∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))

Theoremeuf 1950* 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 1951 Formula-building rule for unique existential quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Theoremeubid 1952 Formula-building rule for unique existential quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Theoremeubidv 1953* Formula-building rule for unique existential quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Theoremeubii 1954 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 1955 Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.)
(∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑)

Theoremnfeu1 1956 Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥∃!𝑥𝜑

Theoremnfmo1 1957 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥∃*𝑥𝜑

Theoremsb8eu 1958 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 1959 Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)
𝑦𝜑       (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)

Theoremnfeudv 1960* Deduction version of nfeu 1964. Similar to nfeud 1961 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 1961 Deduction version of nfeu 1964. (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 1962 Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Theoremnfeuv 1963* Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 1964 but has the additional constraint that 𝑥 and 𝑦 must be distinct. (Contributed by Jim Kingdon, 23-May-2018.)
𝑥𝜑       𝑥∃!𝑦𝜑

Theoremnfeu 1964 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 1965 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)
𝑥𝜑       𝑥∃*𝑦𝜑

Theoremhbeu 1966 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 1967 Deduction version of hbeu 1966. (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 1968 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 1969 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 1970* 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 1971 Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 21-Oct-2005.)
(𝜑 → ∀𝑥𝜑)       ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))

Theoremeuorv 1972* Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 23-Mar-1995.)
((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))

Theoremmo2n 1973* There is at most one of something which does not exist. (Contributed by Jim Kingdon, 2-Jul-2018.)
𝑦𝜑       (¬ ∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))

Theoremmon 1974 There is at most one of something which does not exist. (Contributed by Jim Kingdon, 5-Jul-2018.)
(¬ ∃𝑥𝜑 → ∃*𝑥𝜑)

Theoremeuex 1975 Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(∃!𝑥𝜑 → ∃𝑥𝜑)

Theoremeumo0 1976* Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.)
(𝜑 → ∀𝑦𝜑)       (∃!𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))

Theoremeumo 1977 Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.)
(∃!𝑥𝜑 → ∃*𝑥𝜑)

Theoremeumoi 1978 "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)
∃!𝑥𝜑       ∃*𝑥𝜑

Theoremmobidh 1979 Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

Theoremmobid 1980 Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

Theoremmobidv 1981* Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

Theoremmobii 1982 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 1983 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.)
(∃*𝑥𝜑 → ∀𝑥∃*𝑥𝜑)

Theoremhbmo 1984 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)
(𝜑 → ∀𝑥𝜑)       (∃*𝑦𝜑 → ∀𝑥∃*𝑦𝜑)

Theoremcbvmo 1985 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 1986* 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 1987* Converse of mo23 1986 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 1988* 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 1989* 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 1990* 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 1991* An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.)
𝑦𝜑       (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))

Theoremeu5 1992 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 1993 Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.)
(∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑))

Theoremmoabs 1994 Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.)
(∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))

Theoremexmodc 1995 If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.)
(DECID𝑥𝜑 → (∃𝑥𝜑 ∨ ∃*𝑥𝜑))

Theoremexmonim 1996 There is at most one of something which does not exist. Unlike exmodc 1995 there is no decidability condition. (Contributed by Jim Kingdon, 22-Sep-2018.)
(¬ ∃𝑥𝜑 → ∃*𝑥𝜑)

Theoremmo2r 1997* A condition which implies "at most one." (Contributed by Jim Kingdon, 2-Jul-2018.)
𝑦𝜑       (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑)

Theoremmo3h 1998* 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 1999* 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 2000* Alternate definition of "at most one" where existence is decidable. (Contributed by Jim Kingdon, 2-Jul-2018.)
𝑦𝜑       (DECID𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))

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