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Theorem List for Metamath Proof Explorer - 2101-2200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhbequid 2101 Bound-variable hypothesis builder for  x  =  x. This theorem tells us that any variable, including  x, is effectively not free in  x  =  x, even though  x is technically free according to the traditional definition of free variable. (The proof does not use ax-9o 2079.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  =  x 
 ->  A. y  x  =  x )
 
Theoremnfequid-o 2102 Bound-variable hypothesis builder for  x  =  x. This theorem tells us that any variable, including  x, is effectively not free in  x  =  x, even though  x is technically free according to the traditional definition of free variable. (The proof uses only ax-5 1546, ax-8 1645, ax-12o 2083, and ax-gen 1535. This shows that this can be proved without ax9 1891, even though the theorem equid 1646 cannot be. A shorter proof using ax9 1891 is obtainable from equid 1646 and hbth 1541.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax9v 1638, which is used for the derivation of ax12o 1877, unless we consider ax-12o 2083 the starting axiom rather than ax-12 1868. (Contributed by NM, 13-Jan-2011.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 F/ y  x  =  x
 
Theoremax46 2103 Proof of a single axiom that can replace ax-4 2076 and ax-6o 2078. See ax46to4 2104 and ax46to6 2105 for the re-derivation of those axioms. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( A. x  -.  A. x ph  ->  A. x ph )  ->  ph )
 
Theoremax46to4 2104 Re-derivation of ax-4 2076 from ax46 2103. Only propositional calculus is used for the re-derivation. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x ph  -> 
 ph )
 
Theoremax46to6 2105 Re-derivation of ax-6o 2078 from ax46 2103. Only propositional calculus is used for the re-derivation. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x  -.  A. x ph  ->  ph )
 
Theoremax67 2106 Proof of a single axiom that can replace both ax-6o 2078 and ax-7 1710. See ax67to6 2108 and ax67to7 2109 for the re-derivation of those axioms. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x  -.  A. y A. x ph 
 ->  A. y ph )
 
Theoremnfa1-o 2107  x is not free in  A. x ph. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 F/ x A. x ph
 
Theoremax67to6 2108 Re-derivation of ax-6o 2078 from ax67 2106. Note that ax-6o 2078 and ax-7 1710 are not used by the re-derivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x  -.  A. x ph  ->  ph )
 
Theoremax67to7 2109 Re-derivation of ax-7 1710 from ax67 2106. Note that ax-6o 2078 and ax-7 1710 are not used by the re-derivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremax467 2110 Proof of a single axiom that can replace ax-4 2076, ax-6o 2078, and ax-7 1710 in a subsystem that includes these axioms plus ax-5o 2077 and ax-gen 1535 (and propositional calculus). See ax467to4 2111, ax467to6 2112, and ax467to7 2113 for the re-derivation of those axioms. This theorem extends the idea in Scott Fenton's ax46 2103. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( A. x A. y  -.  A. x A. y ph  ->  A. x ph )  ->  ph )
 
Theoremax467to4 2111 Re-derivation of ax-4 2076 from ax467 2110. Only propositional calculus is used by the re-derivation. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x ph  -> 
 ph )
 
Theoremax467to6 2112 Re-derivation of ax-6o 2078 from ax467 2110. Note that ax-6o 2078 and ax-7 1710 are not used by the re-derivation. The use of alimi 1548 (which uses ax-4 2076) is allowed since we have already proved ax467to4 2111. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x  -.  A. x ph  ->  ph )
 
Theoremax467to7 2113 Re-derivation of ax-7 1710 from ax467 2110. Note that ax-6o 2078 and ax-7 1710 are not used by the re-derivation. The use of alimi 1548 (which uses ax-4 2076) is allowed since we have already proved ax467to4 2111. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremequidqe 2114 equid 1646 with existential quantifier without using ax-4 2076 or ax-17 1605. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) (Proof modification is discouraged.)
 |- 
 -.  A. y  -.  x  =  x
 
Theoremax4sp1 2115 A special case of ax-4 2076 without using ax-4 2076 or ax-17 1605. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.)
 |-  ( A. y  -.  x  =  x  ->  -.  x  =  x )
 
Theoremequidq 2116 equid 1646 with universal quantifier without using ax-4 2076 or ax-17 1605. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 A. y  x  =  x
 
Theoremequid1ALT 2117 Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. Alternate proof of equid1 2099 from older axioms ax-6o 2078 and ax-9o 2079. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  x  =  x
 
Theoremax10from10o 2118 Rederivation of ax-10 2081 from original version ax-10o 2080. See theorem ax10o 1894 for the derivation of ax-10o 2080 from ax-10 2081.

This theorem should not be referenced in any proof. Instead, use ax-10 2081 above so that uses of ax-10 2081 can be more easily identified, or use aecom-o 2092 when this form is needed for studies involving ax-10o 2080 and omitting ax-17 1605. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
Theoremnaecoms-o 2119 A commutation rule for distinct variable specifiers. Version of naecoms 1890 using ax-10o 2080. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  ph )   =>    |-  ( -.  A. y  y  =  x  ->  ph )
 
Theoremhbnae-o 2120 All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). Version of hbnae 1897 using ax-10o 2080. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  A. z  -.  A. x  x  =  y )
 
Theoremdvelimf-o 2121 Proof of dvelimh 1906 that uses ax-10o 2080 but not ax-11o 2082, ax-10 2081, or ax-11 1717. Version of dvelimh 1906 using ax-10o 2080 instead of ax10o 1894. (Contributed by NM, 12-Nov-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. z ps )   &    |-  (
 z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremdral2-o 2122 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Version of dral2 1908 using ax-10o 2080. (Contributed by NM, 27-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  (
 A. z ph  <->  A. z ps )
 )
 
Theoremaev-o 2123* A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 2085. Version of aev 1933 using ax-10o 2080. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  A. z  w  =  v )
 
Theoremax17eq 2124* Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-17 1605 considered as a metatheorem. Do not use it for later proofs - use ax-17 1605 instead, to avoid reference to the redundant axiom ax-16 2085.) (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  =  y 
 ->  A. z  x  =  y )
 
Theoremdveeq2-o 2125* Quantifier introduction when one pair of variables is distinct. Version of dveeq2 1882 using ax-11o 2082. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y ) )
 
Theoremdveeq2-o16 2126* Version of dveeq2 1882 using ax-16 2085 instead of ax-17 1605. TO DO: Recover proof from older set.mm to remove use of ax-17 1605. (Contributed by NM, 29-Apr-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y ) )
 
Theorema16g-o 2127* A generalization of axiom ax-16 2085. Version of a16g 1887 using ax-10o 2080. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph )
 )
 
Theoremdveeq1-o 2128* Quantifier introduction when one pair of variables is distinct. Version of dveeq1 1960 using ax-10o . (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z ) )
 
Theoremdveeq1-o16 2129* Version of dveeq1 1960 using ax-16 2085 instead of ax-17 1605. (Contributed by NM, 29-Apr-2008.) TO DO: Recover proof from older set.mm to remove use of ax-17 1605. (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z ) )
 
Theoremax17el 2130* Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-17 1605 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 )
 
Theoremax10-16 2131* This theorem shows that, given ax-16 2085, we can derive a version of ax-10 2081. However, it is weaker than ax-10 2081 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 27-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  z  ->  A. z  z  =  x )
 
Theoremdveel2ALT 2132* Version of dveel2 1962 using ax-16 2085 instead of ax-17 1605. (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 ) )
 
Theoremax11f 2133 Basis step for constructing a substitution instance of ax-11o 2082 without using ax-11o 2082. We can start with any formula  ph in which  x is not free. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )
 
Theoremax11eq 2134 Basis step for constructing a substitution instance of ax-11o 2082 without using ax-11o 2082. Atomic formula for equality predicate. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( z  =  w  ->  A. x ( x  =  y  ->  z  =  w ) ) ) )
 
Theoremax11el 2135 Basis step for constructing a substitution instance of ax-11o 2082 without using ax-11o 2082. Atomic formula for membership predicate. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( z  e.  w  ->  A. x ( x  =  y  ->  z  e.  w ) ) ) )
 
Theoremax11indn 2136 Induction step for constructing a substitution instance of ax-11o 2082 without using ax-11o 2082. Negation case. (Contributed by NM, 21-Jan-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  ->  ( -.  ph  ->  A. x ( x  =  y  ->  -.  ph ) ) ) )
 
Theoremax11indi 2137 Induction step for constructing a substitution instance of ax-11o 2082 without using ax-11o 2082. Implication case. (Contributed by NM, 21-Jan-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  ->  ( ps  ->  A. x ( x  =  y  ->  ps ) ) ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( ( ph  ->  ps )  ->  A. x ( x  =  y  ->  ( ph  ->  ps ) ) ) ) )
 
Theoremax11indalem 2138 Lemma for ax11inda2 2140 and ax11inda 2141. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  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 2139* A proof of ax11inda2 2140 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 2140* Induction step for constructing a substitution instance of ax-11o 2082 without using ax-11o 2082. Quantification case. When  z and  y are distinct, this theorem avoids the dummy variables needed by the more general ax11inda 2141. (Contributed by NM, 24-Jan-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 ) ) ) )
 
Theoremax11inda 2141* Induction step for constructing a substitution instance of ax-11o 2082 without using ax-11o 2082. Quantification case. (When  z and  y are distinct, ax11inda2 2140 may be used instead to avoid the dummy variable  w in the proof.) (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  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 ) ) ) )
 
Theoremax11v2-o 2142* Recovery of ax-11o 2082 from ax11v 2038 without using ax-11o 2082. The hypothesis is even weaker than ax11v 2038, with  z both distinct from  x and not occurring in  ph. Thus, the hypothesis provides an alternate axiom that can be used in place of ax-11o 2082. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  =  z 
 ->  ( ph  ->  A. x ( x  =  z  -> 
 ph ) ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  (
 ph  ->  A. x ( x  =  y  ->  ph )
 ) ) )
 
Theoremax11a2-o 2143* Derive ax-11o 2082 from a hypothesis in the form of ax-11 1717, without using ax-11 1717 or ax-11o 2082. The hypothesis is even weaker than ax-11 1717, with  z both distinct from  x and not occurring in  ph. Thus, the hypothesis provides an alternate axiom that can be used in place of ax-11 1717, if we also hvae ax-10o 2080 which this proof uses . As theorem ax11 2096 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-10 2081 instead of ax-10o 2080. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  =  z 
 ->  ( A. z ph  ->  A. x ( x  =  z  ->  ph )
 ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )
 
Theoremax10o-o 2144 Show that ax-10o 2080 can be derived from ax-10 2081. An open problem is whether this theorem can be derived from ax-10 2081 and the others when ax-11 1717 is replaced with ax-11o 2082. See theorem ax10from10o 2118 for the rederivation of ax-10 2081 from ax10o 1894.

Normally, ax10o 1894 should be used rather than ax-10o 2080 or ax10o-o 2144, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ph ) )
 
1.7  Existential uniqueness
 
Syntaxweu 2145 Extend wff definition to include existential uniqueness ("there exists a unique  x such that  ph").
 wff  E! x ph
 
Syntaxwmo 2146 Extend wff definition to include uniqueness ("there exists at most one  x such that  ph").
 wff  E* x ph
 
Theoremeujust 2147* A soundness justification theorem for df-eu 2149, 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 2148 for a proof that provides an example of how it can be achieved through the use of dvelim 1958. (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 2148* A soundness justification theorem for df-eu 2149, 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 1958. (Contributed by NM, 11-Mar-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( E. y A. x ( ph  <->  x  =  y
 ) 
 <-> 
 E. z A. x ( ph  <->  x  =  z
 ) )
 
Definitiondf-eu 2149* 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 2166, eu2 2170, eu3 2171, and eu5 2183 (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 2228). (Contributed by NM, 12-Aug-1993.)
 |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
 
Definitiondf-mo 2150 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 2176. For other possible definitions see mo2 2174 and mo4 2178. (Contributed by NM, 8-Mar-1995.)
 |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
 
Theoremeuf 2151* 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 2152 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 2153* 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 2154 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 2155 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 2156 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 2157 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 2158 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 2159 Deduction version of nfeu 2161. (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 2160 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 2161 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 2162 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)
 |- 
 F/ x ph   =>    |- 
 F/ x E* y ph
 
Theoremsb8eu 2163 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 2164 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 2165 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 2166* 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 2167* 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 2168 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 2169* 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 2170* 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 2171* 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 2172 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 2173* Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)
 |-  ( ( -.  ph  /\ 
 E! x ps )  ->  E! x ( ph  \/  ps ) )
 
Theoremmo2 2174* 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 2175* Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( [ y  /  x ] E* z ph  <->  E* z [ y  /  x ] ph )
 
Theoremmo3 2176* 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 2177* "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 2178* "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 2179 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 2180* 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 2181 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 2182 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 2183 Uniqueness in terms of "at most one." (Contributed by NM, 23-Mar-1995.)
 |-  ( E! x ph  <->  ( E. x ph  /\  E* x ph ) )
 
Theoremeu4 2184* 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 2185 Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.)
 |-  ( E! x ph  ->  E* x ph )
 
Theoremeumoi 2186 "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)
 |- 
 E! x ph   =>    |- 
 E* x ph
 
Theoremexmoeu 2187 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 2188 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 2189 Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.)
 |-  ( E* x ph  <->  ( E. x ph  ->  E* x ph ) )
 
Theoremexmo 2190 Something exists or at most one exists. (Contributed by NM, 8-Mar-1995.)
 |-  ( E. x ph  \/  E* x ph )
 
Theoremmoim 2191 "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 2192 "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 2193* Move antecedent outside of "at most one." (Contributed by NM, 28-Jul-1995.)
 |-  ( E* x (
 ph  ->  ps )  ->  ( ph  ->  E* x ps )
 )
 
Theoremeuimmo 2194 Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( E! x ps  ->  E* x ph ) )
 
Theoremeuim 2195 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 2196 "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 2197 "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 2198 "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 2199 "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 2200 "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 )
 )
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