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Theorem List for Metamath Proof Explorer - 2101-2200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdveeq1ALT 2101* Version of dveeq1 2099 using ax-16 1927 instead of ax-17 1628. (Contributed by NM, 29-Apr-2008.) (Proof modification is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z ) )
 
Theoremdveel1 2102* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ( y  e.  z  ->  A. x  y  e.  z ) )
 
Theoremdveel2 2103* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ( z  e.  y  ->  A. x  z  e.  y ) )
 
Theoremsbal2 2104* 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 )
 )
 
Theoremax15 2105 Axiom ax-15 2106 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 2103 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 2106 below so that theorems needing ax-15 2106 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 2106 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 2105. 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 2106 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 2107* 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 2108* Version of dveel2 2103 using ax-16 1927 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 2109 Basis step for constructing a substitution instance of ax-11o 1941 without using ax-11o 1941. 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 2110 Basis step for constructing a substitution instance of ax-11o 1941 without using ax-11o 1941. 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 2111 Basis step for constructing a substitution instance of ax-11o 1941 without using ax-11o 1941. 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 2112 Induction step for constructing a substitution instance of ax-11o 1941 without using ax-11o 1941. 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 2113 Induction step for constructing a substitution instance of ax-11o 1941 without using ax-11o 1941. 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 2114 Lemma for ax11inda2 2116 and ax11inda 2117. (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 2115* A proof of ax11inda2 2116 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 2116* Induction step for constructing a substitution instance of ax-11o 1941 without using ax-11o 1941. Quantification case. When  z and  y are distinct, this theorem avoids the dummy variables needed by the more general ax11inda 2117. (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 2117* Induction step for constructing a substitution instance of ax-11o 1941 without using ax-11o 1941. Quantification case. (When  z and  y are distinct, ax11inda2 2116 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 2118 Extend wff definition to include existential uniqueness ("there exists a unique  x such that  ph").
 wff  E! x ph
 
Syntaxwmo 2119 Extend wff definition to include uniqueness ("there exists at most one  x such that  ph").
 wff  E* x ph
 
Theoremeujust 2120* A soundness justification theorem for df-eu 2122, 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 2121 for a proof that provides an example of how it can be achieved through the use of dvelim 2096. (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 2121* A soundness justification theorem for df-eu 2122, 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 2096. (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 2122* 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 2139, eu2 2143, eu3 2144, and eu5 2156 (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 2201). (Contributed by NM, 12-Aug-1993.)
 |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
 
Definitiondf-mo 2123 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 2149. For other possible definitions see mo2 2147 and mo4 2151. (Contributed by NM, 8-Mar-1995.)
 |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
 
Theoremeuf 2124* 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 2125 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 2126* 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 2127 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 2128 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 2129 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 2130 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 2131 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 2132 Deduction version of nfeu 2134. (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 2133 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 2134 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 2135 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)
 |- 
 F/ x ph   =>    |- 
 F/ x E* y ph
 
Theoremsb8eu 2136 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 )
 
Theoremsb8mo 2137 Variable substitution in uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |- 
 F/ y ph   =>    |-  ( E* x ph  <->  E* y [ y  /  x ] ph )
 
Theoremcbveu 2138 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 2139* 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 2140* 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 2141 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 2142* 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 2143* 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 2144* 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 2145 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 2146* Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)
 |-  ( ( -.  ph  /\ 
 E! x ps )  ->  E! x ( ph  \/  ps ) )
 
Theoremmo2 2147* 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 2148* Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( [ y  /  x ] E* z ph  <->  E* z [ y  /  x ] ph )
 
Theoremmo3 2149* 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 2150* "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 2151* "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 2152 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 2153* 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 2154 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 2155 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 2156 Uniqueness in terms of "at most one." (Contributed by NM, 23-Mar-1995.)
 |-  ( E! x ph  <->  ( E. x ph  /\  E* x ph ) )
 
Theoremeu4 2157* 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 2158 Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.)
 |-  ( E! x ph  ->  E* x ph )
 
Theoremeumoi 2159 "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)
 |- 
 E! x ph   =>    |- 
 E* x ph
 
Theoremexmoeu 2160 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 2161 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 2162 Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.)
 |-  ( E* x ph  <->  ( E. x ph  ->  E* x ph ) )
 
Theoremexmo 2163 Something exists or at most one exists. (Contributed by NM, 8-Mar-1995.)
 |-  ( E. x ph  \/  E* x ph )
 
Theoremmoim 2164 "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 ) )
 
Theoremmoimi 2165 "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 15-Feb-2006.)
 |-  ( ph  ->  ps )   =>    |-  ( E* x ps  ->  E* x ph )
 
Theoremmorimv 2166* Move antecedent outside of "at most one." (Contributed by NM, 28-Jul-1995.)
 |-  ( E* x (
 ph  ->  ps )  ->  ( ph  ->  E* x ps )
 )
 
Theoremeuimmo 2167 Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( E! x ps  ->  E* x ph ) )
 
Theoremeuim 2168 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 2169 "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 2170 "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 2171 "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 2172 "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 2173 "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 2174 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 2175 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 2176* Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 23-Mar-1995.)
 |-  ( E* x (
 ph  /\  ps )  <->  (
 ph  ->  E* x ps )
 )
 
Theoremmoaneu 2177 Nested "at most one" and uniqueness quantifiers. (Contributed by NM, 25-Jan-2006.)
 |- 
 E* x ( ph  /\ 
 E! x ph )
 
Theoremmoanmo 2178 Nested "at most one" quantifiers. (Contributed by NM, 25-Jan-2006.)
 |- 
 E* x ( ph  /\ 
 E* x ph )
 
Theoremeuanv 2179* Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)
 |-  ( E! x (
 ph  /\  ps )  <->  (
 ph  /\  E! x ps ) )
 
Theoremmopick 2180 "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 2181 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 2182 Version of eupick 2181 with closed formulas. (Contributed by NM, 6-Sep-2008.)
 |-  ( ( E! x ph 
 /\  E. x ( ph  /\ 
 ps ) )  ->  A. x ( ph  ->  ps ) )
 
Theoremeupickb 2183 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 2184 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 2185 "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 2186 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 2187 "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 2188* "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 2189 Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.)
 |-  ( E* x E. y ph  ->  A. y E* x ph )
 
Theorem2euex 2190 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 2191 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 2192 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
 |-  ( E! x E! y ph  ->  E. x E. y ph )
 
Theorem2moswap 2193 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 2194 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 2195 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 2196* 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 2197* 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 2198 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 2199 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 2200 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 ) ) )
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