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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | drsb2 2001 | Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.) |
Theorem | sbn 2002 | Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbi1 2003 | Removal of implication from substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbi2 2004 | Introduction of implication into substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbim 2005 | Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbor 2006 | Logical OR inside and outside of substitution are equivalent. (Contributed by NM, 29-Sep-2002.) |
Theorem | sbrim 2007 | Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Theorem | sblim 2008 | Substitution with a variable not free in consequent affects only the antecedent. (Contributed by NM, 14-Nov-2013.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Theorem | sban 2009 | Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb3an 2010 | Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-Dec-2006.) |
Theorem | sbbi 2011 | Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.) |
Theorem | sblbis 2012 | Introduce left biconditional inside of a substitution. (Contributed by NM, 19-Aug-1993.) |
Theorem | sbrbis 2013 | Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.) |
Theorem | sbrbif 2014 | Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Theorem | spsbe 2015 | A specialization theorem. (Contributed by NM, 5-Aug-1993.) |
Theorem | spsbim 2016 | Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | spsbbi 2017 | Specialization of biconditional. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbbid 2018 | Deduction substituting both sides of a biconditional. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbequ8 2019 | Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | nfsb4t 2020 | A variable not free remains so after substitution with a distinct variable (closed form of nfsb4 2021). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Theorem | nfsb4 2021 | A variable not free remains so after substitution with a distinct variable. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Theorem | dvelimdf 2022 | Deduction form of dvelimf 1937. This version may be useful if we want to avoid ax-17 1603 and use ax-16 2083 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sbco 2023 | A composition law for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbid2 2024 | An identity law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sbidm 2025 | An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | sbco2 2026 | A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sbco2d 2027 | A composition law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sbco3 2028 | A composition law for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbcom 2029 | A commutativity law for substitution. (Contributed by NM, 27-May-1997.) |
Theorem | sb5rf 2030 | Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sb6rf 2031 | Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sb8 2032 | Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sb8e 2033 | Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sb9i 2034 | Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb9 2035 | Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) |
Theorem | ax11v 2036* | This is a version of ax-11o 2080 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. See theorem ax11v2 1932 for the rederivation of ax-11o 2080 from this theorem. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb56 2037* | Two equivalent ways of expressing the proper substitution of for in , when and are distinct. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 1630. (Contributed by NM, 14-Apr-2008.) |
Theorem | sb6 2038* | Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. (Contributed by NM, 18-Aug-1993.) |
Theorem | sb5 2039* | Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.) |
Theorem | equsb3lem 2040* | Lemma for equsb3 2041. (Contributed by Raph Levien and FL, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
Theorem | equsb3 2041* | Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.) |
Theorem | elsb3 2042* | Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
Theorem | elsb4 2043* | Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
Theorem | hbs1 2044* | is not free in when and are distinct. (Contributed by NM, 5-Aug-1993.) |
Theorem | nfs1v 2045* | is not free in when and are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | sbhb 2046* | Two ways of expressing " is (effectively) not free in ." (Contributed by NM, 29-May-2009.) |
Theorem | sbnf2 2047* | Two ways of expressing " is (effectively) not free in ." (Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | nfsb 2048* | If is not free in , it is not free in when and are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbsb 2049* | If is not free in , it is not free in when and are distinct. (Contributed by NM, 12-Aug-1993.) |
Theorem | nfsbd 2050* | Deduction version of nfsb 2048. (Contributed by NM, 15-Feb-2013.) |
Theorem | 2sb5 2051* | Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.) |
Theorem | 2sb6 2052* | Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.) |
Theorem | sbcom2 2053* | Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.) |
Theorem | pm11.07 2054* | Theorem *11.07 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.) |
Theorem | sb6a 2055* | Equivalence for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | 2sb5rf 2056* | Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | 2sb6rf 2057* | Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | dfsb7 2058* | An alternate definition of proper substitution df-sb 1630. By introducing a dummy variable in the definiens, we are able to eliminate any distinct variable restrictions among the variables , , and of the definiendum. No distinct variable conflicts arise because effectively insulates from . To achieve this, we use a chain of two substitutions in the form of sb5 2039, first for then for . Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 2270. Theorem sb7h 2060 provides a version where and don't have to be distinct. (Contributed by NM, 28-Jan-2004.) |
Theorem | sb7f 2059* | This version of dfsb7 2058 does not require that and be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-17 1603 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1630 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sb7h 2060* | This version of dfsb7 2058 does not require that and be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-17 1603 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1630 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | sb10f 2061* | Hao Wang's identity axiom P6 in Irving Copi, Symbolic Logic (5th ed., 1979), p. 328. In traditional predicate calculus, this is a sole axiom for identity from which the usual ones can be derived. (Contributed by NM, 9-May-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sbid2v 2062* | An identity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.) |
Theorem | sbelx 2063* | Elimination of substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbel2x 2064* | Elimination of double substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbal1 2065* | A theorem used in elimination of disjoint variable restriction on and by replacing it with a distinctor . (Contributed by NM, 5-Aug-1993.) |
Theorem | sbal 2066* | Move universal quantifier in and out of substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbex 2067* | Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.) |
Theorem | sbalv 2068* | Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.) |
Theorem | exsb 2069* | An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.) |
Theorem | exsbOLD 2070* | An equivalent expression for existence. Obsolete as of 19-Jun-2017. (Contributed by NM, 2-Feb-2005.) (New usage is discouraged.) |
Theorem | 2exsb 2071* | An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.) |
Theorem | dvelimALT 2072* | Version of dvelim 1956 that doesn't use ax-10 2079. (See dvelimh 1904 for a version that doesn't use ax-11 1715.) (Contributed by NM, 17-May-2008.) (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | sbal2 2073* | 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 2075, ax-4 2074, ax-7 1708, ax-6o 2076, ax-8 1643, ax-12o 2081, ax-9o 2077, ax-10o 2078, ax-13 1686, ax-14 1688, ax-15 2082, ax-11o 2080, ax-16 2083, and ax-17 1603. Like ours, it includes the rule of generalization (ax-gen 1533). The ones we need to prove from our axioms are ax-5o 2075, ax-4 2074, ax-6o 2076, ax-12o 2081, ax-9o 2077, ax-10o 2078, ax-15 2082, ax-11o 2080, and ax-16 2083. The theorems showing the derivations of those axioms, which have all been proved earlier, are ax5o 1717, ax4 2084 (also called sp 1716), ax6o 1723, ax12o 1875, ax9o 1890, ax10o 1892, ax15 1961, ax11o 1934, ax16 1985, and ax10 1884. In addition, ax-10 2079 was an intermediate axiom we adopted at one time, and we show its proof in this section as ax10from10o 2116. This section also includes a few miscellaneous legacy theorems such as hbequid 2099 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 1533, ax-17 1603, ax-8 1643, ax-9 1635, ax-13 1686, and ax-14 1688 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 1544, ax-6 1703, ax-9 1635, ax-11 1715, and ax-12 1866. However, once we have rederived an axiom (e.g. theorem ax5 2085 for axiom ax-5 1544), we may make use of theorems outside of this section that make use of the rederived axiom (e.g. we may use theorem alimi 1546, which uses ax-5 1544, after proving ax5 2085). | ||
These older axiom schemes are obsolete and should not be used outside of this section. They are proved above as theorems ax5o , sp 1716, ax6o 1723, ax9o 1890, ax10o 1892, ax10 1884, ax11o 1934, ax12o 1875, ax15 1961, and ax16 1985. | ||
Axiom | ax-4 2074 |
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 2074
through ax-7 1708 plus rule
ax-gen 1533). 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 1533. Conditional forms of the converse are given by ax-12 1866, ax-15 2082, ax-16 2083, and ax-17 1603. 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 1964. An interesting alternate axiomatization uses ax467 2108 and ax-5o 2075 in place of ax-4 2074, ax-5 1544, ax-6 1703, and ax-7 1708. This axiom is obsolete and should no longer be used. It is proved above as theorem sp 1716. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Axiom | ax-5o 2075 |
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 1717. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Axiom | ax-6o 2076 |
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 2108 in place of ax-4 2074,
ax-6o 2076,
and ax-7 1708.
This axiom is obsolete and should no longer be used. It is proved above as theorem ax6o 1723. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Axiom | ax-9o 2077 |
A variant of ax9 1889. 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 1890. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Axiom | ax-10o 2078 |
Axiom ax-10o 2078 ("o" for "old") was the
original version of ax-10 2079,
before it was discovered (in May 2008) that the shorter ax-10 2079 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 1892. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Axiom | ax-10 2079 |
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 2078 ("o" for "old") and was replaced with this shorter ax-10 2079 in May 2008. The old axiom is proved from this one as theorem ax10o 1892. Conversely, this axiom is proved from ax-10o 2078 as theorem ax10from10o 2116. This axiom was proved redundant in July 2015. See theorem ax10 1884. This axiom is obsolete and should no longer be used. It is proved above as theorem ax10 1884. (Contributed by NM, 16-May-2008.) (New usage is discouraged.) |
Axiom | ax-11o 2080 |
Axiom ax-11o 2080 ("o" for "old") was the
original version of ax-11 1715,
before it was discovered (in Jan. 2007) that the shorter ax-11 1715 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 2080 (from which the ax-11 1715 instance follows by theorem ax11 2094.) The proof is by induction on formula length, using ax11eq 2132 and ax11el 2133 for the basis steps and ax11indn 2134, ax11indi 2135, and ax11inda 2139 for the induction steps. (This paragraph is true provided we use ax-10o 2078 in place of ax-10 2079.) This axiom is obsolete and should no longer be used. It is proved above as theorem ax11o 1934. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Axiom | ax-12o 2081 |
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 1875. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Axiom | ax-15 2082 |
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 1603; see theorem ax15 1961.
Alternately,
ax-17 1603 becomes unnecessary in principle with this
axiom, but we lose the
more powerful metalogic afforded by ax-17 1603. We retain ax-15 2082 here to
provide completeness for systems with the simpler metalogic that results
from omitting ax-17 1603, 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 1961. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Axiom | ax-16 2083* |
Axiom of Distinct Variables. The only axiom of predicate calculus
requiring that variables be distinct (if we consider ax-17 1603 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 4201), but nonetheless it is technically
necessary as you can see from its uses.
This axiom is redundant if we include ax-17 1603; see theorem ax16 1985. Alternately, ax-17 1603 becomes logically redundant in the presence of this axiom, but without ax-17 1603 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 2083 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-17 1603, 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 1985. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorems ax11 2094 and ax12 2095 require some intermediate theorems that are included in this section. | ||
Theorem | ax4 2084 | This theorem repeats sp 1716 under the name ax4 2084, so that the metamath program's "verify markup" command will check that it matches axiom scheme ax-4 2074. It is preferred that references to this theorem use the name sp 1716. (Contributed by NM, 18-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | ax5 2085 | Rederivation of axiom ax-5 1544 from ax-5o 2075 and other older axioms. See ax5o 1717 for the derivation of ax-5o 2075 from ax-5 1544. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | ax6 2086 | Rederivation of axiom ax-6 1703 from ax-6o 2076 and other older axioms. See ax6o 1723 for the derivation of ax-6o 2076 from ax-6 1703. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | ax9from9o 2087 | Rederivation of axiom ax-9 1635 from ax-9o 2077 and other older axioms. See ax9o 1890 for the derivation of ax-9o 2077 from ax-9 1635. 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 2088 | 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 2089 | Inference version of ax-5o 2075. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorem | aecom-o 2090 | 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 1886 using ax-10o 2078. Unlike ax10from10o 2116, this version does not require ax-17 1603. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorem | aecoms-o 2091 | A commutation rule for identical variable specifiers. Version of aecoms 1887 using ax-10o . (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorem | hbae-o 2092 | All variables are effectively bound in an identical variable specifier. Version of hbae 1893 using ax-10o 2078. (Contributed by NM, 5-Aug-1993.) (Proof modification is disccouraged.) (New usage is discouraged.) |
Theorem | dral1-o 2093 | 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 1905 using ax-10o 2078. (Contributed by NM, 24-Nov-1994.) (New usage is discouraged.) |
Theorem | ax11 2094 |
Rederivation of axiom ax-11 1715 from ax-11o 2080, ax-10o 2078, and other older
axioms. The proof does not require ax-16 2083 or ax-17 1603. See theorem
ax11o 1934 for the derivation of ax-11o 2080 from ax-11 1715.
An open problem is whether we can prove this using ax-10 2079 instead of ax-10o 2078. This proof uses newer axioms ax-5 1544 and ax-9 1635, 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 2075 and ax-9o 2077. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | ax12 2095 |
Derive ax-12 1866 from ax-12o 2081 and other older axioms.
This proof uses newer axioms ax-5 1544 and ax-9 1635, 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 2075 and ax-9o 2077. (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 2096* |
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 1603 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 1533, ax-5o 2075, ax-4 2074, ax-7 1708, ax-6o 2076, ax-8 1643, ax-12o 2081, ax-9o 2077, ax-10o 2078, ax-13 1686, ax-14 1688, ax-15 2082, ax-11o 2080, and ax-16 2083: in that system, we can derive any instance of ax-17 1603 not containing wff variables by induction on formula length, using ax17eq 2122 and ax17el 2128 for the basis together hbn 1720, hbal 1710, and hbim 1725. 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 2097 | 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 1603; see the proof of equid 1644. See equid1ALT 2115 for an alternate proof. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | sps-o 2098 | Generalization of antecedent. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | hbequid 2099 | 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 2077.) (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 2100 | 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 1544, ax-8 1643, ax-12o 2081, and ax-gen 1533. This shows that this can be proved without ax9 1889, even though the theorem equid 1644 cannot be. A shorter proof using ax9 1889 is obtainable from equid 1644 and hbth 1539.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax9v 1636, which is used for the derivation of ax12o 1875, unless we consider ax-12o 2081 the starting axiom rather than ax-12 1866. (Contributed by NM, 13-Jan-2011.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
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