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Type | Label | Description |
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Statement | ||
Theorem | sbal1yz 2001* |
Lemma for proving sbal1 2002. Same as sbal1 2002 but with an additional
disjoint variable condition on ![]() ![]() ![]() |
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Theorem | sbal1 2002* |
A theorem used in elimination of disjoint variable conditions on
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Theorem | sbexyz 2003* |
Move existential quantifier in and out of substitution. Identical to
sbex 2004 except that it has an additional disjoint
variable condition on
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Theorem | sbex 2004* | 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 2005* | Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.) |
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Theorem | sbco4lem 2006* |
Lemma for sbco4 2007. It replaces the temporary variable ![]() ![]() |
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Theorem | sbco4 2007* |
Two ways of exchanging two variables. Both sides of the biconditional
exchange ![]() ![]() ![]() ![]() ![]() |
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Theorem | exsb 2008* | An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.) |
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Theorem | 2exsb 2009* | An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.) |
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Theorem | dvelimALT 2010* | Version of dvelim 2017 that doesn't use ax-10 1505. Because it has different distinct variable constraints than dvelim 2017 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 2011* |
Like dvelimf 2015 but with a distinct variable constraint on
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Theorem | hbsb4 2012 | 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 2013 | A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 2012). (Contributed by NM, 7-Apr-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
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Theorem | nfsb4t 2014 | A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 2012). (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 2015 | Version of dvelim 2017 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) |
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Theorem | dvelimdf 2016 | Deduction form of dvelimf 2015. This version may be useful if we want to avoid ax-17 1526 and use ax-16 1814 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 2017* |
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 2015 (with no distinct variable restrictions) and dvelimALT 2010 (that avoids ax-10 1505). (Contributed by NM, 23-Nov-1994.) |
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Theorem | dvelimor 2018* |
Disjunctive distinct variable constraint elimination. A user of this
theorem starts with a formula ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | dveeq1 2019* | 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 2020* | Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.) |
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Theorem | nfsb4or 2021 | 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 2022 |
Deduce that ![]() ![]() |
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Theorem | hbe1a 2023 | Dual statement of hbe1 1495. (Contributed by Wolf Lammen, 15-Sep-2021.) |
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Theorem | nf5-1 2024 | One direction of nf5 . (Contributed by Wolf Lammen, 16-Sep-2021.) |
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Theorem | nf5d 2025 |
Deduce that ![]() ![]() |
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Syntax | weu 2026 |
Extend wff definition to include existential uniqueness ("there exists a
unique ![]() ![]() |
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Syntax | wmo 2027 |
Extend wff definition to include uniqueness ("there exists at most one
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Theorem | eujust 2028* |
A soundness justification theorem for df-eu 2029, showing that the
definition is equivalent to itself with its dummy variable renamed.
Note that ![]() ![]() |
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Definition | df-eu 2029* |
Define existential uniqueness, i.e., "there exists exactly one ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Definition | df-mo 2030 |
Define "there exists at most one ![]() ![]() |
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Theorem | euf 2031* | 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 2032 | Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) |
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Theorem | eubid 2033 | Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) |
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Theorem | eubidv 2034* | Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) |
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Theorem | eubii 2035 | 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 2036 | Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) |
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Theorem | nfeu1 2037 | 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 2038 | 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 2039 | 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 2040 | Variable substitution for "at most one". (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
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Theorem | nfeudv 2041* |
Deduction version of nfeu 2045. Similar to nfeud 2042 but has the additional
constraint that ![]() ![]() |
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Theorem | nfeud 2042 | Deduction version of nfeu 2045. (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 2043 | Bound-variable hypothesis builder for "at most one". (Contributed by Mario Carneiro, 14-Nov-2016.) |
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Theorem | nfeuv 2044* |
Bound-variable hypothesis builder for existential uniqueness. This is
similar to nfeu 2045 but has the additional condition that ![]() ![]() |
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Theorem | nfeu 2045 |
Bound-variable hypothesis builder for existential uniqueness. Note that
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Theorem | nfmo 2046 | Bound-variable hypothesis builder for "at most one". (Contributed by NM, 9-Mar-1995.) |
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Theorem | hbeu 2047 |
Bound-variable hypothesis builder for uniqueness. Note that ![]() ![]() |
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Theorem | hbeud 2048 | Deduction version of hbeu 2047. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 25-May-2018.) |
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Theorem | sb8euh 2049 | 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 2050 | 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 2051* | 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 2052 | Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 21-Oct-2005.) |
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Theorem | euorv 2053* | Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 23-Mar-1995.) |
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Theorem | mo2n 2054* | There is at most one of something which does not exist. (Contributed by Jim Kingdon, 2-Jul-2018.) |
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Theorem | mon 2055 | There is at most one of something which does not exist. (Contributed by Jim Kingdon, 5-Jul-2018.) |
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Theorem | euex 2056 | Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
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Theorem | eumo0 2057* | Existential uniqueness implies "at most one". (Contributed by NM, 8-Jul-1994.) |
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Theorem | eumo 2058 | 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 2059 | "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.) |
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Theorem | mobidh 2060 | Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) |
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Theorem | mobid 2061 | Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) |
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Theorem | mobidv 2062* | Formula-building rule for "at most one" quantifier (deduction form). (Contributed by Mario Carneiro, 7-Oct-2016.) |
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Theorem | mobii 2063 | 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 2064 | Bound-variable hypothesis builder for "at most one". (Contributed by NM, 8-Mar-1995.) |
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Theorem | hbmo 2065 | Bound-variable hypothesis builder for "at most one". (Contributed by NM, 9-Mar-1995.) |
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Theorem | cbvmo 2066 | 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 2067* | An implication between two definitions of "there exists at most one." (Contributed by Jim Kingdon, 25-Jun-2018.) |
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Theorem | mor 2068* |
Converse of mo23 2067 with an additional ![]() ![]() ![]() |
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Theorem | modc 2069* | Equivalent definitions of "there exists at most one," given decidable existence. (Contributed by Jim Kingdon, 1-Jul-2018.) |
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Theorem | eu2 2070* | 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 2071* | An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.) |
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Theorem | eu3 2072* | An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) |
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Theorem | eu5 2073 | 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 2074 | Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.) |
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Theorem | moabs 2075 | Absorption of existence condition by "at most one". (Contributed by NM, 4-Nov-2002.) |
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Theorem | exmodc 2076 | If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.) |
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Theorem | exmonim 2077 | There is at most one of something which does not exist. Unlike exmodc 2076 there is no decidability condition. (Contributed by Jim Kingdon, 22-Sep-2018.) |
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Theorem | mo2r 2078* | A condition which implies "at most one". (Contributed by Jim Kingdon, 2-Jul-2018.) |
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Theorem | mo3h 2079* |
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 2080* |
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 2081* | Alternate definition of "at most one" where existence is decidable. (Contributed by Jim Kingdon, 2-Jul-2018.) |
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Theorem | euan 2082 | 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 2083* | Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 23-Mar-1995.) |
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Theorem | euor2 2084 | 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 2085* | Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.) |
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Theorem | mo4f 2086* | "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.) |
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Theorem | mo4 2087* | "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.) |
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Theorem | eu4 2088* | Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.) |
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Theorem | exmoeudc 2089 | Existence in terms of "at most one" and uniqueness. (Contributed by Jim Kingdon, 3-Jul-2018.) |
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Theorem | moim 2090 | "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.) |
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Theorem | moimi 2091 | "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 15-Feb-2006.) |
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Theorem | moimv 2092* | Move antecedent outside of "at most one". (Contributed by NM, 28-Jul-1995.) |
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Theorem | euimmo 2093 | Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.) |
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Theorem | euim 2094 | Add existential unique existential quantifiers to an implication. Note the reversed implication in the antecedent. (Contributed by NM, 19-Oct-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
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Theorem | moan 2095 | "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.) |
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Theorem | moani 2096 | "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.) |
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Theorem | moor 2097 | "At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.) |
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Theorem | mooran1 2098 | "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
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Theorem | mooran2 2099 | "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
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Theorem | moanim 2100 | Introduction of a conjunct into at-most-one quantifier. (Contributed by NM, 3-Dec-2001.) |
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