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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 2sb5rf 2001* | Reversed double substitution. (Contributed by NM, 3-Feb-2005.) |
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Theorem | 2sb6rf 2002* | Reversed double substitution. (Contributed by NM, 3-Feb-2005.) |
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Theorem | dfsb7 2003* |
An alternate definition of proper substitution df-sb 1774. By introducing
a dummy variable ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | sb7f 2004* |
This version of dfsb7 2003 does not require that ![]() ![]() |
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Theorem | sb7af 2005* |
An alternate definition of proper substitution df-sb 1774. Similar to
dfsb7a 2006 but does not require that ![]() ![]() ![]() ![]() ![]() |
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Theorem | dfsb7a 2006* |
An alternate definition of proper substitution df-sb 1774. Similar to
dfsb7 2003 in that it involves a dummy variable ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | sb10f 2007* | 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.) |
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Theorem | sbid2v 2008* | 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.) |
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Theorem | sbelx 2009* | Elimination of substitution. (Contributed by NM, 5-Aug-1993.) |
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Theorem | sbel2x 2010* | Elimination of double substitution. (Contributed by NM, 5-Aug-1993.) |
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Theorem | sbalyz 2011* |
Move universal quantifier in and out of substitution. Identical to
sbal 2012 except that it has an additional distinct
variable constraint on
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Theorem | sbal 2012* | Move universal quantifier in and out of substitution. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.) |
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Theorem | sbal1yz 2013* |
Lemma for proving sbal1 2014. Same as sbal1 2014 but with an additional
disjoint variable condition on ![]() ![]() ![]() |
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Theorem | sbal1 2014* |
A theorem used in elimination of disjoint variable conditions on
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Theorem | sbexyz 2015* |
Move existential quantifier in and out of substitution. Identical to
sbex 2016 except that it has an additional disjoint
variable condition on
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Theorem | sbex 2016* | Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.) |
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Theorem | sbalv 2017* | Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.) |
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Theorem | sbco4lem 2018* |
Lemma for sbco4 2019. It replaces the temporary variable ![]() ![]() |
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Theorem | sbco4 2019* |
Two ways of exchanging two variables. Both sides of the biconditional
exchange ![]() ![]() ![]() ![]() ![]() |
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Theorem | exsb 2020* | An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.) |
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Theorem | 2exsb 2021* | An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.) |
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Theorem | dvelimALT 2022* | Version of dvelim 2029 that doesn't use ax-10 1516. Because it has different distinct variable constraints than dvelim 2029 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.) |
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Theorem | dvelimfv 2023* |
Like dvelimf 2027 but with a distinct variable constraint on
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Theorem | hbsb4 2024 | 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.) |
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Theorem | hbsb4t 2025 | A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 2024). (Contributed by NM, 7-Apr-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
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Theorem | nfsb4t 2026 | A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 2024). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof rewritten by Jim Kingdon, 9-May-2018.) |
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Theorem | dvelimf 2027 | Version of dvelim 2029 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) |
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Theorem | dvelimdf 2028 | Deduction form of dvelimf 2027. This version may be useful if we want to avoid ax-17 1537 and use ax-16 1825 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.) |
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Theorem | dvelim 2029* |
This theorem can be used to eliminate a distinct variable restriction on
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To obtain a closed-theorem form of this inference, prefix the hypotheses
with Other variants of this theorem are dvelimf 2027 (with no distinct variable restrictions) and dvelimALT 2022 (that avoids ax-10 1516). (Contributed by NM, 23-Nov-1994.) |
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Theorem | dvelimor 2030* |
Disjunctive distinct variable constraint elimination. A user of this
theorem starts with a formula ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | dveeq1 2031* | Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) (Proof rewritten by Jim Kingdon, 19-Feb-2018.) |
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Theorem | sbal2 2032* | Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.) |
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Theorem | nfsb4or 2033 | A variable not free remains so after substitution with a distinct variable. (Contributed by Jim Kingdon, 11-May-2018.) |
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Theorem | nfd2 2034 |
Deduce that ![]() ![]() |
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Theorem | hbe1a 2035 | Dual statement of hbe1 1506. (Contributed by Wolf Lammen, 15-Sep-2021.) |
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Theorem | nf5-1 2036 | One direction of nf5 . (Contributed by Wolf Lammen, 16-Sep-2021.) |
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Theorem | nf5d 2037 |
Deduce that ![]() ![]() |
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Syntax | weu 2038 |
Extend wff definition to include existential uniqueness ("there exists a
unique ![]() ![]() |
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Syntax | wmo 2039 |
Extend wff definition to include uniqueness ("there exists at most one
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Theorem | eujust 2040* |
A soundness justification theorem for df-eu 2041, showing that the
definition is equivalent to itself with its dummy variable renamed.
Note that ![]() ![]() |
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Definition | df-eu 2041* |
Define existential uniqueness, i.e., "there exists exactly one ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Definition | df-mo 2042 |
Define "there exists at most one ![]() ![]() |
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Theorem | euf 2043* | A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.) |
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Theorem | eubidh 2044 | Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) |
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Theorem | eubid 2045 | Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) |
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Theorem | eubidv 2046* | Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) |
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Theorem | eubii 2047 | Introduce unique existential quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) |
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Theorem | hbeu1 2048 | Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) |
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Theorem | nfeu1 2049 | Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
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Theorem | nfmo1 2050 | Bound-variable hypothesis builder for "at most one". (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) |
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Theorem | sb8eu 2051 | Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
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Theorem | sb8mo 2052 | Variable substitution for "at most one". (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
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Theorem | nfeudv 2053* |
Deduction version of nfeu 2057. Similar to nfeud 2054 but has the additional
constraint that ![]() ![]() |
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Theorem | nfeud 2054 | Deduction version of nfeu 2057. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim Kingdon, 25-May-2018.) |
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Theorem | nfmod 2055 | Bound-variable hypothesis builder for "at most one". (Contributed by Mario Carneiro, 14-Nov-2016.) |
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Theorem | nfeuv 2056* |
Bound-variable hypothesis builder for existential uniqueness. This is
similar to nfeu 2057 but has the additional condition that ![]() ![]() |
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Theorem | nfeu 2057 |
Bound-variable hypothesis builder for existential uniqueness. Note that
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Theorem | nfmo 2058 | Bound-variable hypothesis builder for "at most one". (Contributed by NM, 9-Mar-1995.) |
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Theorem | hbeu 2059 |
Bound-variable hypothesis builder for uniqueness. Note that ![]() ![]() |
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Theorem | hbeud 2060 | Deduction version of hbeu 2059. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 25-May-2018.) |
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Theorem | sb8euh 2061 | Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Andrew Salmon, 9-Jul-2011.) |
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Theorem | cbveu 2062 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
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Theorem | eu1 2063* | An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.) |
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Theorem | euor 2064 | Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 21-Oct-2005.) |
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Theorem | euorv 2065* | Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 23-Mar-1995.) |
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Theorem | mo2n 2066* | There is at most one of something which does not exist. (Contributed by Jim Kingdon, 2-Jul-2018.) |
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Theorem | mon 2067 | There is at most one of something which does not exist. (Contributed by Jim Kingdon, 5-Jul-2018.) |
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Theorem | euex 2068 | Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
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Theorem | eumo0 2069* | Existential uniqueness implies "at most one". (Contributed by NM, 8-Jul-1994.) |
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Theorem | eumo 2070 | Existential uniqueness implies "at most one". (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.) |
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Theorem | eumoi 2071 | "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.) |
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Theorem | mobidh 2072 | Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) |
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Theorem | mobid 2073 | Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) |
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Theorem | mobidv 2074* | Formula-building rule for "at most one" quantifier (deduction form). (Contributed by Mario Carneiro, 7-Oct-2016.) |
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Theorem | mobii 2075 | Formula-building rule for "at most one" quantifier (inference form). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.) |
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Theorem | hbmo1 2076 | Bound-variable hypothesis builder for "at most one". (Contributed by NM, 8-Mar-1995.) |
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Theorem | hbmo 2077 | Bound-variable hypothesis builder for "at most one". (Contributed by NM, 9-Mar-1995.) |
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Theorem | cbvmo 2078 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.) |
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Theorem | mo23 2079* | An implication between two definitions of "there exists at most one." (Contributed by Jim Kingdon, 25-Jun-2018.) |
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Theorem | mor 2080* |
Converse of mo23 2079 with an additional ![]() ![]() ![]() |
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Theorem | modc 2081* | Equivalent definitions of "there exists at most one," given decidable existence. (Contributed by Jim Kingdon, 1-Jul-2018.) |
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Theorem | eu2 2082* | An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.) |
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Theorem | eu3h 2083* | An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.) |
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Theorem | eu3 2084* | An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) |
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Theorem | eu5 2085 | Uniqueness in terms of "at most one". (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.) |
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Theorem | exmoeu2 2086 | Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.) |
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Theorem | moabs 2087 | Absorption of existence condition by "at most one". (Contributed by NM, 4-Nov-2002.) |
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Theorem | exmodc 2088 | If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.) |
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Theorem | exmonim 2089 | There is at most one of something which does not exist. Unlike exmodc 2088 there is no decidability condition. (Contributed by Jim Kingdon, 22-Sep-2018.) |
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Theorem | mo2r 2090* | A condition which implies "at most one". (Contributed by Jim Kingdon, 2-Jul-2018.) |
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Theorem | mo3h 2091* |
Alternate definition of "at most one". Definition of [BellMachover]
p. 460, except that definition has the side condition that ![]() ![]() |
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Theorem | mo3 2092* |
Alternate definition of "at most one". Definition of [BellMachover]
p. 460, except that definition has the side condition that ![]() ![]() |
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Theorem | mo2dc 2093* | Alternate definition of "at most one" where existence is decidable. (Contributed by Jim Kingdon, 2-Jul-2018.) |
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Theorem | euan 2094 | Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
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Theorem | euanv 2095* | Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 23-Mar-1995.) |
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Theorem | euor2 2096 | Introduce or eliminate a disjunct in a unique existential quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
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Theorem | sbmo 2097* | Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.) |
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Theorem | mo4f 2098* | "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.) |
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Theorem | mo4 2099* | "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.) |
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Theorem | eu4 2100* | Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.) |
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