Home Metamath Proof ExplorerTheorem List (p. 22 of 320) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-21553) Hilbert Space Explorer (21554-23076) Users' Mathboxes (23077-31972)

Theorem List for Metamath Proof Explorer - 2101-2200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsbalv 2101* Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)

Theoremexsb 2102* An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.)

TheoremexsbOLD 2103* An equivalent expression for existence. Obsolete as of 19-Jun-2017. (Contributed by NM, 2-Feb-2005.) (New usage is discouraged.)

Theorem2exsb 2104* An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.)

TheoremdvelimALT 2105* Version of dvelim 1988 that doesn't use ax-10 2112. (See dvelimh 1936 for a version that doesn't use ax-11 1732.) (Contributed by NM, 17-May-2008.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremsbal2 2106* 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 2108, ax-4 2107, ax-7 1725, ax-6o 2109, ax-8 1666, ax-12o 2114, ax-9o 2110, ax-10o 2111, ax-13 1703, ax-14 1705, ax-15 2115, ax-11o 2113, ax-16 2116, and ax-17 1607. Like ours, it includes the rule of generalization (ax-gen 1537).

The ones we need to prove from our axioms are ax-5o 2108, ax-4 2107, ax-6o 2109, ax-12o 2114, ax-9o 2110, ax-10o 2111, ax-15 2115, ax-11o 2113, and ax-16 2116. The theorems showing the derivations of those axioms, which have all been proved earlier, are ax5o 1738, ax4 2117 (also called sp 1733), ax6o 1740, ax12o 1906, ax9o 1922, ax10o 1924, ax15 1993, ax11o 1966, ax16 2017, and ax10 1916. In addition, ax-10 2112 was an intermediate axiom we adopted at one time, and we show its proof in this section as ax10from10o 2149.

This section also includes a few miscellaneous legacy theorems such as hbequid 2132 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 1537, ax-17 1607, ax-8 1666, ax-9 1645, ax-13 1703, and ax-14 1705 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 1548, ax-6 1720, ax-9 1645, ax-11 1732, and ax-12 1897. However, once we have rederived an axiom (e.g. theorem ax5 2118 for axiom ax-5 1548), we may make use of theorems outside of this section that make use of the rederived axiom (e.g. we may use theorem alimi 1550, which uses ax-5 1548, after proving ax5 2118).

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 1733, ax6o 1740, ax9o 1922, ax10o 1924, ax10 1916, ax11o 1966, ax12o 1906, ax15 1993, and ax16 2017.

Axiomax-4 2107 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 .) This is one of the axioms of what we call "pure" predicate calculus (ax-4 2107 through ax-7 1725 plus rule ax-gen 1537). 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 1537. Conditional forms of the converse are given by ax-12 1897, ax-15 2115, ax-16 2116, and ax-17 1607.

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 1996.

An interesting alternate axiomatization uses ax467 2141 and ax-5o 2108 in place of ax-4 2107, ax-5 1548, ax-6 1720, and ax-7 1725.

This axiom is obsolete and should no longer be used. It is proved above as theorem sp 1733. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Axiomax-5o 2108 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 . One of the 4 axioms of "pure" predicate calculus. 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 1738. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Axiomax-6o 2109 Axiom of Quantified Negation. This axiom is used to manipulate negated quantifiers. One of the 4 axioms of pure predicate calculus. Equivalent to axiom scheme C7' in [Megill] p. 448 (p. 16 of the preprint). An alternate axiomatization could use ax467 2141 in place of ax-4 2107, ax-6o 2109, and ax-7 1725.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax6o 1740. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Axiomax-9o 2110 A variant of ax9 1921. 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 1922. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Axiomax-10o 2111 Axiom ax-10o 2111 ("o" for "old") was the original version of ax-10 2112, before it was discovered (in May 2008) that the shorter ax-10 2112 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 1924. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Axiomax-10 2112 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 2111 ("o" for "old") and was replaced with this shorter ax-10 2112 in May 2008. The old axiom is proved from this one as theorem ax10o 1924. Conversely, this axiom is proved from ax-10o 2111 as theorem ax10from10o 2149.

This axiom was proved redundant in July 2015. See theorem ax10 1916.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax10 1916. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)

Axiomax-11o 2113 Axiom ax-11o 2113 ("o" for "old") was the original version of ax-11 1732, before it was discovered (in Jan. 2007) that the shorter ax-11 1732 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 2113 (from which the ax-11 1732 instance follows by theorem ax11 2127.) The proof is by induction on formula length, using ax11eq 2165 and ax11el 2166 for the basis steps and ax11indn 2167, ax11indi 2168, and ax11inda 2172 for the induction steps. (This paragraph is true provided we use ax-10o 2111 in place of ax-10 2112.)

This axiom is obsolete and should no longer be used. It is proved above as theorem ax11o 1966. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Axiomax-12o 2114 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 1906. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Axiomax-15 2115 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 1607; see theorem ax15 1993. Alternately, ax-17 1607 becomes unnecessary in principle with this axiom, but we lose the more powerful metalogic afforded by ax-17 1607. We retain ax-15 2115 here to provide completeness for systems with the simpler metalogic that results from omitting ax-17 1607, 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 1993. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Axiomax-16 2116* Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1607 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 4238), but nonetheless it is technically necessary as you can see from its uses.

This axiom is redundant if we include ax-17 1607; see theorem ax16 2017. Alternately, ax-17 1607 becomes logically redundant in the presence of this axiom, but without ax-17 1607 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 2116 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-17 1607, 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 2017. (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 2127 and ax12from12o 2128 require some intermediate theorems that are included in this section.

Theoremax4 2117 This theorem repeats sp 1733 under the name ax4 2117, so that the metamath program's "verify markup" command will check that it matches axiom scheme ax-4 2107. It is preferred that references to this theorem use the name sp 1733. (Contributed by NM, 18-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremax5 2118 Rederivation of axiom ax-5 1548 from ax-5o 2108 and other older axioms. See ax5o 1738 for the derivation of ax-5o 2108 from ax-5 1548. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax6 2119 Rederivation of axiom ax-6 1720 from ax-6o 2109 and other older axioms. See ax6o 1740 for the derivation of ax-6o 2109 from ax-6 1720. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax9from9o 2120 Rederivation of axiom ax-9 1645 from ax-9o 2110 and other older axioms. See ax9o 1922 for the derivation of ax-9o 2110 from ax-9 1645. 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 2121 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 2122 Inference version of ax-5o 2108. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Theoremaecom-o 2123 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 1918 using ax-10o 2111. Unlike ax10from10o 2149, this version does not require ax-17 1607. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Theoremaecoms-o 2124 A commutation rule for identical variable specifiers. Version of aecoms 1919 using ax-10o . (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Theoremhbae-o 2125 All variables are effectively bound in an identical variable specifier. Version of hbae 1925 using ax-10o 2111. (Contributed by NM, 5-Aug-1993.) (Proof modification is disccouraged.) (New usage is discouraged.)

Theoremdral1-o 2126 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 1937 using ax-10o 2111. (Contributed by NM, 24-Nov-1994.) (New usage is discouraged.)

Theoremax11 2127 Rederivation of axiom ax-11 1732 from ax-11o 2113, ax-10o 2111, and other older axioms. The proof does not require ax-16 2116 or ax-17 1607. See theorem ax11o 1966 for the derivation of ax-11o 2113 from ax-11 1732.

An open problem is whether we can prove this using ax-10 2112 instead of ax-10o 2111.

This proof uses newer axioms ax-5 1548 and ax-9 1645, 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 2108 and ax-9o 2110. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax12from12o 2128 Derive ax-12 1897 from ax-12o 2114 and other older axioms.

This proof uses newer axioms ax-5 1548 and ax-9 1645, 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 2108 and ax-9o 2110. (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 2129* 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 1607 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 1537, ax-5o 2108, ax-4 2107, ax-7 1725, ax-6o 2109, ax-8 1666, ax-12o 2114, ax-9o 2110, ax-10o 2111, ax-13 1703, ax-14 1705, ax-15 2115, ax-11o 2113, and ax-16 2116: in that system, we can derive any instance of ax-17 1607 not containing wff variables by induction on formula length, using ax17eq 2155 and ax17el 2161 for the basis together hbn 1743, hbal 1727, and hbim 1750. 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 2130 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 1607; see the proof of equid 1667. See equid1ALT 2148 for an alternate proof. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsps-o 2131 Generalization of antecedent. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremhbequid 2132 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 2110.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnfequid-o 2133 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 1548, ax-8 1666, ax-12o 2114, and ax-gen 1537. This shows that this can be proved without ax9 1921, even though the theorem equid 1667 cannot be. A shorter proof using ax9 1921 is obtainable from equid 1667 and hbth 1543.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax9v 1646, which is used for the derivation of ax12o 1906, unless we consider ax-12o 2114 the starting axiom rather than ax-12 1897. (Contributed by NM, 13-Jan-2011.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax46 2134 Proof of a single axiom that can replace ax-4 2107 and ax-6o 2109. See ax46to4 2135 and ax46to6 2136 for the re-derivation of those axioms. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax46to4 2135 Re-derivation of ax-4 2107 from ax46 2134. 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 2136 Re-derivation of ax-6o 2109 from ax46 2134. 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 2137 Proof of a single axiom that can replace both ax-6o 2109 and ax-7 1725. See ax67to6 2139 and ax67to7 2140 for the re-derivation of those axioms. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnfa1-o 2138 is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax67to6 2139 Re-derivation of ax-6o 2109 from ax67 2137. Note that ax-6o 2109 and ax-7 1725 are not used by the re-derivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax67to7 2140 Re-derivation of ax-7 1725 from ax67 2137. Note that ax-6o 2109 and ax-7 1725 are not used by the re-derivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax467 2141 Proof of a single axiom that can replace ax-4 2107, ax-6o 2109, and ax-7 1725 in a subsystem that includes these axioms plus ax-5o 2108 and ax-gen 1537 (and propositional calculus). See ax467to4 2142, ax467to6 2143, and ax467to7 2144 for the re-derivation of those axioms. This theorem extends the idea in Scott Fenton's ax46 2134. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax467to4 2142 Re-derivation of ax-4 2107 from ax467 2141. Only propositional calculus is used by the re-derivation. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax467to6 2143 Re-derivation of ax-6o 2109 from ax467 2141. Note that ax-6o 2109 and ax-7 1725 are not used by the re-derivation. The use of alimi 1550 (which uses ax-4 2107) is allowed since we have already proved ax467to4 2142. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax467to7 2144 Re-derivation of ax-7 1725 from ax467 2141. Note that ax-6o 2109 and ax-7 1725 are not used by the re-derivation. The use of alimi 1550 (which uses ax-4 2107) is allowed since we have already proved ax467to4 2142. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremequidqe 2145 equid 1667 with existential quantifier without using ax-4 2107 or ax-17 1607. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) (Proof modification is discouraged.)

Theoremax4sp1 2146 A special case of ax-4 2107 without using ax-4 2107 or ax-17 1607. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.)

Theoremequidq 2147 equid 1667 with universal quantifier without using ax-4 2107 or ax-17 1607. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremequid1ALT 2148 Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. Alternate proof of equid1 2130 from older axioms ax-6o 2109 and ax-9o 2110. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax10from10o 2149 Rederivation of ax-10 2112 from original version ax-10o 2111. See theorem ax10o 1924 for the derivation of ax-10o 2111 from ax-10 2112.

This theorem should not be referenced in any proof. Instead, use ax-10 2112 above so that uses of ax-10 2112 can be more easily identified, or use aecom-o 2123 when this form is needed for studies involving ax-10o 2111 and omitting ax-17 1607. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnaecoms-o 2150 A commutation rule for distinct variable specifiers. Version of naecoms 1920 using ax-10o 2111. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremhbnae-o 2151 All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). Version of hbnae 1927 using ax-10o 2111. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdvelimf-o 2152 Proof of dvelimh 1936 that uses ax-10o 2111 but not ax-11o 2113, ax-10 2112, or ax-11 1732. Version of dvelimh 1936 using ax-10o 2111 instead of ax10o 1924. (Contributed by NM, 12-Nov-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdral2-o 2153 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 1938 using ax-10o 2111. (Contributed by NM, 27-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremaev-o 2154* A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 2116. Version of aev 1963 using ax-10o 2111. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax17eq 2155* Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-17 1607 considered as a metatheorem. Do not use it for later proofs - use ax-17 1607 instead, to avoid reference to the redundant axiom ax-16 2116.) (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdveeq2-o 2156* Quantifier introduction when one pair of variables is distinct. Version of dveeq2 1912 using ax-11o 2113. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdveeq2-o16 2157* Version of dveeq2 1912 using ax-16 2116 instead of ax-17 1607. TO DO: Recover proof from older set.mm to remove use of ax-17 1607. (Contributed by NM, 29-Apr-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorema16g-o 2158* A generalization of axiom ax-16 2116. Version of a16g 1917 using ax-10o 2111. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdveeq1-o 2159* Quantifier introduction when one pair of variables is distinct. Version of dveeq1 1990 using ax-10o . (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdveeq1-o16 2160* Version of dveeq1 1990 using ax-16 2116 instead of ax-17 1607. (Contributed by NM, 29-Apr-2008.) TO DO: Recover proof from older set.mm to remove use of ax-17 1607. (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax17el 2161* Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-17 1607 considered as a metatheorem.) (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax10-16 2162* This theorem shows that, given ax-16 2116, we can derive a version of ax-10 2112. However, it is weaker than ax-10 2112 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 27-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdveel2ALT 2163* Version of dveel2 1992 using ax-16 2116 instead of ax-17 1607. (Contributed by NM, 10-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11f 2164 Basis step for constructing a substitution instance of ax-11o 2113 without using ax-11o 2113. 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 2165 Basis step for constructing a substitution instance of ax-11o 2113 without using ax-11o 2113. Atomic formula for equality predicate. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11el 2166 Basis step for constructing a substitution instance of ax-11o 2113 without using ax-11o 2113. Atomic formula for membership predicate. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11indn 2167 Induction step for constructing a substitution instance of ax-11o 2113 without using ax-11o 2113. Negation case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11indi 2168 Induction step for constructing a substitution instance of ax-11o 2113 without using ax-11o 2113. Implication case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11indalem 2169 Lemma for ax11inda2 2171 and ax11inda 2172. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11inda2ALT 2170* A proof of ax11inda2 2171 that is slightly more direct. (Contributed by NM, 4-May-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11inda2 2171* Induction step for constructing a substitution instance of ax-11o 2113 without using ax-11o 2113. Quantification case. When and are distinct, this theorem avoids the dummy variables needed by the more general ax11inda 2172. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11inda 2172* Induction step for constructing a substitution instance of ax-11o 2113 without using ax-11o 2113. Quantification case. (When and are distinct, ax11inda2 2171 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 2173* Recovery of ax-11o 2113 from ax11v 2068 without using ax-11o 2113. The hypothesis is even weaker than ax11v 2068, with both distinct from and not occurring in . Thus, the hypothesis provides an alternate axiom that can be used in place of ax-11o 2113. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11a2-o 2174* Derive ax-11o 2113 from a hypothesis in the form of ax-11 1732, without using ax-11 1732 or ax-11o 2113. The hypothesis is even weaker than ax-11 1732, with both distinct from and not occurring in . Thus, the hypothesis provides an alternate axiom that can be used in place of ax-11 1732, if we also hvae ax-10o 2111 which this proof uses . As theorem ax11 2127 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-10 2112 instead of ax-10o 2111. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax10o-o 2175 Show that ax-10o 2111 can be derived from ax-10 2112. An open problem is whether this theorem can be derived from ax-10 2112 and the others when ax-11 1732 is replaced with ax-11o 2113. See theorem ax10from10o 2149 for the rederivation of ax-10 2112 from ax10o 1924.

Normally, ax10o 1924 should be used rather than ax-10o 2111 or ax10o-o 2175, 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 2176 Extend wff definition to include existential uniqueness ("there exists a unique such that ").

Syntaxwmo 2177 Extend wff definition to include uniqueness ("there exists at most one such that ").

Theoremeujust 2178* A soundness justification theorem for df-eu 2180, showing that the definition is equivalent to itself with its dummy variable renamed. Note that and needn't be distinct variables. See eujustALT 2179 for a proof that provides an example of how it can be achieved through the use of dvelim 1988. (Contributed by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

TheoremeujustALT 2179* A soundness justification theorem for df-eu 2180, 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 1988. (Contributed by NM, 11-Mar-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

Definitiondf-eu 2180* 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 2197, eu2 2201, eu3 2202, and eu5 2214 (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 2259). (Contributed by NM, 12-Aug-1993.)

Definitiondf-mo 2181 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 2207. For other possible definitions see mo2 2205 and mo4 2209. (Contributed by NM, 8-Mar-1995.)

Theoremeuf 2182* A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.)

Theoremeubid 2183 Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)

Theoremeubidv 2184* Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)

Theoremeubii 2185 Introduce uniqueness quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremnfeu1 2186 Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfmo1 2187 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfeud2 2188 Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.)

Theoremnfmod2 2189 Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.)

Theoremnfeud 2190 Deduction version of nfeu 2192. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfmod 2191 Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.)

Theoremnfeu 2192 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 2193 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)

Theoremsb8eu 2194 Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremsb8mo 2195 Variable substitution in uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theoremcbveu 2196 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremeu1 2197* 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.)

Theoremmo 2198* Equivalent definitions of "there exists at most one." (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremeuex 2199 Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremeumo0 2200* Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-31972
 Copyright terms: Public domain < Previous  Next >