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Theorem List for Metamath Proof Explorer - 2101-2200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremax15 2101 Axiom ax-15 2102 is redundant if we assume ax-17 1628. Remark 9.6 in [Megill] p. 448 (p. 16 of the preprint), regarding axiom scheme C14'.

Note that  w is a dummy variable introduced in the proof. On the web page, it is implicitly assumed to be distinct from all other variables. (This is made explicit in the database file set.mm). Its purpose is to satisfy the distinct variable requirements of dveel2 2099 and ax-17 1628. By the end of the proof it has vanished, and the final theorem has no distinct variable requirements.

This theorem should not be referenced in any proof. Instead, use ax-15 2102 below so that theorems needing ax-15 2102 can be more easily identified. (Contributed by NM, 29-Jun-1995.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  ( -.  A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  e.  y  ->  A. z  x  e.  y
 ) ) )
 
Axiomax-15 2102 Axiom of Quantifier Introduction. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. Axiom scheme C14' in [Megill] p. 448 (p. 16 of the preprint). It is redundant if we include ax-17 1628; see theorem ax15 2101. Alternately, ax-17 1628 becomes unnecessary in principle with this axiom, but we lose the more powerful metalogic afforded by ax-17 1628. We retain ax-15 2102 here to provide completeness for systems with the simpler metalogic that results from omitting ax-17 1628, which might be easier to study for some theoretical purposes. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
 |-  ( -.  A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  e.  y  ->  A. z  x  e.  y
 ) ) )
 
Theoremax17el 2103* Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-17 1628 considered as a metatheorem.) (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  e.  y  ->  A. z  x  e.  y )
 
Theoremdveel2ALT 2104* Version of dveel2 2099 using ax-16 1926 instead of ax-17 1628. (Contributed by NM, 10-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  ( z  e.  y  ->  A. x  z  e.  y ) )
 
Theoremax11eq 2105 Basis step for constructing a substitution instance of ax-11o 1940 without using ax-11o 1940. Atomic formula for equality predicate. (Contributed by NM, 22-Jan-2007.)
 |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( z  =  w  ->  A. x ( x  =  y  ->  z  =  w ) ) ) )
 
Theoremax11el 2106 Basis step for constructing a substitution instance of ax-11o 1940 without using ax-11o 1940. Atomic formula for membership predicate. (Contributed by NM, 22-Jan-2007.)
 |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( z  e.  w  ->  A. x ( x  =  y  ->  z  e.  w ) ) ) )
 
Theoremax11f 2107 Basis step for constructing a substitution instance of ax-11o 1940 without using ax-11o 1940. We can start with any formula  ph in which  x is not free. (Contributed by NM, 21-Jan-2007.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )
 
Theoremax11indn 2108 Induction step for constructing a substitution instance of ax-11o 1940 without using ax-11o 1940. Negation case. (Contributed by NM, 21-Jan-2007.)
 |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( -.  ph  ->  A. x ( x  =  y  ->  -.  ph ) ) ) )
 
Theoremax11indi 2109 Induction step for constructing a substitution instance of ax-11o 1940 without using ax-11o 1940. Implication case. (Contributed by NM, 21-Jan-2007.)
 |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )   &    |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( ps  ->  A. x ( x  =  y  ->  ps ) ) ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( ( ph  ->  ps )  ->  A. x ( x  =  y  ->  ( ph  ->  ps ) ) ) ) )
 
Theoremax11indalem 2110 Lemma for ax11inda2 2112 and ax11inda 2113. (Contributed by NM, 24-Jan-2007.)
 |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )   =>    |-  ( -.  A. y  y  =  z  ->  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( A. z ph  ->  A. x ( x  =  y  ->  A. z ph ) ) ) ) )
 
Theoremax11inda2ALT 2111* A proof of ax11inda2 2112 that is slightly more direct. (Contributed by NM, 4-May-2007.) (Proof modification is discouraged.) (New usage is discouraged.
 |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  (
 A. z ph  ->  A. x ( x  =  y  ->  A. z ph ) ) ) )
 
Theoremax11inda2 2112* Induction step for constructing a substitution instance of ax-11o 1940 without using ax-11o 1940. Quantification case. When  z and  y are distinct, this theorem avoids the dummy variables needed by the more general ax11inda 2113. (Contributed by NM, 24-Jan-2007.)
 |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  (
 A. z ph  ->  A. x ( x  =  y  ->  A. z ph ) ) ) )
 
Theoremax11inda 2113* Induction step for constructing a substitution instance of ax-11o 1940 without using ax-11o 1940. Quantification case. (When  z and  y are distinct, ax11inda2 2112 may be used instead to avoid the dummy variable  w in the proof.) (Contributed by NM, 24-Jan-2007.)
 |-  ( -.  A. x  x  =  w  ->  ( x  =  w  ->  ( ph  ->  A. x ( x  =  w  ->  ph ) ) ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  (
 A. z ph  ->  A. x ( x  =  y  ->  A. z ph ) ) ) )
 
1.6.6  Existential uniqueness
 
Syntaxweu 2114 Extend wff definition to include existential uniqueness ("there exists a unique  x such that  ph").
 wff  E! x ph
 
Syntaxwmo 2115 Extend wff definition to include uniqueness ("there exists at most one  x such that  ph").
 wff  E* x ph
 
Theoremeujust 2116* A soundness justification theorem for df-eu 2118, showing that the definition is equivalent to itself with its dummy variable renamed. Note that  y and  z needn't be distinct variables. See eujustALT 2117 for a proof that provides an example of how it can be achieved through the use of dvelim 2092. (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
 ) )
 
TheoremeujustALT 2117* A soundness justification theorem for df-eu 2118, showing that the definition is equivalent to itself with its dummy variable renamed. Note that  y and  z needn't be distinct variables. While this isn't strictly necessary for soundness, the proof provides an example of how it can be achieved through the use of dvelim 2092. (Contributed by NM, 11-Mar-2010.) (Proof modification is discouraged.)
 |-  ( E. y A. x ( ph  <->  x  =  y
 ) 
 <-> 
 E. z A. x ( ph  <->  x  =  z
 ) )
 
Definitiondf-eu 2118* 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 2134, eu2 2138, eu3 2139, and eu5 2151 (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 2196). (Contributed by NM, 12-Aug-1993.)
 |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
 
Definitiondf-mo 2119 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 2144. For other possible definitions see mo2 2142 and mo4 2146. (Contributed by NM, 8-Mar-1995.)
 |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
 
Theoremeuf 2120* A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.)
 |- 
 F/ y ph   =>    |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
 
Theoremeubid 2121 Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E! x ps  <->  E! x ch )
 )
 
Theoremeubidv 2122* Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E! x ps  <->  E! x ch )
 )
 
Theoremeubii 2123 Introduce uniqueness 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 )
 
Theoremnfeu1 2124 Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
 |- 
 F/ x E! x ph
 
Theoremnfmo1 2125 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
 
Theoremnfeud2 2126 Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- 
 F/ y ph   &    |-  ( ( ph  /\ 
 -.  A. x  x  =  y )  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x E! y ps )
 
Theoremnfmod2 2127 Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- 
 F/ y ph   &    |-  ( ( ph  /\ 
 -.  A. x  x  =  y )  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x E* y ps )
 
Theoremnfeud 2128 Deduction version of nfeu 2130. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x E! y ps )
 
Theoremnfmod 2129 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 )
 
Theoremnfeu 2130 Bound-variable hypothesis builder for "at most one." Note that  x and  y needn't be distinct (this makes the proof more difficult). (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.)
 |- 
 F/ x ph   =>    |- 
 F/ x E! y ph
 
Theoremnfmo 2131 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)
 |- 
 F/ x ph   =>    |- 
 F/ x E* y ph
 
Theoremsb8eu 2132 Variable substitution in uniqueness 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 )
 
Theoremcbveu 2133 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E! x ph  <->  E! y ps )
 
Theoremeu1 2134* An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
 |- 
 F/ y ph   =>    |-  ( E! x ph  <->  E. x ( ph  /\  A. y ( [ y  /  x ] ph  ->  x  =  y ) ) )
 
Theoremmo 2135* Equivalent definitions of "there exists at most one." (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
 |- 
 F/ y ph   =>    |-  ( E. y A. x ( ph  ->  x  =  y )  <->  A. x A. y
 ( ( ph  /\  [
 y  /  x ] ph )  ->  x  =  y ) )
 
Theoremeuex 2136 Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( E! x ph  ->  E. x ph )
 
Theoremeumo0 2137* Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.)
 |- 
 F/ y ph   =>    |-  ( E! x ph  ->  E. y A. x ( ph  ->  x  =  y ) )
 
Theoremeu2 2138* An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.)
 |- 
 F/ y ph   =>    |-  ( E! x ph  <->  ( E. x ph  /\  A. x A. y ( (
 ph  /\  [ y  /  x ] ph )  ->  x  =  y ) ) )
 
Theoremeu3 2139* An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.)
 |- 
 F/ y ph   =>    |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
 
Theoremeuor 2140 Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.)
 |- 
 F/ x ph   =>    |-  ( ( -.  ph  /\ 
 E! x ps )  ->  E! x ( ph  \/  ps ) )
 
Theoremeuorv 2141* Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)
 |-  ( ( -.  ph  /\ 
 E! x ps )  ->  E! x ( ph  \/  ps ) )
 
Theoremmo2 2142* Alternate definition of "at most one." (Contributed by NM, 8-Mar-1995.)
 |- 
 F/ y ph   =>    |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
 
Theoremsbmo 2143* Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( [ y  /  x ] E* z ph  <->  E* z [ y  /  x ] ph )
 
Theoremmo3 2144* Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that  y not occur in  ph in place of our hypothesis. (Contributed by NM, 8-Mar-1995.)
 |- 
 F/ y ph   =>    |-  ( E* x ph  <->  A. x A. y ( (
 ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
 
Theoremmo4f 2145* "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)
 |- 
 F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E* x ph  <->  A. x A. y ( ( ph  /\  ps )  ->  x  =  y ) )
 
Theoremmo4 2146* "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E* x ph  <->  A. x A. y ( ( ph  /\  ps )  ->  x  =  y ) )
 
Theoremmobid 2147 Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E* x ps  <->  E* x ch )
 )
 
Theoremmobidv 2148* Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by Mario Carneiro, 7-Oct-2016.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E* x ps  <->  E* x ch )
 )
 
Theoremmobii 2149 Formula-building rule for "at most one" quantifier (inference rule). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.)
 |-  ( ps  <->  ch )   =>    |-  ( E* x ps  <->  E* x ch )
 
Theoremcbvmo 2150 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E* x ph  <->  E* y ps )
 
Theoremeu5 2151 Uniqueness in terms of "at most one." (Contributed by NM, 23-Mar-1995.)
 |-  ( E! x ph  <->  ( E. x ph  /\  E* x ph ) )
 
Theoremeu4 2152* Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E! x ph  <->  ( E. x ph  /\  A. x A. y ( (
 ph  /\  ps )  ->  x  =  y ) ) )
 
Theoremeumo 2153 Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.)
 |-  ( E! x ph  ->  E* x ph )
 
Theoremeumoi 2154 "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)
 |- 
 E! x ph   =>    |- 
 E* x ph
 
Theoremexmoeu 2155 Existence in terms of "at most one" and uniqueness. (Contributed by NM, 5-Apr-2004.)
 |-  ( E. x ph  <->  ( E* x ph  ->  E! x ph ) )
 
Theoremexmoeu2 2156 Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.)
 |-  ( E. x ph  ->  ( E* x ph  <->  E! x ph ) )
 
Theoremmoabs 2157 Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.)
 |-  ( E* x ph  <->  ( E. x ph  ->  E* x ph ) )
 
Theoremexmo 2158 Something exists or at most one exists. (Contributed by NM, 8-Mar-1995.)
 |-  ( E. x ph  \/  E* x ph )
 
Theoremimmo 2159 "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( E* x ps  ->  E* x ph ) )
 
Theoremimmoi 2160 "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 15-Feb-2006.)
 |-  ( ph  ->  ps )   =>    |-  ( E* x ps  ->  E* x ph )
 
Theoremmoimv 2161* Move antecedent outside of "at most one." (Contributed by NM, 28-Jul-1995.)
 |-  ( E* x (
 ph  ->  ps )  ->  ( ph  ->  E* x ps )
 )
 
Theoremeuimmo 2162 Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( E! x ps  ->  E* x ph ) )
 
Theoremeuim 2163 Add existential uniqueness 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.)
 |-  ( ( E. x ph 
 /\  A. x ( ph  ->  ps ) )  ->  ( E! x ps  ->  E! x ph ) )
 
Theoremmoan 2164 "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.)
 |-  ( E* x ph  ->  E* x ( ps 
 /\  ph ) )
 
Theoremmoani 2165 "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.)
 |- 
 E* x ph   =>    |- 
 E* x ( ps 
 /\  ph )
 
Theoremmoor 2166 "At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.)
 |-  ( E* x (
 ph  \/  ps )  ->  E* x ph )
 
Theoremmooran1 2167 "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( ( E* x ph 
 \/  E* x ps )  ->  E* x ( ph  /\ 
 ps ) )
 
Theoremmooran2 2168 "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( E* x (
 ph  \/  ps )  ->  ( E* x ph  /\ 
 E* x ps )
 )
 
Theoremmoanim 2169 Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.)
 |- 
 F/ x ph   =>    |-  ( E* x (
 ph  /\  ps )  <->  (
 ph  ->  E* x ps )
 )
 
Theoremeuan 2170 Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |- 
 F/ x ph   =>    |-  ( E! x (
 ph  /\  ps )  <->  (
 ph  /\  E! x ps ) )
 
Theoremmoanimv 2171* Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 23-Mar-1995.)
 |-  ( E* x (
 ph  /\  ps )  <->  (
 ph  ->  E* x ps )
 )
 
Theoremmoaneu 2172 Nested "at most one" and uniqueness quantifiers. (Contributed by NM, 25-Jan-2006.)
 |- 
 E* x ( ph  /\ 
 E! x ph )
 
Theoremmoanmo 2173 Nested "at most one" quantifiers. (Contributed by NM, 25-Jan-2006.)
 |- 
 E* x ( ph  /\ 
 E* x ph )
 
Theoremeuanv 2174* Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)
 |-  ( E! x (
 ph  /\  ps )  <->  (
 ph  /\  E! x ps ) )
 
Theoremmopick 2175 "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.)
 |-  ( ( E* x ph 
 /\  E. x ( ph  /\ 
 ps ) )  ->  ( ph  ->  ps )
 )
 
Theoremeupick 2176 Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing  x such that 
ph is true, and there is also an  x (actually the same one) such that  ph and  ps are both true, then  ph implies  ps regardless of  x. This theorem can be useful for eliminating existential quantifiers in a hypothesis. Compare Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by NM, 10-Jul-1994.)
 |-  ( ( E! x ph 
 /\  E. x ( ph  /\ 
 ps ) )  ->  ( ph  ->  ps )
 )
 
Theoremeupicka 2177 Version of eupick 2176 with closed formulas. (Contributed by NM, 6-Sep-2008.)
 |-  ( ( E! x ph 
 /\  E. x ( ph  /\ 
 ps ) )  ->  A. x ( ph  ->  ps ) )
 
Theoremeupickb 2178 Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.)
 |-  ( ( E! x ph 
 /\  E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( ph  <->  ps ) )
 
Theoremeupickbi 2179 Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( E. x (
 ph  /\  ps )  <->  A. x ( ph  ->  ps ) ) )
 
Theoremmopick2 2180 "At most one" can show the existence of a common value. In this case we can infer existence of conjunction from a conjunction of existence, and it is one way to achieve the converse of 19.40 1608. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( ( E* x ph 
 /\  E. x ( ph  /\ 
 ps )  /\  E. x ( ph  /\  ch ) )  ->  E. x ( ph  /\  ps  /\  ch ) )
 
Theoremeuor2 2181 Introduce or eliminate a disjunct in a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( -.  E. x ph 
 ->  ( E! x (
 ph  \/  ps )  <->  E! x ps ) )
 
Theoremmoexex 2182 "At most one" double quantification. (Contributed by NM, 3-Dec-2001.)
 |- 
 F/ y ph   =>    |-  ( ( E* x ph 
 /\  A. x E* y ps )  ->  E* y E. x ( ph  /\  ps ) )
 
Theoremmoexexv 2183* "At most one" double quantification. (Contributed by NM, 26-Jan-1997.)
 |-  ( ( E* x ph 
 /\  A. x E* y ps )  ->  E* y E. x ( ph  /\  ps ) )
 
Theorem2moex 2184 Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.)
 |-  ( E* x E. y ph  ->  A. y E* x ph )
 
Theorem2euex 2185 Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( E! x E. y ph  ->  E. y E! x ph )
 
Theorem2eumo 2186 Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.)
 |-  ( E! x E* y ph  ->  E* x E! y ph )
 
Theorem2eu2ex 2187 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
 |-  ( E! x E! y ph  ->  E. x E. y ph )
 
Theorem2moswap 2188 A condition allowing swap of "at most one" and existential quantifiers. (Contributed by NM, 10-Apr-2004.)
 |-  ( A. x E* y ph  ->  ( E* x E. y ph  ->  E* y E. x ph ) )
 
Theorem2euswap 2189 A condition allowing swap of uniqueness and existential quantifiers. (Contributed by NM, 10-Apr-2004.)
 |-  ( A. x E* y ph  ->  ( E! x E. y ph  ->  E! y E. x ph ) )
 
Theorem2exeu 2190 Double existential uniqueness implies double uniqueness quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.)
 |-  ( ( E! x E. y ph  /\  E! y E. x ph )  ->  E! x E! y ph )
 
Theorem2mo 2191* Two equivalent expressions for double "at most one." (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.)
 |-  ( E. z E. w A. x A. y
 ( ph  ->  ( x  =  z  /\  y  =  w ) )  <->  A. x A. y A. z A. w ( ( ph  /\  [
 z  /  x ] [ w  /  y ] ph )  ->  ( x  =  z  /\  y  =  w )
 ) )
 
Theorem2mos 2192* Double "exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005.)
 |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph 
 <->  ps ) )   =>    |-  ( E. z E. w A. x A. y ( ph  ->  ( x  =  z  /\  y  =  w )
 ) 
 <-> 
 A. x A. y A. z A. w ( ( ph  /\  ps )  ->  ( x  =  z  /\  y  =  w ) ) )
 
Theorem2eu1 2193 Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. (Contributed by NM, 3-Dec-2001.)
 |-  ( A. x E* y ph  ->  ( E! x E! y ph  <->  ( E! x E. y ph  /\  E! y E. x ph )
 ) )
 
Theorem2eu2 2194 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
 |-  ( E! y E. x ph  ->  ( E! x E! y ph  <->  E! x E. y ph ) )
 
Theorem2eu3 2195 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
 |-  ( A. x A. y ( E* x ph 
 \/  E* y ph )  ->  ( ( E! x E! y ph  /\  E! y E! x ph )  <->  ( E! x E. y ph  /\  E! y E. x ph ) ) )
 
Theorem2eu4 2196* This theorem provides us with a definition of double existential uniqueness ("exactly one 
x and exactly one  y"). Naively one might think (incorrectly) that it could be defined by  E! x E! y ph. See 2eu1 2193 for a condition under which the naive definition holds and 2exeu 2190 for a one-way implication. See 2eu5 2197 and 2eu8 2200 for alternate definitions. (Contributed by NM, 3-Dec-2001.)
 |-  ( ( E! x E. y ph  /\  E! y E. x ph )  <->  ( E. x E. y ph  /\  E. z E. w A. x A. y
 ( ph  ->  ( x  =  z  /\  y  =  w ) ) ) )
 
Theorem2eu5 2197* An alternate definition of double existential uniqueness (see 2eu4 2196). A mistake sometimes made in the literature is to use  E! x E! y to mean "exactly one  x and exactly one  y." (For example, see Proposition 7.53 of [TakeutiZaring] p. 53.) It turns out that this is actually a weaker assertion, as can be seen by expanding out the formal definitions. This theorem shows that the erroneous definition can be repaired by conjoining 
A. x E* y ph as an additional condition. The correct definition apparently has never been published. ( E* means "exists at most one.") (Contributed by NM, 26-Oct-2003.)
 |-  ( ( E! x E! y ph  /\  A. x E* y ph )  <->  ( E. x E. y ph  /\  E. z E. w A. x A. y
 ( ph  ->  ( x  =  z  /\  y  =  w ) ) ) )
 
Theorem2eu6 2198* Two equivalent expressions for double existential uniqueness. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.)
 |-  ( ( E! x E. y ph  /\  E! y E. x ph )  <->  E. z E. w A. x A. y ( ph  <->  ( x  =  z  /\  y  =  w )
 ) )
 
Theorem2eu7 2199 Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.)
 |-  ( ( E! x E. y ph  /\  E! y E. x ph )  <->  E! x E! y ( E. x ph  /\  E. y ph ) )
 
Theorem2eu8 2200 Two equivalent expressions for double existential uniqueness. Curiously, we can put  E! on either of the internal conjuncts but not both. We can also commute  E! x E! y using 2eu7 2199. (Contributed by NM, 20-Feb-2005.)
 |-  ( E! x E! y ( E. x ph 
 /\  E. y ph )  <->  E! x E! y ( E! x ph  /\  E. y ph ) )
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