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Theorem List for Metamath Proof Explorer - 2201-2300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaecom-o 2201 Commutation law for identical variable specifiers. The antecedent and consequent are true when  x and  y are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). Version of aecom 2002 using ax-10o 2189. Unlike ax10from10o 2227, this version does not require ax-17 1623. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
Theoremaecoms-o 2202 A commutation rule for identical variable specifiers. Version of aecoms 2003 using ax-10o . (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  ph )   =>    |-  ( A. y  y  =  x  ->  ph )
 
Theoremhbae-o 2203 All variables are effectively bound in an identical variable specifier. Version of hbae 2005 using ax-10o 2189. (Contributed by NM, 5-Aug-1993.) (Proof modification is disccouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )
 
Theoremdral1-o 2204 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 dral1 2022 using ax-10o 2189. (Contributed by NM, 24-Nov-1994.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  (
 A. x ph  <->  A. y ps )
 )
 
Theoremax11 2205 Rederivation of axiom ax-11 1757 from ax-11o 2191, ax-10o 2189, and other older axioms. The proof does not require ax-16 2194 or ax-17 1623. See theorem ax11o 2047 for the derivation of ax-11o 2191 from ax-11 1757.

An open problem is whether we can prove this using ax-10 2190 instead of ax-10o 2189.

This proof uses newer axioms ax-5 1563 and ax-9 1662, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-5o 2186 and ax-9o 2188. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  ( x  =  y 
 ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremax12from12o 2206 Derive ax-12 1946 from ax-12o 2192 and other older axioms.

This proof uses newer axioms ax-5 1563 and ax-9 1662, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-5o 2186 and ax-9o 2188. (Contributed by NM, 21-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 )
 
1.6.3  Legacy theorems using obsolete axioms

These theorems were mostly intended to study properties of the older axiom schemes and are not useful outside of this section. They should not be used outside of this section. They may be deleted when they are deemed to no longer be of interest.

 
Theoremax17o 2207* Axiom to quantify a variable over a formula in which it does not occur. Axiom C5 in [Megill] p. 444 (p. 11 of the preprint). Also appears as Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of [Monk2] p. 113.

(This theorem simply repeats ax-17 1623 so that we can include the following note, which applies only to the obsolete axiomatization.)

This axiom is logically redundant in the (logically complete) predicate calculus axiom system consisting of ax-gen 1552, ax-5o 2186, ax-4 2185, ax-7 1745, ax-6o 2187, ax-8 1683, ax-12o 2192, ax-9o 2188, ax-10o 2189, ax-13 1723, ax-14 1725, ax-15 2193, ax-11o 2191, and ax-16 2194: in that system, we can derive any instance of ax-17 1623 not containing wff variables by induction on formula length, using ax17eq 2233 and ax17el 2239 for the basis together hbn 1797, hbal 1747, and hbim 1832. However, if we omit this axiom, our development would be quite inconvenient since we could work only with specific instances of wffs containing no wff variables - this axiom introduces the concept of a set variable not occurring in a wff (as opposed to just two set variables being distinct). (Contributed by NM, 19-Aug-2017.) (New usage is discouraged.) (Proof modification discouraged.)

 |-  ( ph  ->  A. x ph )
 
Theoremequid1 2208 Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. This is often an axiom of equality in textbook systems, but we don't need it as an axiom since it can be proved from our other axioms (although the proof, as you can see below, is not as obvious as you might think). This proof uses only axioms without distinct variable conditions and thus requires no dummy variables. A simpler proof, similar to Tarki's, is possible if we make use of ax-17 1623; see the proof of equid 1684. See equid1ALT 2226 for an alternate proof. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  x  =  x
 
Theoremsps-o 2209 Generalization of antecedent. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   =>    |-  ( A. x ph  ->  ps )
 
Theoremhbequid 2210 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 2188.) (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 2211 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 1563, ax-8 1683, ax-12o 2192, and ax-gen 1552. This shows that this can be proved without ax9 1949, even though the theorem equid 1684 cannot be. A shorter proof using ax9 1949 is obtainable from equid 1684 and hbth 1558.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax9v 1663, which is used for the derivation of ax12o 1976, unless we consider ax-12o 2192 the starting axiom rather than ax-12 1946. (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 2212 Proof of a single axiom that can replace ax-4 2185 and ax-6o 2187. See ax46to4 2213 and ax46to6 2214 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 2213 Re-derivation of ax-4 2185 from ax46 2212. 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 2214 Re-derivation of ax-6o 2187 from ax46 2212. 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 2215 Proof of a single axiom that can replace both ax-6o 2187 and ax-7 1745. See ax67to6 2217 and ax67to7 2218 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 2216  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 2217 Re-derivation of ax-6o 2187 from ax67 2215. Note that ax-6o 2187 and ax-7 1745 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 2218 Re-derivation of ax-7 1745 from ax67 2215. Note that ax-6o 2187 and ax-7 1745 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 2219 Proof of a single axiom that can replace ax-4 2185, ax-6o 2187, and ax-7 1745 in a subsystem that includes these axioms plus ax-5o 2186 and ax-gen 1552 (and propositional calculus). See ax467to4 2220, ax467to6 2221, and ax467to7 2222 for the re-derivation of those axioms. This theorem extends the idea in Scott Fenton's ax46 2212. (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 2220 Re-derivation of ax-4 2185 from ax467 2219. 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 2221 Re-derivation of ax-6o 2187 from ax467 2219. Note that ax-6o 2187 and ax-7 1745 are not used by the re-derivation. The use of alimi 1565 (which uses ax-4 2185) is allowed since we have already proved ax467to4 2220. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x  -.  A. x ph  ->  ph )
 
Theoremax467to7 2222 Re-derivation of ax-7 1745 from ax467 2219. Note that ax-6o 2187 and ax-7 1745 are not used by the re-derivation. The use of alimi 1565 (which uses ax-4 2185) is allowed since we have already proved ax467to4 2220. (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 2223 equid 1684 with existential quantifier without using ax-4 2185 or ax-17 1623. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) (Proof modification is discouraged.)
 |- 
 -.  A. y  -.  x  =  x
 
Theoremax4sp1 2224 A special case of ax-4 2185 without using ax-4 2185 or ax-17 1623. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.)
 |-  ( A. y  -.  x  =  x  ->  -.  x  =  x )
 
Theoremequidq 2225 equid 1684 with universal quantifier without using ax-4 2185 or ax-17 1623. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 A. y  x  =  x
 
Theoremequid1ALT 2226 Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. Alternate proof of equid1 2208 from older axioms ax-6o 2187 and ax-9o 2188. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  x  =  x
 
Theoremax10from10o 2227 Rederivation of ax-10 2190 from original version ax-10o 2189. See theorem ax10o 2001 for the derivation of ax-10o 2189 from ax-10 2190.

This theorem should not be referenced in any proof. Instead, use ax-10 2190 above so that uses of ax-10 2190 can be more easily identified, or use aecom-o 2201 when this form is needed for studies involving ax-10o 2189 and omitting ax-17 1623. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
Theoremnaecoms-o 2228 A commutation rule for distinct variable specifiers. Version of naecoms 2004 using ax-10o 2189. (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 2229 All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). Version of hbnae 2007 using ax-10o 2189. (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 2230 Proof of dvelimh 2015 that uses ax-10o 2189 but not ax-11o 2191, ax-10 2190, or ax-11 1757. Version of dvelimh 2015 using ax-10o 2189 instead of ax10o 2001. (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 2231 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 2020 using ax-10o 2189. (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 2232* A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 2194. Version of aev 2011 using ax-10o 2189. (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 2233* Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-17 1623 considered as a metatheorem. Do not use it for later proofs - use ax-17 1623 instead, to avoid reference to the redundant axiom ax-16 2194.) (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  =  y 
 ->  A. z  x  =  y )
 
Theoremdveeq2-o 2234* Quantifier introduction when one pair of variables is distinct. Version of dveeq2 2019 using ax-11o 2191. (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 2235* Version of dveeq2 2019 using ax-16 2194 instead of ax-17 1623. TO DO: Recover proof from older set.mm to remove use of ax-17 1623. (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 2236* A generalization of axiom ax-16 2194. Version of a16g 2012 using ax-10o 2189. (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 2237* Quantifier introduction when one pair of variables is distinct. Version of dveeq1 1987 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 2238* Version of dveeq1 1987 using ax-16 2194 instead of ax-17 1623. (Contributed by NM, 29-Apr-2008.) TO DO: Recover proof from older set.mm to remove use of ax-17 1623. (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z ) )
 
Theoremax17el 2239* Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-17 1623 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 2240* This theorem shows that, given ax-16 2194, we can derive a version of ax-10 2190. However, it is weaker than ax-10 2190 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 2241* Version of dveel2 2069 using ax-16 2194 instead of ax-17 1623. (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 2242 Basis step for constructing a substitution instance of ax-11o 2191 without using ax-11o 2191. 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 2243 Basis step for constructing a substitution instance of ax-11o 2191 without using ax-11o 2191. 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 2244 Basis step for constructing a substitution instance of ax-11o 2191 without using ax-11o 2191. 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 2245 Induction step for constructing a substitution instance of ax-11o 2191 without using ax-11o 2191. 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 2246 Induction step for constructing a substitution instance of ax-11o 2191 without using ax-11o 2191. 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 2247 Lemma for ax11inda2 2249 and ax11inda 2250. (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 2248* A proof of ax11inda2 2249 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 2249* Induction step for constructing a substitution instance of ax-11o 2191 without using ax-11o 2191. Quantification case. When  z and  y are distinct, this theorem avoids the dummy variables needed by the more general ax11inda 2250. (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 2250* Induction step for constructing a substitution instance of ax-11o 2191 without using ax-11o 2191. Quantification case. (When  z and  y are distinct, ax11inda2 2249 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 2251* Recovery of ax-11o 2191 from ax11v 2145 without using ax-11o 2191. The hypothesis is even weaker than ax11v 2145, 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 2191. (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 2252* Derive ax-11o 2191 from a hypothesis in the form of ax-11 1757, without using ax-11 1757 or ax-11o 2191. The hypothesis is even weaker than ax-11 1757, 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 1757, if we also hvae ax-10o 2189 which this proof uses . As theorem ax11 2205 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-10 2190 instead of ax-10o 2189. (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 2253 Show that ax-10o 2189 can be derived from ax-10 2190. An open problem is whether this theorem can be derived from ax-10 2190 and the others when ax-11 1757 is replaced with ax-11o 2191. See theorem ax10from10o 2227 for the rederivation of ax-10 2190 from ax10o 2001.

Normally, ax10o 2001 should be used rather than ax-10o 2189 or ax10o-o 2253, 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 2254 Extend wff definition to include existential uniqueness ("there exists a unique  x such that  ph").
 wff  E! x ph
 
Syntaxwmo 2255 Extend wff definition to include uniqueness ("there exists at most one  x such that  ph").
 wff  E* x ph
 
Theoremeujust 2256* A soundness justification theorem for df-eu 2258, 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 2257 for a proof that provides an example of how it can be achieved through the use of dvelim 2066. (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 2257* A soundness justification theorem for df-eu 2258, 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 2066. (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 2258* 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 2275, eu2 2279, eu3 2280, and eu5 2292 (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 2337). (Contributed by NM, 12-Aug-1993.)
 |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
 
Definitiondf-mo 2259 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 2285. For other possible definitions see mo2 2283 and mo4 2287. (Contributed by NM, 8-Mar-1995.)
 |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
 
Theoremeuf 2260* 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 2261 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 2262* 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 2263 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 2264 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 2265 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 2266 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 2267 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 2268 Deduction version of nfeu 2270. (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 2269 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 2270 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 2271 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)
 |- 
 F/ x ph   =>    |- 
 F/ x E* y ph
 
Theoremsb8eu 2272 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 2273 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 2274 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 2275* 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 2276* 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 2277 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 2278* 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 2279* 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 2280* 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 2281 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 2282* Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)
 |-  ( ( -.  ph  /\ 
 E! x ps )  ->  E! x ( ph  \/  ps ) )
 
Theoremmo2 2283* 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 2284* Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( [ y  /  x ] E* z ph  <->  E* z [ y  /  x ] ph )
 
Theoremmo3 2285* 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 2286* "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 2287* "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 2288 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 2289* 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 2290 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 2291 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 2292 Uniqueness in terms of "at most one." (Contributed by NM, 23-Mar-1995.)
 |-  ( E! x ph  <->  ( E. x ph  /\  E* x ph ) )
 
Theoremeu4 2293* 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 2294 Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.)
 |-  ( E! x ph  ->  E* x ph )
 
Theoremeumoi 2295 "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)
 |- 
 E! x ph   =>    |- 
 E* x ph
 
Theoremexmoeu 2296 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 2297 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 2298 Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.)
 |-  ( E* x ph  <->  ( E. x ph  ->  E* x ph ) )
 
Theoremexmo 2299 Something exists or at most one exists. (Contributed by NM, 8-Mar-1995.)
 |-  ( E. x ph  \/  E* x ph )
 
Theoremmoim 2300 "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 ) )
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