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Theorem List for Metamath Proof Explorer - 2201-2300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsbel2x 2201* Elimination of double substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbal1 2202* A theorem used in elimination of disjoint variable restriction on and by replacing it with a distinctor . (Contributed by NM, 5-Aug-1993.)

Theoremsbal 2203* Move universal quantifier in and out of substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbex 2204* Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.)

Theoremsbalv 2205* Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)

Theoremexsb 2206* An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.)

TheoremexsbOLD 2207* An equivalent expression for existence. Obsolete as of 19-Jun-2017. (Contributed by NM, 2-Feb-2005.) (New usage is discouraged.) (Proof modification is discouraged.)

Theorem2exsb 2208* An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.)

TheoremdvelimALT 2209* Version of dvelim 2069 that doesn't use ax-10 2216. (See dvelimh 2067 for a version that doesn't use ax-11 1761.) (Contributed by NM, 17-May-2008.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremsbal2 2210* Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.)

1.6  Predicate calculus with equality: Older axiomatization (1 rule, 14 schemes)

The "metalogical completeness theorem", Theorem 9.7 of [Megill] p. 448, uses a different but (logically and metalogically) equivalent set of axiom schemes for its proof. In order to show that our axiomatization is also metalogically complete, we derive the axiom schemes of that paper in this section (or mention where they are derived, if they have already been derived as therorems above). Additionally, we re-derive our axiomatization from the one in the paper, showing that the two systems are equivalent.

The 14 predicate calculus axioms used by the paper are ax-5o 2212, ax-4 2211, ax-7 1749, ax-6o 2213, ax-8 1687, ax-12o 2218, ax-9o 2214, ax-10o 2215, ax-13 1727, ax-14 1729, ax-15 2219, ax-11o 2217, ax-16 2220, and ax-17 1626. Like ours, it includes the rule of generalization (ax-gen 1555).

The ones we need to prove from our axioms are ax-5o 2212, ax-4 2211, ax-6o 2213, ax-12o 2218, ax-9o 2214, ax-10o 2215, ax-15 2219, ax-11o 2217, and ax-16 2220. The theorems showing the derivations of those axioms, which have all been proved earlier, are ax5o 1765, ax4 2221 (also called sp 1763), ax6o 1766, ax12o 2010, ax9o 1954, ax10o 2038, ax15 2101, ax11o 2077, ax16 2129, and ax10 2025. In addition, ax-10 2216 was an intermediate axiom we adopted at one time, and we show its proof in this section as ax10from10o 2253.

This section also includes a few miscellaneous legacy theorems such as hbequid 2236 use the older axioms.

Note: The axioms and theorems in this section should not be used outside of this section. Inside this section, we may use the external axioms ax-gen 1555, ax-17 1626, ax-8 1687, ax-9 1666, ax-13 1727, and ax-14 1729 since they are common to both our current and the older axiomatizations. (These are the ones that were never revised.)

The following newer axioms may NOT be used in this section until we have proved them from the older axioms: ax-5 1566, ax-6 1744, ax-9 1666, ax-11 1761, and ax-12 1950. However, once we have rederived an axiom (e.g. theorem ax5 2222 for axiom ax-5 1566), we may make use of theorems outside of this section that make use of the rederived axiom (e.g. we may use theorem alimi 1568, which uses ax-5 1566, after proving ax5 2222).

1.6.1  Obsolete schemes ax-5o ax-4 ax-6o ax-9o ax-10o ax-10 ax-11o ax-12o ax-15 ax-16

These older axiom schemes are obsolete and should not be used outside of this section. They are proved above as theorems ax5o , sp 1763, ax6o 1766, ax9o 1954, ax10o 2038, ax10 2025, ax11o 2077, ax12o 2010, ax15 2101, and ax16 2129.

Axiomax-4 2211 Axiom of Specialization. A quantified wff implies the wff without a quantifier (i.e. an instance, or special case, of the generalized wff). In other words if something is true for all , it is true for any specific (that would typically occur as a free variable in the wff substituted for ). (A free variable is one that does not occur in the scope of a quantifier: and are both free in , but only is free in .) Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski] p. 67 (under his system S2, defined in the last paragraph on p. 77).

Note that the converse of this axiom does not hold in general, but a weaker inference form of the converse holds and is expressed as rule ax-gen 1555. Conditional forms of the converse are given by ax-12 1950, ax-15 2219, ax-16 2220, and ax-17 1626.

Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from for the special case. For use, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution - see stdpc4 2087.

An interesting alternate axiomatization uses ax467 2245 and ax-5o 2212 in place of ax-4 2211, ax-5 1566, ax-6 1744, and ax-7 1749.

This axiom is obsolete and should no longer be used. It is proved above as theorem sp 1763. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Axiomax-5o 2212 Axiom of Quantified Implication. This axiom moves a quantifier from outside to inside an implication, quantifying . Notice that must not be a free variable in the antecedent of the quantified implication, and we express this by binding to "protect" the axiom from a containing a free . Axiom scheme C4' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Lemma 5 of [Monk2] p. 108 and Axiom 5 of [Mendelson] p. 69.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax5o 1765. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Axiomax-6o 2213 Axiom of Quantified Negation. This axiom is used to manipulate negated quantifiers. Equivalent to axiom scheme C7' in [Megill] p. 448 (p. 16 of the preprint). An alternate axiomatization could use ax467 2245 in place of ax-4 2211, ax-6o 2213, and ax-7 1749.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax6o 1766. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Axiomax-9o 2214 A variant of ax9 1953. Axiom scheme C10' in [Megill] p. 448 (p. 16 of the preprint).

This axiom is obsolete and should no longer be used. It is proved above as theorem ax9o 1954. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Axiomax-10o 2215 Axiom ax-10o 2215 ("o" for "old") was the original version of ax-10 2216, before it was discovered (in May 2008) that the shorter ax-10 2216 could replace it. It appears as Axiom scheme C11' in [Megill] p. 448 (p. 16 of the preprint).

This axiom is obsolete and should no longer be used. It is proved above as theorem ax10o 2038. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Axiomax-10 2216 Axiom of Quantifier Substitution. One of the equality and substitution axioms of predicate calculus with equality. Appears as Lemma L12 in [Megill] p. 445 (p. 12 of the preprint).

The original version of this axiom was ax-10o 2215 ("o" for "old") and was replaced with this shorter ax-10 2216 in May 2008. The old axiom is proved from this one as theorem ax10o 2038. Conversely, this axiom is proved from ax-10o 2215 as theorem ax10from10o 2253.

This axiom was proved redundant in July 2015. See theorem ax10 2025.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax10 2025. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)

Axiomax-11o 2217 Axiom ax-11o 2217 ("o" for "old") was the original version of ax-11 1761, before it was discovered (in Jan. 2007) that the shorter ax-11 1761 could replace it. It appears as Axiom scheme C15' in [Megill] p. 448 (p. 16 of the preprint). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases. To understand this theorem more easily, think of " ..." as informally meaning "if and are distinct variables then..." The antecedent becomes false if the same variable is substituted for and , ensuring the theorem is sound whenever this is the case. In some later theorems, we call an antecedent of the form a "distinctor."

Interestingly, if the wff expression substituted for contains no wff variables, the resulting statement can be proved without invoking this axiom. This means that even though this axiom is metalogically independent from the others, it is not logically independent. Specifically, we can prove any wff-variable-free instance of axiom ax-11o 2217 (from which the ax-11 1761 instance follows by theorem ax11 2231.) The proof is by induction on formula length, using ax11eq 2269 and ax11el 2270 for the basis steps and ax11indn 2271, ax11indi 2272, and ax11inda 2276 for the induction steps. (This paragraph is true provided we use ax-10o 2215 in place of ax-10 2216.)

This axiom is obsolete and should no longer be used. It is proved above as theorem ax11o 2077. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Axiomax-12o 2218 Axiom of Quantifier Introduction. One of the equality and substitution axioms of predicate calculus with equality. Informally, it says that whenever is distinct from and , and is true, then quantified with is also true. In other words, is irrelevant to the truth of . Axiom scheme C9' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax12o 2010. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Axiomax-15 2219 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 1626; see theorem ax15 2101. Alternately, ax-17 1626 becomes unnecessary in principle with this axiom, but we lose the more powerful metalogic afforded by ax-17 1626. We retain ax-15 2219 here to provide completeness for systems with the simpler metalogic that results from omitting ax-17 1626, which might be easier to study for some theoretical purposes.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax15 2101. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Axiomax-16 2220* Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1626 to be a metatheorem and not an axiom). Axiom scheme C16' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases. It is a somewhat bizarre axiom since the antecedent is always false in set theory (see dtru 4382), but nonetheless it is technically necessary as you can see from its uses.

This axiom is redundant if we include ax-17 1626; see theorem ax16 2129. Alternately, ax-17 1626 becomes logically redundant in the presence of this axiom, but without ax-17 1626 we lose the more powerful metalogic that results from being able to express the concept of a set variable not occurring in a wff (as opposed to just two set variables being distinct). We retain ax-16 2220 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-17 1626, which might be easier to study for some theoretical purposes.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax16 2129. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

1.6.2  Rederive new axioms from old: ax5 , ax6 , ax9from9o , ax11 , ax12from12o

Theorems ax11 2231 and ax12from12o 2232 require some intermediate theorems that are included in this section.

Theoremax4 2221 This theorem repeats sp 1763 under the name ax4 2221, so that the metamath program's "verify markup" command will check that it matches axiom scheme ax-4 2211. It is preferred that references to this theorem use the name sp 1763. (Contributed by NM, 18-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremax5 2222 Rederivation of axiom ax-5 1566 from ax-5o 2212 and other older axioms. See ax5o 1765 for the derivation of ax-5o 2212 from ax-5 1566. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax6 2223 Rederivation of axiom ax-6 1744 from ax-6o 2213 and other older axioms. See ax6o 1766 for the derivation of ax-6o 2213 from ax-6 1744. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax9from9o 2224 Rederivation of axiom ax-9 1666 from ax-9o 2214 and other older axioms. See ax9o 1954 for the derivation of ax-9o 2214 from ax-9 1666. Lemma L18 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremhba1-o 2225 is not free in . Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Theorema5i-o 2226 Inference version of ax-5o 2212. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Theoremaecom-o 2227 Commutation law for identical variable specifiers. The antecedent and consequent are true when and are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). Version of aecom 2035 using ax-10o 2215. Unlike ax10from10o 2253, this version does not require ax-17 1626. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Theoremaecoms-o 2228 A commutation rule for identical variable specifiers. Version of aecoms 2036 using ax-10o . (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Theoremhbae-o 2229 All variables are effectively bound in an identical variable specifier. Version of hbae 2040 using ax-10o 2215. (Contributed by NM, 5-Aug-1993.) (Proof modification is disccouraged.) (New usage is discouraged.)

Theoremdral1-o 2230 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 2053 using ax-10o 2215. (Contributed by NM, 24-Nov-1994.) (New usage is discouraged.)

Theoremax11 2231 Rederivation of axiom ax-11 1761 from ax-11o 2217, ax-10o 2215, and other older axioms. The proof does not require ax-16 2220 or ax-17 1626. See theorem ax11o 2077 for the derivation of ax-11o 2217 from ax-11 1761.

An open problem is whether we can prove this using ax-10 2216 instead of ax-10o 2215.

This proof uses newer axioms ax-5 1566 and ax-9 1666, 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 2212 and ax-9o 2214. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax12from12o 2232 Derive ax-12 1950 from ax-12o 2218 and other older axioms.

This proof uses newer axioms ax-5 1566 and ax-9 1666, 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 2212 and ax-9o 2214. (Contributed by NM, 21-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

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 2233* 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 1626 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 1555, ax-5o 2212, ax-4 2211, ax-7 1749, ax-6o 2213, ax-8 1687, ax-12o 2218, ax-9o 2214, ax-10o 2215, ax-13 1727, ax-14 1729, ax-15 2219, ax-11o 2217, and ax-16 2220: in that system, we can derive any instance of ax-17 1626 not containing wff variables by induction on formula length, using ax17eq 2259 and ax17el 2265 for the basis together hbn 1801, hbal 1751, and hbim 1836. 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.)

Theoremequid1 2234 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 1626; see the proof of equid 1688. See equid1ALT 2252 for an alternate proof. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsps-o 2235 Generalization of antecedent. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremhbequid 2236 Bound-variable hypothesis builder for . This theorem tells us that any variable, including , is effectively not free in , even though is technically free according to the traditional definition of free variable. (The proof does not use ax-9o 2214.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnfequid-o 2237 Bound-variable hypothesis builder for . This theorem tells us that any variable, including , is effectively not free in , even though is technically free according to the traditional definition of free variable. (The proof uses only ax-5 1566, ax-8 1687, ax-12o 2218, and ax-gen 1555. This shows that this can be proved without ax9 1953, even though the theorem equid 1688 cannot be. A shorter proof using ax9 1953 is obtainable from equid 1688 and hbth 1561.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax9v 1667, which is used for the derivation of ax12o 2010, unless we consider ax-12o 2218 the starting axiom rather than ax-12 1950. (Contributed by NM, 13-Jan-2011.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax46 2238 Proof of a single axiom that can replace ax-4 2211 and ax-6o 2213. See ax46to4 2239 and ax46to6 2240 for the re-derivation of those axioms. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax46to4 2239 Re-derivation of ax-4 2211 from ax46 2238. Only propositional calculus is used for the re-derivation. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax46to6 2240 Re-derivation of ax-6o 2213 from ax46 2238. Only propositional calculus is used for the re-derivation. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax67 2241 Proof of a single axiom that can replace both ax-6o 2213 and ax-7 1749. See ax67to6 2243 and ax67to7 2244 for the re-derivation of those axioms. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnfa1-o 2242 is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax67to6 2243 Re-derivation of ax-6o 2213 from ax67 2241. Note that ax-6o 2213 and ax-7 1749 are not used by the re-derivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax67to7 2244 Re-derivation of ax-7 1749 from ax67 2241. Note that ax-6o 2213 and ax-7 1749 are not used by the re-derivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax467 2245 Proof of a single axiom that can replace ax-4 2211, ax-6o 2213, and ax-7 1749 in a subsystem that includes these axioms plus ax-5o 2212 and ax-gen 1555 (and propositional calculus). See ax467to4 2246, ax467to6 2247, and ax467to7 2248 for the re-derivation of those axioms. This theorem extends the idea in Scott Fenton's ax46 2238. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax467to4 2246 Re-derivation of ax-4 2211 from ax467 2245. Only propositional calculus is used by the re-derivation. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax467to6 2247 Re-derivation of ax-6o 2213 from ax467 2245. Note that ax-6o 2213 and ax-7 1749 are not used by the re-derivation. The use of alimi 1568 (which uses ax-4 2211) is allowed since we have already proved ax467to4 2246. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax467to7 2248 Re-derivation of ax-7 1749 from ax467 2245. Note that ax-6o 2213 and ax-7 1749 are not used by the re-derivation. The use of alimi 1568 (which uses ax-4 2211) is allowed since we have already proved ax467to4 2246. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremequidqe 2249 equid 1688 with existential quantifier without using ax-4 2211 or ax-17 1626. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax4sp1 2250 A special case of ax-4 2211 without using ax-4 2211 or ax-17 1626. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremequidq 2251 equid 1688 with universal quantifier without using ax-4 2211 or ax-17 1626. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremequid1ALT 2252 Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. Alternate proof of equid1 2234 from older axioms ax-6o 2213 and ax-9o 2214. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax10from10o 2253 Rederivation of ax-10 2216 from original version ax-10o 2215. See theorem ax10o 2038 for the derivation of ax-10o 2215 from ax-10 2216.

This theorem should not be referenced in any proof. Instead, use ax-10 2216 above so that uses of ax-10 2216 can be more easily identified, or use aecom-o 2227 when this form is needed for studies involving ax-10o 2215 and omitting ax-17 1626. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnaecoms-o 2254 A commutation rule for distinct variable specifiers. Version of naecoms 2037 using ax-10o 2215. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremhbnae-o 2255 All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). Version of hbnae 2043 using ax-10o 2215. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdvelimf-o 2256 Proof of dvelimh 2067 that uses ax-10o 2215 but not ax-11o 2217, ax-10 2216, or ax-11 1761. Version of dvelimh 2067 using ax-10o 2215 instead of ax10o 2038. (Contributed by NM, 12-Nov-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdral2-o 2257 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 2051 using ax-10o 2215. (Contributed by NM, 27-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremaev-o 2258* A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 2220. Version of aev 2047 using ax-10o 2215. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax17eq 2259* Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-17 1626 considered as a metatheorem. Do not use it for later proofs - use ax-17 1626 instead, to avoid reference to the redundant axiom ax-16 2220.) (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdveeq2-o 2260* Quantifier introduction when one pair of variables is distinct. Version of dveeq2 2073 using ax-11o 2217. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdveeq2-o16 2261* Version of dveeq2 2073 using ax-16 2220 instead of ax-17 1626. TO DO: Recover proof from older set.mm to remove use of ax-17 1626. (Contributed by NM, 29-Apr-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorema16g-o 2262* A generalization of axiom ax-16 2220. Version of a16g 2048 using ax-10o 2215. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdveeq1-o 2263* Quantifier introduction when one pair of variables is distinct. Version of dveeq1 2021 using ax-10o . (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdveeq1-o16 2264* Version of dveeq1 2021 using ax-16 2220 instead of ax-17 1626. (Contributed by NM, 29-Apr-2008.) TO DO: Recover proof from older set.mm to remove use of ax-17 1626. (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax17el 2265* Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-17 1626 considered as a metatheorem.) (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax10-16 2266* This theorem shows that, given ax-16 2220, we can derive a version of ax-10 2216. However, it is weaker than ax-10 2216 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 27-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdveel2ALT 2267* Version of dveel2 2100 using ax-16 2220 instead of ax-17 1626. (Contributed by NM, 10-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11f 2268 Basis step for constructing a substitution instance of ax-11o 2217 without using ax-11o 2217. We can start with any formula in which is not free. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11eq 2269 Basis step for constructing a substitution instance of ax-11o 2217 without using ax-11o 2217. Atomic formula for equality predicate. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11el 2270 Basis step for constructing a substitution instance of ax-11o 2217 without using ax-11o 2217. Atomic formula for membership predicate. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11indn 2271 Induction step for constructing a substitution instance of ax-11o 2217 without using ax-11o 2217. Negation case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11indi 2272 Induction step for constructing a substitution instance of ax-11o 2217 without using ax-11o 2217. Implication case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11indalem 2273 Lemma for ax11inda2 2275 and ax11inda 2276. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11inda2ALT 2274* A proof of ax11inda2 2275 that is slightly more direct. (Contributed by NM, 4-May-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11inda2 2275* Induction step for constructing a substitution instance of ax-11o 2217 without using ax-11o 2217. Quantification case. When and are distinct, this theorem avoids the dummy variables needed by the more general ax11inda 2276. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11inda 2276* Induction step for constructing a substitution instance of ax-11o 2217 without using ax-11o 2217. Quantification case. (When and are distinct, ax11inda2 2275 may be used instead to avoid the dummy variable in the proof.) (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11v2-o 2277* Recovery of ax-11o 2217 from ax11v 2171 without using ax-11o 2217. The hypothesis is even weaker than ax11v 2171, with both distinct from and not occurring in . Thus, the hypothesis provides an alternate axiom that can be used in place of ax-11o 2217. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11a2-o 2278* Derive ax-11o 2217 from a hypothesis in the form of ax-11 1761, without using ax-11 1761 or ax-11o 2217. The hypothesis is even weaker than ax-11 1761, with both distinct from and not occurring in . Thus, the hypothesis provides an alternate axiom that can be used in place of ax-11 1761, if we also hvae ax-10o 2215 which this proof uses . As theorem ax11 2231 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-10 2216 instead of ax-10o 2215. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax10o-o 2279 Show that ax-10o 2215 can be derived from ax-10 2216. An open problem is whether this theorem can be derived from ax-10 2216 and the others when ax-11 1761 is replaced with ax-11o 2217. See theorem ax10from10o 2253 for the rederivation of ax-10 2216 from ax10o 2038.

Normally, ax10o 2038 should be used rather than ax-10o 2215 or ax10o-o 2279, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

1.7  Existential uniqueness

Syntaxweu 2280 Extend wff definition to include existential uniqueness ("there exists a unique such that ").

Syntaxwmo 2281 Extend wff definition to include uniqueness ("there exists at most one such that ").

Theoremeujust 2282* A soundness justification theorem for df-eu 2284, showing that the definition is equivalent to itself with its dummy variable renamed. Note that and needn't be distinct variables. See eujustALT 2283 for a proof that provides an example of how it can be achieved through the use of dvelim 2069. (Contributed by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

TheoremeujustALT 2283* A soundness justification theorem for df-eu 2284, showing that the definition is equivalent to itself with its dummy variable renamed. Note that and 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 2069. (Contributed by NM, 11-Mar-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

Definitiondf-eu 2284* Define existential uniqueness, i.e. "there exists exactly one such that ." Definition 10.1 of [BellMachover] p. 97; also Definition *14.02 of [WhiteheadRussell] p. 175. Other possible definitions are given by eu1 2301, eu2 2305, eu3 2306, and eu5 2318 (which in some cases we show with a hypothesis in place of a distinct variable condition on and ). Double uniqueness is tricky: does not mean "exactly one and one " (see 2eu4 2363). (Contributed by NM, 12-Aug-1993.)

Definitiondf-mo 2285 Define "there exists at most one such that ." Here we define it in terms of existential uniqueness. Notation of [BellMachover] p. 460, whose definition we show as mo3 2311. For other possible definitions see mo2 2309 and mo4 2313. (Contributed by NM, 8-Mar-1995.)

Theoremeuf 2286* A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.)

Theoremeubid 2287 Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)

Theoremeubidv 2288* Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)

Theoremeubii 2289 Introduce uniqueness quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremnfeu1 2290 Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfmo1 2291 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfeud2 2292 Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.)

Theoremnfmod2 2293 Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.)

Theoremnfeud 2294 Deduction version of nfeu 2296. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfmod 2295 Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.)

Theoremnfeu 2296 Bound-variable hypothesis builder for "at most one." Note that and needn't be distinct (this makes the proof more difficult). (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfmo 2297 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)

Theoremsb8eu 2298 Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremsb8mo 2299 Variable substitution in uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theoremcbveu 2300 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

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