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Theorem List for Intuitionistic Logic Explorer - 1801-1900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcbvexdva 1801* Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
((φ x = y) → (ψχ))       (φ → (xψyχ))

Theoremcbvex4v 1802* Rule used to change bound variables, using implicit substitition. (Contributed by NM, 26-Jul-1995.)
((x = v y = u) → (φψ))    &   ((z = f w = g) → (ψχ))       (xyzwφvufgχ)

Theoremeean 1803 Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Revised by Mario Carneiro, 6-Oct-2016.)
yφ    &   xψ       (xy(φ ψ) ↔ (xφ yψ))

Theoremeeanv 1804* Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.)
(xy(φ ψ) ↔ (xφ yψ))

Theoremeeeanv 1805* Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(xyz(φ ψ χ) ↔ (xφ yψ zχ))

Theoremee4anv 1806* Rearrange existential quantifiers. (Contributed by NM, 31-Jul-1995.)
(xyzw(φ ψ) ↔ (xyφ zwψ))

Theoremee8anv 1807* Rearrange existential quantifiers. (Contributed by Jim Kingdon, 23-Nov-2019.)
(xyzwvu𝑡𝑠(φ ψ) ↔ (xyzwφ vu𝑡𝑠ψ))

Theoremnexdv 1808* Deduction for generalization rule for negated wff. (Contributed by NM, 5-Aug-1993.)
(φ → ¬ ψ)       (φ → ¬ xψ)

Theoremchvarv 1809* Implicit substitution of y for x into a theorem. (Contributed by NM, 20-Apr-1994.)
(x = y → (φψ))    &   φ       ψ

Theoremcleljust 1810* When the class variables of set theory are replaced with setvar variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the setvar variables in wel 1391 with the class variables in wcel 1390. (Contributed by NM, 28-Jan-2004.)
(x yz(z = x z y))

1.4.5  More substitution theorems

Theoremhbs1 1811* x is not free in [y / x]φ when x and y are distinct. (Contributed by NM, 5-Aug-1993.) (Proof by Jim Kingdon, 16-Dec-2017.) (New usage is discouraged.)
([y / x]φx[y / x]φ)

Theoremnfs1v 1812* x is not free in [y / x]φ when x and y are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.)
x[y / x]φ

Theoremsbhb 1813* Two ways of expressing "x is (effectively) not free in φ." (Contributed by NM, 29-May-2009.)
((φxφ) ↔ y(φ → [y / x]φ))

Theoremhbsbv 1814* This is a version of hbsb 1820 with an extra distinct variable constraint, on z and x. (Contributed by Jim Kingdon, 25-Dec-2017.)
(φzφ)       ([y / x]φz[y / x]φ)

Theoremnfsbxy 1815* Similar to hbsb 1820 but with an extra distinct variable constraint, on x and y. (Contributed by Jim Kingdon, 19-Mar-2018.)
zφ       z[y / x]φ

Theoremnfsbxyt 1816* Closed form of nfsbxy 1815. (Contributed by Jim Kingdon, 9-May-2018.)
(xzφ → Ⅎz[y / x]φ)

Theoremsbco2vlem 1817* This is a version of sbco2 1836 where z is distinct from x and from y. It is a lemma on the way to proving sbco2v 1818 which only requires that z and x be distinct. (Contributed by Jim Kingdon, 25-Dec-2017.) (One distinct variable constraint removed by Jim Kingdon, 3-Feb-2018.)
(φzφ)       ([y / z][z / x]φ ↔ [y / x]φ)

Theoremsbco2v 1818* This is a version of sbco2 1836 where z is distinct from x. (Contributed by Jim Kingdon, 12-Feb-2018.)
(φzφ)       ([y / z][z / x]φ ↔ [y / x]φ)

Theoremnfsb 1819* 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.) (Proof rewritten by Jim Kingdon, 19-Mar-2018.)
zφ       z[y / x]φ

Theoremhbsb 1820* 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.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.)
(φzφ)       ([y / x]φz[y / x]φ)

Theoremequsb3lem 1821* Lemma for equsb3 1822. (Contributed by NM, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([x / y]y = zx = z)

Theoremequsb3 1822* Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)
([x / y]y = zx = z)

Theoremsbn 1823 Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.)
([y / x] ¬ φ ↔ ¬ [y / x]φ)

Theoremsbim 1824 Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.)
([y / x](φψ) ↔ ([y / x]φ → [y / x]ψ))

Theoremsbor 1825 Logical OR inside and outside of substitution are equivalent. (Contributed by NM, 29-Sep-2002.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.)
([y / x](φ ψ) ↔ ([y / x]φ [y / x]ψ))

Theoremsban 1826 Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.)
([y / x](φ ψ) ↔ ([y / x]φ [y / x]ψ))

Theoremsbrim 1827 Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 5-Aug-1993.)
(φxφ)       ([y / x](φψ) ↔ (φ → [y / x]ψ))

Theoremsblim 1828 Substitution with a variable not free in consequent affects only the antecedent. (Contributed by NM, 14-Nov-2013.) (Revised by Mario Carneiro, 4-Oct-2016.)
xψ       ([y / x](φψ) ↔ ([y / x]φψ))

Theoremsb3an 1829 Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-Dec-2006.)
([y / x](φ ψ χ) ↔ ([y / x]φ [y / x]ψ [y / x]χ))

Theoremsbbi 1830 Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
([y / x](φψ) ↔ ([y / x]φ ↔ [y / x]ψ))

Theoremsblbis 1831 Introduce left biconditional inside of a substitution. (Contributed by NM, 19-Aug-1993.)
([y / x]φψ)       ([y / x](χφ) ↔ ([y / x]χψ))

Theoremsbrbis 1832 Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.)
([y / x]φψ)       ([y / x](φχ) ↔ (ψ ↔ [y / x]χ))

Theoremsbrbif 1833 Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.)
(χxχ)    &   ([y / x]φψ)       ([y / x](φχ) ↔ (ψχ))

Theoremsbco2yz 1834* This is a version of sbco2 1836 where z is distinct from y. It is a lemma on the way to proving sbco2 1836 which has no distinct variable constraints. (Contributed by Jim Kingdon, 19-Mar-2018.)
zφ       ([y / z][z / x]φ ↔ [y / x]φ)

Theoremsbco2h 1835 A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 19-Mar-2018.)
(φzφ)       ([y / z][z / x]φ ↔ [y / x]φ)

Theoremsbco2 1836 A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)
zφ       ([y / z][z / x]φ ↔ [y / x]φ)

Theoremsbco2d 1837 A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
(φxφ)    &   (φzφ)    &   (φ → (ψzψ))       (φ → ([y / z][z / x]ψ ↔ [y / x]ψ))

Theoremsbco2vd 1838* Version of sbco2d 1837 with a distinct variable constraint between x and z. (Contributed by Jim Kingdon, 19-Feb-2018.)
(φxφ)    &   (φzφ)    &   (φ → (ψzψ))       (φ → ([y / z][z / x]ψ ↔ [y / x]ψ))

Theoremsbco 1839 A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
([y / x][x / y]φ ↔ [y / x]φ)

Theoremsbco3v 1840* Version of sbco3 1845 with a distinct variable constraint between x and y. (Contributed by Jim Kingdon, 19-Feb-2018.)
([z / y][y / x]φ ↔ [z / x][x / y]φ)

Theoremsbcocom 1841 Relationship between composition and commutativity for substitution. (Contributed by Jim Kingdon, 28-Feb-2018.)
([z / y][y / x]φ ↔ [z / y][z / x]φ)

Theoremsbcomv 1842* Version of sbcom 1846 with a distinct variable constraint between x and z. (Contributed by Jim Kingdon, 28-Feb-2018.)
([y / z][y / x]φ ↔ [y / x][y / z]φ)

Theoremsbcomxyyz 1843* Version of sbcom 1846 with distinct variable constraints between x and y, and y and z. (Contributed by Jim Kingdon, 21-Mar-2018.)
([y / z][y / x]φ ↔ [y / x][y / z]φ)

Theoremsbco3xzyz 1844* Version of sbco3 1845 with distinct variable constraints between x and z, and y and z. Lemma for proving sbco3 1845. (Contributed by Jim Kingdon, 22-Mar-2018.)
([z / y][y / x]φ ↔ [z / x][x / y]φ)

Theoremsbco3 1845 A composition law for substitution. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.)
([z / y][y / x]φ ↔ [z / x][x / y]φ)

Theoremsbcom 1846 A commutativity law for substitution. (Contributed by NM, 27-May-1997.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.)
([y / z][y / x]φ ↔ [y / x][y / z]φ)

Theoremnfsbt 1847* Closed form of nfsb 1819. (Contributed by Jim Kingdon, 9-May-2018.)
(xzφ → Ⅎz[y / x]φ)

Theoremnfsbd 1848* Deduction version of nfsb 1819. (Contributed by NM, 15-Feb-2013.)
xφ    &   (φ → Ⅎzψ)       (φ → Ⅎz[y / x]ψ)

Theoremelsb3 1849* Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([x / y]y zx z)

Theoremelsb4 1850* 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 yz x)

Theoremsb9v 1851* Like sb9 1852 but with a distinct variable constraint between x and y. (Contributed by Jim Kingdon, 28-Feb-2018.)
(x[x / y]φy[y / x]φ)

Theoremsb9 1852 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)
(x[x / y]φy[y / x]φ)

Theoremsb9i 1853 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)
(x[x / y]φy[y / x]φ)

Theoremsbnf2 1854* 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.)
(Ⅎxφyz([y / x]φ ↔ [z / x]φ))

Theoremhbsbd 1855* Deduction version of hbsb 1820. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)
(φxφ)    &   (φzφ)    &   (φ → (ψzψ))       (φ → ([y / x]ψz[y / x]ψ))

Theorem2sb5 1856* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
([z / x][w / y]φxy((x = z y = w) φ))

Theorem2sb6 1857* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
([z / x][w / y]φxy((x = z y = w) → φ))

Theoremsbcom2v 1858* Lemma for proving sbcom2 1860. It is the same as sbcom2 1860 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 1859* Lemma for proving sbcom2 1860. It is the same as sbcom2v 1858 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 1860* 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 1861* Equivalence for substitution. (Contributed by NM, 5-Aug-1993.)
([y / x]φx(x = y → [x / y]φ))

Theorem2sb5rf 1862* Reversed double substitution. (Contributed by NM, 3-Feb-2005.)
(φzφ)    &   (φwφ)       (φzw((z = x w = y) [z / x][w / y]φ))

Theorem2sb6rf 1863* Reversed double substitution. (Contributed by NM, 3-Feb-2005.)
(φzφ)    &   (φwφ)       (φzw((z = x w = y) → [z / x][w / y]φ))

Theoremdfsb7 1864* An alternate definition of proper substitution df-sb 1643. 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 1764, first z for x then y for z. Compare Definition 2.1'' of [Quine] p. 17. Theorem sb7f 1865 provides a version where φ and z don't have to be distinct. (Contributed by NM, 28-Jan-2004.)
([y / x]φz(z = y x(x = z φ)))

Theoremsb7f 1865* This version of dfsb7 1864 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 1416 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1643 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.)
(φzφ)       ([y / x]φz(z = y x(x = z φ)))

Theoremsb7af 1866* An alternative definition of proper substitution df-sb 1643. Similar to dfsb7a 1867 but does not require that φ and z be distinct. Similar to sb7f 1865 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]φz(z = yx(x = zφ)))

Theoremdfsb7a 1867* An alternative definition of proper substitution df-sb 1643. Similar to dfsb7 1864 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 1866. (Contributed by Jim Kingdon, 5-Feb-2018.)
([y / x]φz(z = yx(x = zφ)))

Theoremsb10f 1868* 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.)
(φxφ)       ([y / z]φx(x = y [x / z]φ))

Theoremsbid2v 1869* 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 1870* Elimination of substitution. (Contributed by NM, 5-Aug-1993.)
(φx(x = y [x / y]φ))

Theoremsbel2x 1871* Elimination of double substitution. (Contributed by NM, 5-Aug-1993.)
(φxy((x = z y = w) [y / w][x / z]φ))

Theoremsbalyz 1872* Move universal quantifier in and out of substitution. Identical to sbal 1873 except that it has an additional distinct variable constraint on y and z. (Contributed by Jim Kingdon, 29-Dec-2017.)
([z / y]xφx[z / y]φ)

Theoremsbal 1873* Move universal quantifier in and out of substitution. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.)
([z / y]xφx[z / y]φ)

Theoremsbal1yz 1874* Lemma for proving sbal1 1875. Same as sbal1 1875 but with an additional distinct variable constraint on y and z. (Contributed by Jim Kingdon, 23-Feb-2018.)
x x = z → ([z / y]xφx[z / y]φ))

Theoremsbal1 1875* 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]xφx[z / y]φ))

Theoremsbexyz 1876* Move existential quantifier in and out of substitution. Identical to sbex 1877 except that it has an additional distinct variable constraint on y and z. (Contributed by Jim Kingdon, 29-Dec-2017.)
([z / y]xφx[z / y]φ)

Theoremsbex 1877* Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.)
([z / y]xφx[z / y]φ)

Theoremsbalv 1878* Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)
([y / x]φψ)       ([y / x]zφzψ)

Theoremsbco4lem 1879* Lemma for sbco4 1880. 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 1880* 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 1881* An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.)
(xφyx(x = yφ))

Theorem2exsb 1882* An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.)
(xyφzwxy((x = z y = w) → φ))

TheoremdvelimALT 1883* Version of dvelim 1890 that doesn't use ax-10 1393. Because it has different distinct variable constraints than dvelim 1890 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.)
(φxφ)    &   (z = y → (φψ))       x x = y → (ψxψ))

Theoremdvelimfv 1884* Like dvelimf 1888 but with a distinct variable constraint on x and z. (Contributed by Jim Kingdon, 6-Mar-2018.)
(φxφ)    &   (ψzψ)    &   (z = y → (φψ))       x x = y → (ψxψ))

Theoremhbsb4 1885 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.)
(φzφ)       z z = y → ([y / x]φz[y / x]φ))

Theoremhbsb4t 1886 A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1885). (Contributed by NM, 7-Apr-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(xz(φzφ) → (¬ z z = y → ([y / x]φz[y / x]φ)))

Theoremnfsb4t 1887 A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1885). (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 1888 Version of dvelim 1890 without any variable restrictions. (Contributed by NM, 1-Oct-2002.)
(φxφ)    &   (ψzψ)    &   (z = y → (φψ))       x x = y → (ψxψ))

Theoremdvelimdf 1889 Deduction form of dvelimf 1888. This version may be useful if we want to avoid ax-17 1416 and use ax-16 1692 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 1890* 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 1889.

Other variants of this theorem are dvelimf 1888 (with no distinct variable restrictions) and dvelimALT 1883 (that avoids ax-10 1393). (Contributed by NM, 23-Nov-1994.)

(φxφ)    &   (z = y → (φψ))       x x = y → (ψxψ))

Theoremdvelimor 1891* 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 xψ)

Theoremdveeq1 1892* 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 = zx y = z))

Theoremdveel1 1893* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
x x = y → (y zx y z))

Theoremdveel2 1894* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
x x = y → (z yx z y))

Theoremsbal2 1895* Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.)
x x = y → ([z / y]xφx[z / y]φ))

Theoremnfsb4or 1896 A variable not free remains so after substitution with a distinct variable. (Contributed by Jim Kingdon, 11-May-2018.)
zφ       (z z = y z[y / x]φ)

1.4.6  Existential uniqueness

Syntaxweu 1897 Extend wff definition to include existential uniqueness ("there exists a unique x such that φ").
wff ∃!xφ

Syntaxwmo 1898 Extend wff definition to include uniqueness ("there exists at most one x such that φ").
wff ∃*xφ

Theoremeujust 1899* A soundness justification theorem for df-eu 1900, 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 1900* 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 1922, eu2 1941, eu3 1943, and eu5 1944 (which in some cases we show with a hypothesis φyφ 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 1990). (Contributed by NM, 5-Aug-1993.)
(∃!xφyx(φx = y))

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