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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | sbalv 2101* | Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.) |
Theorem | exsb 2102* | An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.) |
Theorem | exsbOLD 2103* | An equivalent expression for existence. Obsolete as of 19-Jun-2017. (Contributed by NM, 2-Feb-2005.) (New usage is discouraged.) |
Theorem | 2exsb 2104* | An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.) |
Theorem | dvelimALT 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.) |
Theorem | sbal2 2106* | Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.) |
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). | ||
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. | ||
Axiom | ax-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.) |
Axiom | ax-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.) |
Axiom | ax-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.) |
Axiom | ax-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.) |
Axiom | ax-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.) |
Axiom | ax-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.) |
Axiom | ax-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.) |
Axiom | ax-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.) |
Axiom | ax-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.) |
Axiom | ax-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.) |
Theorems ax11 2127 and ax12from12o 2128 require some intermediate theorems that are included in this section. | ||
Theorem | ax4 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.) |
Theorem | ax5 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.) |
Theorem | ax6 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.) |
Theorem | ax9from9o 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.) |
Theorem | hba1-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.) |
Theorem | a5i-o 2122 | Inference version of ax-5o 2108. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorem | aecom-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.) |
Theorem | aecoms-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.) |
Theorem | hbae-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.) |
Theorem | dral1-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.) |
Theorem | ax11 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.) |
Theorem | ax12from12o 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.) |
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. | ||
Theorem | ax17o 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.) |
Theorem | equid1 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.) |
Theorem | sps-o 2131 | Generalization of antecedent. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | hbequid 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.) |
Theorem | nfequid-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.) |
Theorem | ax46 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.) |
Theorem | ax46to4 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.) |
Theorem | ax46to6 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.) |
Theorem | ax67 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.) |
Theorem | nfa1-o 2138 | is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | ax67to6 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.) |
Theorem | ax67to7 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.) |
Theorem | ax467 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.) |
Theorem | ax467to4 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.) |
Theorem | ax467to6 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.) |
Theorem | ax467to7 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.) |
Theorem | equidqe 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.) |
Theorem | ax4sp1 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.) |
Theorem | equidq 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.) |
Theorem | equid1ALT 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.) |
Theorem | ax10from10o 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.) |
Theorem | naecoms-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.) |
Theorem | hbnae-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.) |
Theorem | dvelimf-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.) |
Theorem | dral2-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.) |
Theorem | aev-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.) |
Theorem | ax17eq 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.) |
Theorem | dveeq2-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.) |
Theorem | dveeq2-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.) |
Theorem | a16g-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.) |
Theorem | dveeq1-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.) |
Theorem | dveeq1-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.) |
Theorem | ax17el 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.) |
Theorem | ax10-16 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.) |
Theorem | dveel2ALT 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.) |
Theorem | ax11f 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.) |
Theorem | ax11eq 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.) |
Theorem | ax11el 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.) |
Theorem | ax11indn 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.) |
Theorem | ax11indi 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.) |
Theorem | ax11indalem 2169 | Lemma for ax11inda2 2171 and ax11inda 2172. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | ax11inda2ALT 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.) |
Theorem | ax11inda2 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.) |
Theorem | ax11inda 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.) |
Theorem | ax11v2-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.) |
Theorem | ax11a2-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.) |
Theorem | ax10o-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.) |
Syntax | weu 2176 | Extend wff definition to include existential uniqueness ("there exists a unique such that "). |
Syntax | wmo 2177 | Extend wff definition to include uniqueness ("there exists at most one such that "). |
Theorem | eujust 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.) |
Theorem | eujustALT 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.) |
Definition | df-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.) |
Definition | df-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.) |
Theorem | euf 2182* | A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.) |
Theorem | eubid 2183 | Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.) |
Theorem | eubidv 2184* | Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.) |
Theorem | eubii 2185 | Introduce uniqueness quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | nfeu1 2186 | Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
Theorem | nfmo1 2187 | Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) |
Theorem | nfeud2 2188 | Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.) |
Theorem | nfmod2 2189 | Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.) |
Theorem | nfeud 2190 | Deduction version of nfeu 2192. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) |
Theorem | nfmod 2191 | Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.) |
Theorem | nfeu 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.) |
Theorem | nfmo 2193 | Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.) |
Theorem | sb8eu 2194 | Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
Theorem | sb8mo 2195 | Variable substitution in uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | cbveu 2196 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
Theorem | eu1 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.) |
Theorem | mo 2198* | Equivalent definitions of "there exists at most one." (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
Theorem | euex 2199 | Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Theorem | eumo0 2200* | Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.) |
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