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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | sb3 2101 | One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb4 2102 | One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb4b 2103 | Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.) |
Theorem | dfsb2 2104 | An alternate definition of proper substitution that, like df-sb 1656, mixes free and bound variables to avoid distinct variable requirements. (Contributed by NM, 17-Feb-2005.) |
Theorem | dfsb3 2105 | An alternate definition of proper substitution df-sb 1656 that uses only primitive connectives (no defined terms) on the right-hand side. (Contributed by NM, 6-Mar-2007.) |
Theorem | hbsb2 2106 | Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | nfsb2 2107 | Bound-variable hypothesis builder for substitution. (Contributed by Mario Carneiro, 4-Oct-2016.) |
Theorem | sbequi 2108 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbequ 2109 | An equality theorem for substitution. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.) |
Theorem | drsb2 2110 | 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 2111 | Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbi1 2112 | Removal of implication from substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbi2 2113 | Introduction of implication into substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbim 2114 | Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbor 2115 | Logical OR inside and outside of substitution are equivalent. (Contributed by NM, 29-Sep-2002.) |
Theorem | sbrim 2116 | 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 2117 | 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 2118 | Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb3an 2119 | Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-Dec-2006.) |
Theorem | sbbi 2120 | Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.) |
Theorem | sblbis 2121 | Introduce left biconditional inside of a substitution. (Contributed by NM, 19-Aug-1993.) |
Theorem | sbrbis 2122 | Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.) |
Theorem | sbrbif 2123 | Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Theorem | spsbe 2124 | A specialization theorem. (Contributed by NM, 5-Aug-1993.) |
Theorem | spsbim 2125 | Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | spsbbi 2126 | Specialization of biconditional. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbbid 2127 | Deduction substituting both sides of a biconditional. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbequ8 2128 | Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | nfsb4t 2129 | A variable not free remains so after substitution with a distinct variable (closed form of nfsb4 2130). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Theorem | nfsb4 2130 | 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 2131 | Deduction form of dvelimf 2050. This version may be useful if we want to avoid ax-17 1623 and use ax-16 2194 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sbco 2132 | A composition law for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbid2 2133 | An identity law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sbidm 2134 | An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | sbco2 2135 | A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sbco2d 2136 | A composition law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sbco3 2137 | A composition law for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbcom 2138 | A commutativity law for substitution. (Contributed by NM, 27-May-1997.) |
Theorem | sb5rf 2139 | Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sb6rf 2140 | Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sb8 2141 | Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.) |
Theorem | sb8e 2142 | Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.) |
Theorem | sb9i 2143 | Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb9 2144 | Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) |
Theorem | ax11v 2145* | This is a version of ax-11o 2191 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. See theorem ax11v2 2045 for the rederivation of ax-11o 2191 from this theorem. (Contributed by NM, 5-Aug-1993.) |
Theorem | ax11vALT 2146* | Alternate proof of ax11v 2145 that avoids theorem ax16 2094 and is proved directly from ax-11 1757 rather than via ax11o 2047. (Contributed by Jim Kingdon, 15-Dec-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | sb56 2147* | 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 1656. (Contributed by NM, 14-Apr-2008.) |
Theorem | sb6 2148* | 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 2149* | Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.) |
Theorem | equsb3lem 2150* | Lemma for equsb3 2151. (Contributed by Raph Levien and FL, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
Theorem | equsb3 2151* | Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.) |
Theorem | elsb3 2152* | Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
Theorem | elsb4 2153* | Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
Theorem | hbs1 2154* | is not free in when and are distinct. (Contributed by NM, 5-Aug-1993.) |
Theorem | nfs1v 2155* | is not free in when and are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | sbhb 2156* | Two ways of expressing " is (effectively) not free in ." (Contributed by NM, 29-May-2009.) |
Theorem | sbnf2 2157* | 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 2158* | If is not free in , it is not free in when and are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbsb 2159* | If is not free in , it is not free in when and are distinct. (Contributed by NM, 12-Aug-1993.) |
Theorem | nfsbd 2160* | Deduction version of nfsb 2158. (Contributed by NM, 15-Feb-2013.) |
Theorem | 2sb5 2161* | Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.) |
Theorem | 2sb6 2162* | Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.) |
Theorem | sbcom2 2163* | 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 2164* | (Probably not) Axiom *11.07 in [WhiteheadRussell] p. 159. The original confusingly reads: *11.07 "Whatever possible argument may be, is true whatever possible argument may be" implies the corresponding statement with and interchanged except in " ". This theorem will be deleted after 22-Feb-2018 if no one is able to determine the correct interpretation. See https://groups.google.com/d/msg/metamath/iS0fOvSemC8/YzrRyX70AgAJ. (Contributed by Andrew Salmon, 17-Jun-2011.) (Proof shortened by Jim Kingdon, 22-Jan-2018.) (New usage is discouraged.) |
Theorem | pm11.07OLD 2165* | Obsolete proof of pm11.07 2164 as of 22-Jan-2018. (Contributed by Andrew Salmon, 17-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | sb6a 2166* | Equivalence for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | 2sb5rf 2167* | Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | 2sb6rf 2168* | Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sb7f 2169* | This version of dfsb7 2171 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 1623 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1656 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 2170* | This version of dfsb7 2171 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 1623 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1656 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 | dfsb7 2171* | An alternate definition of proper substitution df-sb 1656. 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 2149, first for then for . Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 2391. Theorem sb7h 2170 provides a version where and don't have to be distinct. (Contributed by NM, 28-Jan-2004.) |
Theorem | sb10f 2172* | 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 2173* | 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 2174* | Elimination of substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbel2x 2175* | Elimination of double substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbal1 2176* | 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 2177* | Move universal quantifier in and out of substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbex 2178* | Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.) |
Theorem | sbalv 2179* | Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.) |
Theorem | exsb 2180* | An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.) |
Theorem | exsbOLD 2181* | An equivalent expression for existence. Obsolete as of 19-Jun-2017. (Contributed by NM, 2-Feb-2005.) (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | 2exsb 2182* | An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.) |
Theorem | dvelimALT 2183* | Version of dvelim 2066 that doesn't use ax-10 2190. (See dvelimh 2015 for a version that doesn't use ax-11 1757.) (Contributed by NM, 17-May-2008.) (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | sbal2 2184* | 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 2186, ax-4 2185, ax-7 1745, ax-6o 2187, ax-8 1683, ax-12o 2192, ax-9o 2188, ax-10o 2189, ax-13 1723, ax-14 1725, ax-15 2193, ax-11o 2191, ax-16 2194, and ax-17 1623. Like ours, it includes the rule of generalization (ax-gen 1552). The ones we need to prove from our axioms are ax-5o 2186, ax-4 2185, ax-6o 2187, ax-12o 2192, ax-9o 2188, ax-10o 2189, ax-15 2193, ax-11o 2191, and ax-16 2194. The theorems showing the derivations of those axioms, which have all been proved earlier, are ax5o 1761, ax4 2195 (also called sp 1759), ax6o 1762, ax12o 1976, ax9o 1950, ax10o 2001, ax15 2070, ax11o 2047, ax16 2094, and ax10 1991. In addition, ax-10 2190 was an intermediate axiom we adopted at one time, and we show its proof in this section as ax10from10o 2227. This section also includes a few miscellaneous legacy theorems such as hbequid 2210 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 1552, ax-17 1623, ax-8 1683, ax-9 1662, ax-13 1723, and ax-14 1725 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 1563, ax-6 1740, ax-9 1662, ax-11 1757, and ax-12 1946. However, once we have rederived an axiom (e.g. theorem ax5 2196 for axiom ax-5 1563), we may make use of theorems outside of this section that make use of the rederived axiom (e.g. we may use theorem alimi 1565, which uses ax-5 1563, after proving ax5 2196). | ||
These older axiom schemes are obsolete and should not be used outside of this section. They are proved above as theorems ax5o , sp 1759, ax6o 1762, ax9o 1950, ax10o 2001, ax10 1991, ax11o 2047, ax12o 1976, ax15 2070, and ax16 2094. | ||
Axiom | ax-4 2185 |
Axiom of Specialization. A quantified wff implies the wff without a
quantifier (i.e. an instance, or special case, of the generalized wff).
In other words if something is true for all , it is true for any
specific (that
would typically occur as a free variable in the wff
substituted for ). (A free variable is one that does not occur in
the scope of a quantifier: and are both
free in ,
but only is free
in .) Axiom
scheme C5' in [Megill]
p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski]
p. 67 (under his system S2, defined in the last paragraph on p. 77).
Note that the converse of this axiom does not hold in general, but a weaker inference form of the converse holds and is expressed as rule ax-gen 1552. Conditional forms of the converse are given by ax-12 1946, ax-15 2193, ax-16 2194, and ax-17 1623. 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 2073. An interesting alternate axiomatization uses ax467 2219 and ax-5o 2186 in place of ax-4 2185, ax-5 1563, ax-6 1740, and ax-7 1745. This axiom is obsolete and should no longer be used. It is proved above as theorem sp 1759. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Axiom | ax-5o 2186 |
Axiom of Quantified Implication. This axiom moves a quantifier from
outside to inside an implication, quantifying . Notice that
must not be a free variable in the antecedent of the quantified
implication, and we express this by binding to "protect" the axiom
from a
containing a free .
Axiom scheme C4' in [Megill]
p. 448 (p. 16 of the preprint). It is a special case of Lemma 5 of
[Monk2] p. 108 and Axiom 5 of [Mendelson] p. 69.
This axiom is obsolete and should no longer be used. It is proved above as theorem ax5o 1761. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Axiom | ax-6o 2187 |
Axiom of Quantified Negation. This axiom is used to manipulate negated
quantifiers. Equivalent to axiom scheme C7' in [Megill] p. 448 (p. 16 of
the preprint). An alternate axiomatization could use ax467 2219 in place of
ax-4 2185, ax-6o 2187, and ax-7 1745.
This axiom is obsolete and should no longer be used. It is proved above as theorem ax6o 1762. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Axiom | ax-9o 2188 |
A variant of ax9 1949. 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 1950. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Axiom | ax-10o 2189 |
Axiom ax-10o 2189 ("o" for "old") was the
original version of ax-10 2190,
before it was discovered (in May 2008) that the shorter ax-10 2190 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 2001. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Axiom | ax-10 2190 |
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 2189 ("o" for "old") and was replaced with this shorter ax-10 2190 in May 2008. The old axiom is proved from this one as theorem ax10o 2001. Conversely, this axiom is proved from ax-10o 2189 as theorem ax10from10o 2227. This axiom was proved redundant in July 2015. See theorem ax10 1991. This axiom is obsolete and should no longer be used. It is proved above as theorem ax10 1991. (Contributed by NM, 16-May-2008.) (New usage is discouraged.) |
Axiom | ax-11o 2191 |
Axiom ax-11o 2191 ("o" for "old") was the
original version of ax-11 1757,
before it was discovered (in Jan. 2007) that the shorter ax-11 1757 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 2191 (from which the ax-11 1757 instance follows by theorem ax11 2205.) The proof is by induction on formula length, using ax11eq 2243 and ax11el 2244 for the basis steps and ax11indn 2245, ax11indi 2246, and ax11inda 2250 for the induction steps. (This paragraph is true provided we use ax-10o 2189 in place of ax-10 2190.) This axiom is obsolete and should no longer be used. It is proved above as theorem ax11o 2047. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Axiom | ax-12o 2192 |
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 1976. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Axiom | ax-15 2193 |
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 1623; see theorem ax15 2070.
Alternately,
ax-17 1623 becomes unnecessary in principle with this
axiom, but we lose the
more powerful metalogic afforded by ax-17 1623. We retain ax-15 2193 here to
provide completeness for systems with the simpler metalogic that results
from omitting ax-17 1623, 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 2070. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Axiom | ax-16 2194* |
Axiom of Distinct Variables. The only axiom of predicate calculus
requiring that variables be distinct (if we consider ax-17 1623 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 4350), but nonetheless it is technically
necessary as you can see from its uses.
This axiom is redundant if we include ax-17 1623; see theorem ax16 2094. Alternately, ax-17 1623 becomes logically redundant in the presence of this axiom, but without ax-17 1623 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 2194 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-17 1623, 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 2094. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorems ax11 2205 and ax12from12o 2206 require some intermediate theorems that are included in this section. | ||
Theorem | ax4 2195 | This theorem repeats sp 1759 under the name ax4 2195, so that the metamath program's "verify markup" command will check that it matches axiom scheme ax-4 2185. It is preferred that references to this theorem use the name sp 1759. (Contributed by NM, 18-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | ax5 2196 | Rederivation of axiom ax-5 1563 from ax-5o 2186 and other older axioms. See ax5o 1761 for the derivation of ax-5o 2186 from ax-5 1563. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | ax6 2197 | Rederivation of axiom ax-6 1740 from ax-6o 2187 and other older axioms. See ax6o 1762 for the derivation of ax-6o 2187 from ax-6 1740. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | ax9from9o 2198 | Rederivation of axiom ax-9 1662 from ax-9o 2188 and other older axioms. See ax9o 1950 for the derivation of ax-9o 2188 from ax-9 1662. 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 2199 | 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 2200 | Inference version of ax-5o 2186. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
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