P. 21 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2001-2100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nfsbv 2001*	If z is not free in ph, it is not free in [ y / x ] ph when z is distinct from x and y. Version of nfsb 2000 requiring more disjoint variables. (Contributed by Wolf Lammen, 7-Feb-2023.) Remove disjoint variable condition on x , y. (Revised by Steven Nguyen, 13-Aug-2023.) Reduce axiom usage. (Revised by GG, 25-Aug-2024.)
|- F/ z ph   =>    |- F/ z [ y / x ] ph
Theorem	sbco2v 2002*	Version of sbco2 2019 with disjoint variable conditions. (Contributed by Wolf Lammen, 29-Apr-2023.)
|- F/ z ph   =>    |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
Theorem	hbsb 2003*	If z is not free in ph, it is not free in [ y / x ] ph when y and z are distinct. (Contributed by NM, 12-Aug-1993.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.)
|- ( ph -> A. z ph )   =>    |- ( [ y / x ] ph -> A. z [ y / x ] ph )
Theorem	equsb3lem 2004*	Lemma for equsb3 2005. (Contributed by NM, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
|- ( [ y / x ] x = z <-> y = z )
Theorem	equsb3 2005*	Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)
|- ( [ y / x ] x = z <-> y = z )
Theorem	sbn 2006	Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.)
|- ( [ y / x ] -. ph <-> -. [ y / x ] ph )
Theorem	sbim 2007	Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.)
|- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
Theorem	sbor 2008	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 ] ( ph \/ ps ) <-> ( [ y / x ] ph \/ [ y / x ] ps ) )
Theorem	sban 2009	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 ] ( ph /\ ps ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) )
Theorem	sbrim 2010	Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A. x ph )   =>    |- ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) )
Theorem	sblim 2011	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.)
|- F/ x ps   =>    |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> ps ) )
Theorem	sb3an 2012	Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-Dec-2006.)
|- ( [ y / x ] ( ph /\ ps /\ ch ) <-> ( [ y / x ] ph /\ [ y / x ] ps /\ [ y / x ] ch ) )
Theorem	sbbi 2013	Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
|- ( [ y / x ] ( ph <-> ps ) <-> ( [ y / x ] ph <-> [ y / x ] ps ) )
Theorem	sblbis 2014	Introduce left biconditional inside of a substitution. (Contributed by NM, 19-Aug-1993.)
|- ( [ y / x ] ph <-> ps )   =>    |- ( [ y / x ] ( ch <-> ph ) <-> ( [ y / x ] ch <-> ps ) )
Theorem	sbrbis 2015	Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.)
|- ( [ y / x ] ph <-> ps )   =>    |- ( [ y / x ] ( ph <-> ch ) <-> ( ps <-> [ y / x ] ch ) )
Theorem	sbrbif 2016	Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.)
|- ( ch -> A. x ch )   &    |- ( [ y / x ] ph <-> ps )   =>    |- ( [ y / x ] ( ph <-> ch ) <-> ( ps <-> ch ) )
Theorem	sbco2yz 2017*	This is a version of sbco2 2019 where z is distinct from y. It is a lemma on the way to proving sbco2 2019 which has no distinct variable constraints. (Contributed by Jim Kingdon, 19-Mar-2018.)
|- F/ z ph   =>    |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
Theorem	sbco2h 2018	A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 19-Mar-2018.)
|- ( ph -> A. z ph )   =>    |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
Theorem	sbco2 2019	A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)
|- F/ z ph   =>    |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
Theorem	sbco2d 2020	A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A. x ph )   &    |- ( ph -> A. z ph )   &    |- ( ph -> ( ps -> A. z ps ) )   =>    |- ( ph -> ( [ y / z ] [ z / x ] ps <-> [ y / x ] ps ) )
Theorem	sbco2vd 2021*	Version of sbco2d 2020 with a distinct variable constraint between x and z. (Contributed by Jim Kingdon, 19-Feb-2018.)
|- ( ph -> A. x ph )   &    |- ( ph -> A. z ph )   &    |- ( ph -> ( ps -> A. z ps ) )   =>    |- ( ph -> ( [ y / z ] [ z / x ] ps <-> [ y / x ] ps ) )
Theorem	sbco 2022	A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph )
Theorem	sbco3v 2023*	Version of sbco3 2028 with a distinct variable constraint between x and y. (Contributed by Jim Kingdon, 19-Feb-2018.)
|- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )
Theorem	sbcocom 2024	Relationship between composition and commutativity for substitution. (Contributed by Jim Kingdon, 28-Feb-2018.)
|- ( [ z / y ] [ y / x ] ph <-> [ z / y ] [ z / x ] ph )
Theorem	sbcomv 2025*	Version of sbcom 2029 with a distinct variable constraint between x and z. (Contributed by Jim Kingdon, 28-Feb-2018.)
|- ( [ y / z ] [ y / x ] ph <-> [ y / x ] [ y / z ] ph )
Theorem	sbcomxyyz 2026*	Version of sbcom 2029 with distinct variable constraints between x and y, and y and z. (Contributed by Jim Kingdon, 21-Mar-2018.)
|- ( [ y / z ] [ y / x ] ph <-> [ y / x ] [ y / z ] ph )
Theorem	sbco3xzyz 2027*	Version of sbco3 2028 with distinct variable constraints between x and z, and y and z. Lemma for proving sbco3 2028. (Contributed by Jim Kingdon, 22-Mar-2018.)
|- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )
Theorem	sbco3 2028	A composition law for substitution. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.)
|- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )
Theorem	sbcom 2029	A commutativity law for substitution. (Contributed by NM, 27-May-1997.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.)
|- ( [ y / z ] [ y / x ] ph <-> [ y / x ] [ y / z ] ph )
Theorem	nfsbt 2030*	Closed form of nfsb 2000. (Contributed by Jim Kingdon, 9-May-2018.)
|- ( A. x F/ z ph -> F/ z [ y / x ] ph )
Theorem	nfsbd 2031*	Deduction version of nfsb 2000. (Contributed by NM, 15-Feb-2013.)
|- F/ x ph   &    |- ( ph -> F/ z ps )   =>    |- ( ph -> F/ z [ y / x ] ps )
Theorem	sb9v 2032*	Like sb9 2033 but with a distinct variable constraint between x and y. (Contributed by Jim Kingdon, 28-Feb-2018.)
|- ( A. x [ x / y ] ph <-> A. y [ y / x ] ph )
Theorem	sb9 2033	Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)
|- ( A. x [ x / y ] ph <-> A. y [ y / x ] ph )
Theorem	sb9i 2034	Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)
|- ( A. x [ x / y ] ph -> A. y [ y / x ] ph )
Theorem	sbnf2 2035*	Two ways of expressing " x is (effectively) not free in ph." (Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2016.)
|- ( F/ x ph <-> A. y A. z ( [ y / x ] ph <-> [ z / x ] ph ) )
Theorem	hbsbd 2036*	Deduction version of hbsb 2003. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)
|- ( ph -> A. x ph )   &    |- ( ph -> A. z ph )   &    |- ( ph -> ( ps -> A. z ps ) )   =>    |- ( ph -> ( [ y / x ] ps -> A. z [ y / x ] ps ) )
Theorem	2sb5 2037*	Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
|- ( [ z / x ] [ w / y ] ph <-> E. x E. y ( ( x = z /\ y = w ) /\ ph ) )
Theorem	2sb6 2038*	Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
|- ( [ z / x ] [ w / y ] ph <-> A. x A. y ( ( x = z /\ y = w ) -> ph ) )
Theorem	sbcom2v 2039*	Lemma for proving sbcom2 2041. It is the same as sbcom2 2041 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 ] ph <-> [ y / x ] [ w / z ] ph )
Theorem	sbcom2v2 2040*	Lemma for proving sbcom2 2041. It is the same as sbcom2v 2039 but removes the distinct variable constraint on x and y. (Contributed by Jim Kingdon, 19-Feb-2018.)
|- ( [ w / z ] [ y / x ] ph <-> [ y / x ] [ w / z ] ph )
Theorem	sbcom2 2041*	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 ] ph <-> [ y / x ] [ w / z ] ph )
Theorem	sb6a 2042*	Equivalence for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( [ y / x ] ph <-> A. x ( x = y -> [ x / y ] ph ) )
Theorem	2sb5rf 2043*	Reversed double substitution. (Contributed by NM, 3-Feb-2005.)
|- ( ph -> A. z ph )   &    |- ( ph -> A. w ph )   =>    |- ( ph <-> E. z E. w ( ( z = x /\ w = y ) /\ [ z / x ] [ w / y ] ph ) )
Theorem	2sb6rf 2044*	Reversed double substitution. (Contributed by NM, 3-Feb-2005.)
|- ( ph -> A. z ph )   &    |- ( ph -> A. w ph )   =>    |- ( ph <-> A. z A. w ( ( z = x /\ w = y ) -> [ z / x ] [ w / y ] ph ) )
Theorem	dfsb7 2045*	An alternate definition of proper substitution df-sb 1812. By introducing a dummy variable z in the definiens, we are able to eliminate any distinct variable restrictions among the variables x, y, and ph 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 1937, first z for x then y for z. Compare Definition 2.1'' of [Quine] p. 17. Theorem sb7f 2046 provides a version where ph and z don't have to be distinct. (Contributed by NM, 28-Jan-2004.)
|- ( [ y / x ] ph <-> E. z ( z = y /\ E. x ( x = z /\ ph ) ) )
Theorem	sb7f 2046*	This version of dfsb7 2045 does not require that ph 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 1575, 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.)
|- ( ph -> A. z ph )   =>    |- ( [ y / x ] ph <-> E. z ( z = y /\ E. x ( x = z /\ ph ) ) )
Theorem	sb7af 2047*	An alternate definition of proper substitution df-sb 1812. Similar to dfsb7a 2048 but does not require that ph and z be distinct. Similar to sb7f 2046 in that it involves a dummy variable z, but expressed in terms of A. rather than E.. (Contributed by Jim Kingdon, 5-Feb-2018.)
|- F/ z ph   =>    |- ( [ y / x ] ph <-> A. z ( z = y -> A. x ( x = z -> ph ) ) )
Theorem	dfsb7a 2048*	An alternate definition of proper substitution df-sb 1812. Similar to dfsb7 2045 in that it involves a dummy variable z, but expressed in terms of A. rather than E.. For a version which only requires F/ z ph rather than z and ph being distinct, see sb7af 2047. (Contributed by Jim Kingdon, 5-Feb-2018.)
|- ( [ y / x ] ph <-> A. z ( z = y -> A. x ( x = z -> ph ) ) )
Theorem	sb10f 2049*	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.)
|- ( ph -> A. x ph )   =>    |- ( [ y / z ] ph <-> E. x ( x = y /\ [ x / z ] ph ) )
Theorem	sbid2v 2050*	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 ] ph <-> ph )
Theorem	sbelx 2051*	Elimination of substitution. (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> E. x ( x = y /\ [ x / y ] ph ) )
Theorem	sbel2x 2052*	Elimination of double substitution. (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> E. x E. y ( ( x = z /\ y = w ) /\ [ y / w ] [ x / z ] ph ) )
Theorem	sbalyz 2053*	Move universal quantifier in and out of substitution. Identical to sbal 2054 except that it has an additional distinct variable constraint on y and z. (Contributed by Jim Kingdon, 29-Dec-2017.)
|- ( [ z / y ] A. x ph <-> A. x [ z / y ] ph )
Theorem	sbal 2054*	Move universal quantifier in and out of substitution. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.)
|- ( [ z / y ] A. x ph <-> A. x [ z / y ] ph )
Theorem	sbal1yz 2055*	Lemma for proving sbal1 2056. Same as sbal1 2056 but with an additional disjoint variable condition on y , z. (Contributed by Jim Kingdon, 23-Feb-2018.)
|- ( -. A. x x = z -> ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) )
Theorem	sbal1 2056*	A theorem used in elimination of disjoint variable conditions on x , y by replacing it with a distinctor -. A. x x = z. (Contributed by NM, 5-Aug-1993.) (Proof rewitten by Jim Kingdon, 24-Feb-2018.)
|- ( -. A. x x = z -> ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) )
Theorem	sbexyz 2057*	Move existential quantifier in and out of substitution. Identical to sbex 2058 except that it has an additional disjoint variable condition on y , z. (Contributed by Jim Kingdon, 29-Dec-2017.)
|- ( [ z / y ] E. x ph <-> E. x [ z / y ] ph )
Theorem	sbex 2058*	Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.)
|- ( [ z / y ] E. x ph <-> E. x [ z / y ] ph )
Theorem	sbalv 2059*	Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)
|- ( [ y / x ] ph <-> ps )   =>    |- ( [ y / x ] A. z ph <-> A. z ps )
Theorem	sbco4lem 2060*	Lemma for sbco4 2061. It replaces the temporary variable v with another temporary variable w. (Contributed by Jim Kingdon, 26-Sep-2018.)
|- ( [ x / v ] [ y / x ] [ v / y ] ph <-> [ x / w ] [ y / x ] [ w / y ] ph )
Theorem	sbco4 2061*	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 ] ph <-> [ x / w ] [ y / x ] [ w / y ] ph )
Theorem	exsb 2062*	An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.)
|- ( E. x ph <-> E. y A. x ( x = y -> ph ) )
Theorem	2exsb 2063*	An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.)
|- ( E. x E. y ph <-> E. z E. w A. x A. y ( ( x = z /\ y = w ) -> ph ) )
Theorem	dvelimALT 2064*	Version of dvelim 2071 that doesn't use ax-10 1554. Because it has different distinct variable constraints than dvelim 2071 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.)
|- ( ph -> A. x ph )   &    |- ( z = y -> ( ph <-> ps ) )   =>    |- ( -. A. x x = y -> ( ps -> A. x ps ) )
Theorem	dvelimfv 2065*	Like dvelimf 2069 but with a distinct variable constraint on x and z. (Contributed by Jim Kingdon, 6-Mar-2018.)
|- ( ph -> A. x ph )   &    |- ( ps -> A. z ps )   &    |- ( z = y -> ( ph <-> ps ) )   =>    |- ( -. A. x x = y -> ( ps -> A. x ps ) )
Theorem	hbsb4 2066	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.)
|- ( ph -> A. z ph )   =>    |- ( -. A. z z = y -> ( [ y / x ] ph -> A. z [ y / x ] ph ) )
Theorem	hbsb4t 2067	A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 2066). (Contributed by NM, 7-Apr-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( A. x A. z ( ph -> A. z ph ) -> ( -. A. z z = y -> ( [ y / x ] ph -> A. z [ y / x ] ph ) ) )
Theorem	nfsb4t 2068	A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 2066). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof rewritten by Jim Kingdon, 9-May-2018.)
|- ( A. x F/ z ph -> ( -. A. z z = y -> F/ z [ y / x ] ph ) )
Theorem	dvelimf 2069	Version of dvelim 2071 without any variable restrictions. (Contributed by NM, 1-Oct-2002.)
|- ( ph -> A. x ph )   &    |- ( ps -> A. z ps )   &    |- ( z = y -> ( ph <-> ps ) )   =>    |- ( -. A. x x = y -> ( ps -> A. x ps ) )
Theorem	dvelimdf 2070	Deduction form of dvelimf 2069. This version may be useful if we want to avoid ax-17 1575 and use ax-16 1863 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)
|- F/ x ph   &    |- F/ z ph   &    |- ( ph -> F/ x ps )   &    |- ( ph -> F/ z ch )   &    |- ( ph -> ( z = y -> ( ps <-> ch ) ) )   =>    |- ( ph -> ( -. A. x x = y -> F/ x ch ) )
Theorem	dvelim 2071*	This theorem can be used to eliminate a distinct variable restriction on x and z and replace it with the "distinctor" -. A. x x = y as an antecedent. ph normally has z free and can be read ph ( z ), and ps substitutes y for z and can be read ph ( 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 A. x A. z, conjoin them, and apply dvelimdf 2070. Other variants of this theorem are dvelimf 2069 (with no distinct variable restrictions) and dvelimALT 2064 (that avoids ax-10 1554). (Contributed by NM, 23-Nov-1994.)
|- ( ph -> A. x ph )   &    |- ( z = y -> ( ph <-> ps ) )   =>    |- ( -. A. x x = y -> ( ps -> A. x ps ) )
Theorem	dvelimor 2072*	Disjunctive distinct variable constraint elimination. A user of this theorem starts with a formula ph (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 A. x x = y ( ps is just a version of ph with y substituted for z). (Contributed by Jim Kingdon, 11-May-2018.)
|- F/ x ph   &    |- ( z = y -> ( ph <-> ps ) )   =>    |- ( A. x x = y \/ F/ x ps )
Theorem	dveeq1 2073*	Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) (Proof rewritten by Jim Kingdon, 19-Feb-2018.)
|- ( -. A. x x = y -> ( y = z -> A. x y = z ) )
Theorem	sbal2 2074*	Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.)
|- ( -. A. x x = y -> ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) )
Theorem	nfsb4or 2075	A variable not free remains so after substitution with a distinct variable. (Contributed by Jim Kingdon, 11-May-2018.)
|- F/ z ph   =>    |- ( A. z z = y \/ F/ z [ y / x ] ph )
Theorem	nfd2 2076	Deduce that x is not free in ps in a context. (Contributed by Wolf Lammen, 16-Sep-2021.)
|- ( ph -> ( E. x ps -> A. x ps ) )   =>    |- ( ph -> F/ x ps )
Theorem	hbe1a 2077	Dual statement of hbe1 1544. (Contributed by Wolf Lammen, 15-Sep-2021.)
|- ( E. x A. x ph -> A. x ph )
Theorem	nf5-1 2078	One direction of nf5 . (Contributed by Wolf Lammen, 16-Sep-2021.)
|- ( A. x ( ph -> A. x ph ) -> F/ x ph )
Theorem	nf5d 2079	Deduce that x is not free in ps in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- F/ x ph   &    |- ( ph -> ( ps -> A. x ps ) )   =>    |- ( ph -> F/ x ps )
1.4.6  Existential uniqueness
Syntax	weu 2080	Extend wff definition to include existential uniqueness ("there exists a unique x such that ph").
wff E! x ph
Syntax	wmo 2081	Extend wff definition to include uniqueness ("there exists at most one x such that ph").
wff E* x ph
Theorem	eujust 2082*	A soundness justification theorem for df-eu 2083, 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.)
|- ( E. y A. x ( ph <-> x = y ) <-> E. z A. x ( ph <-> x = z ) )
Definition	df-eu 2083*	Define existential uniqueness, i.e., "there exists exactly one x such that ph". Definition 10.1 of [BellMachover] p. 97; also Definition *14.02 of [WhiteheadRussell] p. 175. Other possible definitions are given by eu1 2105, eu2 2125, eu3 2127, and eu5 2128 (which in some cases we show with a hypothesis ph -> A. y ph in place of a distinct variable condition on y and ph). Double uniqueness is tricky: E! x E! y ph does not mean "exactly one x and one y " (see 2eu4 2174). (Contributed by NM, 5-Aug-1993.)
|- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
Definition	df-mo 2084	Define "there exists at most one x such that ph". Here we define it in terms of existential uniqueness. Notation of [BellMachover] p. 460, whose definition we show as mo3 2135. For another possible definition see mo4 2142. (Contributed by NM, 5-Aug-1993.)
|- ( E* x ph <-> ( E. x ph -> E! x ph ) )
Theorem	euf 2085*	A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.)
|- ( ph -> A. y ph )   =>    |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
Theorem	eubidh 2086	Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E! x ps <-> E! x ch ) )
Theorem	eubid 2087	Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.)
|- F/ x ph   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E! x ps <-> E! x ch ) )
Theorem	eubidv 2088*	Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E! x ps <-> E! x ch ) )
Theorem	eubii 2089	Introduce unique existential quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)
|- ( ph <-> ps )   =>    |- ( E! x ph <-> E! x ps )
Theorem	hbeu1 2090	Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.)
|- ( E! x ph -> A. x E! x ph )
Theorem	nfeu1 2091	Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
|- F/ x E! x ph
Theorem	nfmo1 2092	Bound-variable hypothesis builder for "at most one". (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.)
|- F/ x E* x ph
Theorem	sb8eu 2093	Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
|- F/ y ph   =>    |- ( E! x ph <-> E! y [ y / x ] ph )
Theorem	sb8mo 2094	Variable substitution for "at most one". (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- F/ y ph   =>    |- ( E* x ph <-> E* y [ y / x ] ph )
Theorem	nfeudv 2095*	Deduction version of nfeu 2099. Similar to nfeud 2096 but has the additional constraint that x and y must be distinct. (Contributed by Jim Kingdon, 25-May-2018.)
|- F/ y ph   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x E! y ps )
Theorem	nfeud 2096	Deduction version of nfeu 2099. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim Kingdon, 25-May-2018.)
|- F/ y ph   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x E! y ps )
Theorem	nfmod 2097	Bound-variable hypothesis builder for "at most one". (Contributed by Mario Carneiro, 14-Nov-2016.)
|- F/ y ph   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x E* y ps )
Theorem	nfeuv 2098*	Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 2099 but has the additional condition that x and y must be distinct. (Contributed by Jim Kingdon, 23-May-2018.)
|- F/ x ph   =>    |- F/ x E! y ph
Theorem	nfeu 2099	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.)
|- F/ x ph   =>    |- F/ x E! y ph
Theorem	nfmo 2100	Bound-variable hypothesis builder for "at most one". (Contributed by NM, 9-Mar-1995.)
|- F/ x ph   =>    |- F/ x E* y ph