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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | equsb3lem 2101* | Lemma for equsb3 2102. (Contributed by Raph Levien and FL, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
⊢ ([y / x]x = z ↔ y = z) | ||
Theorem | equsb3 2102* | Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.) |
⊢ ([y / x]x = z ↔ y = z) | ||
Theorem | elsb3 2103* | Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
⊢ ([y / x]x ∈ z ↔ y ∈ z) | ||
Theorem | elsb4 2104* | Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
⊢ ([y / x]z ∈ x ↔ z ∈ y) | ||
Theorem | hbs1 2105* | x is not free in [y / x]φ when x and y are distinct. (Contributed by NM, 5-Aug-1993.) |
⊢ ([y / x]φ → ∀x[y / x]φ) | ||
Theorem | nfs1v 2106* | x is not free in [y / x]φ when x and y are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎx[y / x]φ | ||
Theorem | sbhb 2107* | Two ways of expressing "x is (effectively) not free in φ." (Contributed by NM, 29-May-2009.) |
⊢ ((φ → ∀xφ) ↔ ∀y(φ → [y / x]φ)) | ||
Theorem | sbnf2 2108* | Two ways of expressing "x is (effectively) not free in φ." (Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2016.) |
⊢ (Ⅎxφ ↔ ∀y∀z([y / x]φ ↔ [z / x]φ)) | ||
Theorem | nfsb 2109* | If z is not free in φ, it is not free in [y / x]φ when y and z are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎzφ ⇒ ⊢ Ⅎz[y / x]φ | ||
Theorem | hbsb 2110* | If z is not free in φ, it is not free in [y / x]φ when y and z are distinct. (Contributed by NM, 12-Aug-1993.) |
⊢ (φ → ∀zφ) ⇒ ⊢ ([y / x]φ → ∀z[y / x]φ) | ||
Theorem | nfsbd 2111* | Deduction version of nfsb 2109. (Contributed by NM, 15-Feb-2013.) |
⊢ Ⅎxφ & ⊢ (φ → Ⅎzψ) ⇒ ⊢ (φ → Ⅎz[y / x]ψ) | ||
Theorem | 2sb5 2112* | Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.) |
⊢ ([z / x][w / y]φ ↔ ∃x∃y((x = z ∧ y = w) ∧ φ)) | ||
Theorem | 2sb6 2113* | Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.) |
⊢ ([z / x][w / y]φ ↔ ∀x∀y((x = z ∧ y = w) → φ)) | ||
Theorem | sbcom2 2114* | 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.) |
⊢ ([w / z][y / x]φ ↔ [y / x][w / z]φ) | ||
Theorem | pm11.07 2115* | Theorem *11.07 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.) |
⊢ ([w / x][y / z]φ ↔ [y / x][w / z]φ) | ||
Theorem | sb6a 2116* | Equivalence for substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ ([y / x]φ ↔ ∀x(x = y → [x / y]φ)) | ||
Theorem | 2sb5rf 2117* | Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) |
⊢ Ⅎzφ & ⊢ Ⅎwφ ⇒ ⊢ (φ ↔ ∃z∃w((z = x ∧ w = y) ∧ [z / x][w / y]φ)) | ||
Theorem | 2sb6rf 2118* | Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) |
⊢ Ⅎzφ & ⊢ Ⅎwφ ⇒ ⊢ (φ ↔ ∀z∀w((z = x ∧ w = y) → [z / x][w / y]φ)) | ||
Theorem | dfsb7 2119* | An alternate definition of proper substitution df-sb 1649. By introducing a dummy variable z in the definiens, we are able to eliminate any distinct variable restrictions among the variables x, y, and φ of the definiendum. No distinct variable conflicts arise because z effectively insulates x from y. To achieve this, we use a chain of two substitutions in the form of sb5 2100, first z for x then y for z. Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 2340. Theorem sb7h 2121 provides a version where φ and z don't have to be distinct. (Contributed by NM, 28-Jan-2004.) |
⊢ ([y / x]φ ↔ ∃z(z = y ∧ ∃x(x = z ∧ φ))) | ||
Theorem | sb7f 2120* | This version of dfsb7 2119 does not require that φ and z be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-17 1616 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1649 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.) |
⊢ Ⅎzφ ⇒ ⊢ ([y / x]φ ↔ ∃z(z = y ∧ ∃x(x = z ∧ φ))) | ||
Theorem | sb7h 2121* | This version of dfsb7 2119 does not require that φ and z be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-17 1616 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1649 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.) |
⊢ (φ → ∀zφ) ⇒ ⊢ ([y / x]φ ↔ ∃z(z = y ∧ ∃x(x = z ∧ φ))) | ||
Theorem | sb10f 2122* | 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.) |
⊢ Ⅎxφ ⇒ ⊢ ([y / z]φ ↔ ∃x(x = y ∧ [x / z]φ)) | ||
Theorem | sbid2v 2123* | 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.) |
⊢ ([y / x][x / y]φ ↔ φ) | ||
Theorem | sbelx 2124* | Elimination of substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ ↔ ∃x(x = y ∧ [x / y]φ)) | ||
Theorem | sbel2x 2125* | Elimination of double substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ ↔ ∃x∃y((x = z ∧ y = w) ∧ [y / w][x / z]φ)) | ||
Theorem | sbal1 2126* | A theorem used in elimination of disjoint variable restriction on x and y by replacing it with a distinctor ¬ ∀xx = z. (Contributed by NM, 5-Aug-1993.) |
⊢ (¬ ∀x x = z → ([z / y]∀xφ ↔ ∀x[z / y]φ)) | ||
Theorem | sbal 2127* | Move universal quantifier in and out of substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ ([z / y]∀xφ ↔ ∀x[z / y]φ) | ||
Theorem | sbex 2128* | Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.) |
⊢ ([z / y]∃xφ ↔ ∃x[z / y]φ) | ||
Theorem | sbalv 2129* | Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.) |
⊢ ([y / x]φ ↔ ψ) ⇒ ⊢ ([y / x]∀zφ ↔ ∀zψ) | ||
Theorem | exsb 2130* | An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.) |
⊢ (∃xφ ↔ ∃y∀x(x = y → φ)) | ||
Theorem | exsbOLD 2131* | An equivalent expression for existence. Obsolete as of 19-Jun-2017. (Contributed by NM, 2-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∃xφ ↔ ∃y∀x(x = y → φ)) | ||
Theorem | 2exsb 2132* | An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.) |
⊢ (∃x∃yφ ↔ ∃z∃w∀x∀y((x = z ∧ y = w) → φ)) | ||
Theorem | dvelimALT 2133* | Version of dvelim 2016 that doesn't use ax-10 2140. (See dvelimh 1964 for a version that doesn't use ax-11 1746.) (Contributed by NM, 17-May-2008.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (φ → ∀xφ) & ⊢ (z = y → (φ ↔ ψ)) ⇒ ⊢ (¬ ∀x x = y → (ψ → ∀xψ)) | ||
Theorem | sbal2 2134* | Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.) |
⊢ (¬ ∀x x = y → ([z / y]∀xφ ↔ ∀x[z / y]φ)) | ||
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 2136, ax-4 2135, ax-7 1734, ax-6o 2137, ax-8 1675, ax-12o 2142, ax-9o 2138, ax-10o 2139, ax-13 1712, ax-14 1714, ax-15 2143, ax-11o 2141, ax-16 2144, and ax-17 1616. Like ours, it includes the rule of generalization (ax-gen 1546). The ones we need to prove from our axioms are ax-5o 2136, ax-4 2135, ax-6o 2137, ax-12o 2142, ax-9o 2138, ax-10o 2139, ax-15 2143, ax-11o 2141, and ax-16 2144. The theorems showing the derivations of those axioms, which have all been proved earlier, are ax5o 1749, ax4 2145 (also called sp 1747), ax6o 1750, ax12o 1934, ax9o 1950, ax10o 1952, ax15 2021, ax11o 1994, ax16 2045, and ax10 1944. In addition, ax-10 2140 was an intermediate axiom we adopted at one time, and we show its proof in this section as ax10from10o 2177. This section also includes a few miscellaneous legacy theorems such as hbequid 2160 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 1546, ax-17 1616, ax-8 1675, ax-9 1654, ax-13 1712, and ax-14 1714 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 1557, ax-6 1729, ax-9 1654, ax-11 1746, and ax-12 1925. However, once we have rederived an axiom (e.g. theorem ax5 2146 for axiom ax-5 1557), we may make use of theorems outside of this section that make use of the rederived axiom (e.g. we may use theorem alimi 1559, which uses ax-5 1557, after proving ax5 2146). | ||
These older axiom schemes are obsolete and should not be used outside of this section. They are proved above as theorems ax5o , sp 1747, ax6o 1750, ax9o 1950, ax10o 1952, ax10 1944, ax11o 1994, ax12o 1934, ax15 2021, and ax16 2045. | ||
Axiom | ax-4 2135 |
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 x, it is true for any
specific x (that would
typically occur as a free variable in the wff
substituted for φ).
(A free variable is one that does not occur in
the scope of a quantifier: x and y are both free in x = y,
but only x is free in
∀yx = y.) This is one of the axioms of
what we call "pure" predicate calculus (ax-4 2135
through ax-7 1734 plus rule
ax-gen 1546). 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 1546. Conditional forms of the converse are given by ax-12 1925, ax-15 2143, ax-16 2144, and ax-17 1616. Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from x 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 2024. An interesting alternate axiomatization uses ax467 2169 and ax-5o 2136 in place of ax-4 2135, ax-5 1557, ax-6 1729, and ax-7 1734. This axiom is obsolete and should no longer be used. It is proved above as theorem sp 1747. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
⊢ (∀xφ → φ) | ||
Axiom | ax-5o 2136 |
Axiom of Quantified Implication. This axiom moves a quantifier from
outside to inside an implication, quantifying ψ. Notice that x
must not be a free variable in the antecedent of the quantified
implication, and we express this by binding φ to "protect" the axiom
from a φ containing
a free x. 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 1749. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
⊢ (∀x(∀xφ → ψ) → (∀xφ → ∀xψ)) | ||
Axiom | ax-6o 2137 |
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 2169 in place of ax-4 2135,
ax-6o 2137,
and ax-7 1734.
This axiom is obsolete and should no longer be used. It is proved above as theorem ax6o 1750. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
⊢ (¬ ∀x ¬ ∀xφ → φ) | ||
Axiom | ax-9o 2138 |
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.) |
⊢ (∀x(x = y → ∀xφ) → φ) | ||
Axiom | ax-10o 2139 |
Axiom ax-10o 2139 ("o" for "old") was the
original version of ax-10 2140,
before it was discovered (in May 2008) that the shorter ax-10 2140 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 1952. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
⊢ (∀x x = y → (∀xφ → ∀yφ)) | ||
Axiom | ax-10 2140 |
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 2139 ("o" for "old") and was replaced with this shorter ax-10 2140 in May 2008. The old axiom is proved from this one as theorem ax10o 1952. Conversely, this axiom is proved from ax-10o 2139 as theorem ax10from10o 2177. This axiom was proved redundant in July 2015. See theorem ax10 1944. This axiom is obsolete and should no longer be used. It is proved above as theorem ax10 1944. (Contributed by NM, 16-May-2008.) (New usage is discouraged.) |
⊢ (∀x x = y → ∀y y = x) | ||
Axiom | ax-11o 2141 |
Axiom ax-11o 2141 ("o" for "old") was the
original version of ax-11 1746,
before it was discovered (in Jan. 2007) that the shorter ax-11 1746 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 "¬ ∀xx = y
→..." as informally
meaning "if x and
y are distinct variables
then..." The
antecedent becomes false if the same variable is substituted for x and
y, ensuring the theorem
is sound whenever this is the case. In some
later theorems, we call an antecedent of the form ¬ ∀xx = y a
"distinctor."
Interestingly, if the wff expression substituted for φ 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 2141 (from which the ax-11 1746 instance follows by theorem ax11 2155.) The proof is by induction on formula length, using ax11eq 2193 and ax11el 2194 for the basis steps and ax11indn 2195, ax11indi 2196, and ax11inda 2200 for the induction steps. (This paragraph is true provided we use ax-10o 2139 in place of ax-10 2140.) This axiom is obsolete and should no longer be used. It is proved above as theorem ax11o 1994. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ)))) | ||
Axiom | ax-12o 2142 |
Axiom of Quantifier Introduction. One of the equality and substitution
axioms of predicate calculus with equality. Informally, it says that
whenever z is distinct
from x and y, and x = y is
true,
then x = y quantified with z is also true. In other words, z
is irrelevant to the truth of x
= y. 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 1934. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
⊢ (¬ ∀z z = x → (¬ ∀z z = y → (x = y → ∀z x = y))) | ||
Axiom | ax-15 2143 |
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 1616; see theorem ax15 2021.
Alternately,
ax-17 1616 becomes unnecessary in principle with this
axiom, but we lose the
more powerful metalogic afforded by ax-17 1616. We retain ax-15 2143 here to
provide completeness for systems with the simpler metalogic that results
from omitting ax-17 1616, 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 2021. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
⊢ (¬ ∀z z = x → (¬ ∀z z = y → (x ∈ y → ∀z x ∈ y))) | ||
Axiom | ax-16 2144* |
Axiom of Distinct Variables. The only axiom of predicate calculus
requiring that variables be distinct (if we consider ax-17 1616 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 in set.mm), but nonetheless it is
technically necessary as you can see from its uses.
This axiom is redundant if we include ax-17 1616; see theorem ax16 2045. Alternately, ax-17 1616 becomes logically redundant in the presence of this axiom, but without ax-17 1616 we lose the more powerful metalogic that results from being able to express the concept of a setvar variable not occurring in a wff (as opposed to just two setvar variables being distinct). We retain ax-16 2144 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-17 1616, 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 2045. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
⊢ (∀x x = y → (φ → ∀xφ)) | ||
Theorems ax11 2155 and ax12from12o 2156 require some intermediate theorems that are included in this section. | ||
Theorem | ax4 2145 | This theorem repeats sp 1747 under the name ax4 2145, so that the metamath program's "verify markup" command will check that it matches axiom scheme ax-4 2135. It is preferred that references to this theorem use the name sp 1747. (Contributed by NM, 18-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (∀xφ → φ) | ||
Theorem | ax5 2146 | Rederivation of axiom ax-5 1557 from ax-5o 2136 and other older axioms. See ax5o 1749 for the derivation of ax-5o 2136 from ax-5 1557. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀x(φ → ψ) → (∀xφ → ∀xψ)) | ||
Theorem | ax6 2147 | Rederivation of axiom ax-6 1729 from ax-6o 2137 and other older axioms. See ax6o 1750 for the derivation of ax-6o 2137 from ax-6 1729. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀xφ → ∀x ¬ ∀xφ) | ||
Theorem | ax9from9o 2148 | Rederivation of axiom ax-9 1654 from ax-9o 2138 and other older axioms. See ax9o 1950 for the derivation of ax-9o 2138 from ax-9 1654. 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.) |
⊢ ¬ ∀x ¬ x = y | ||
Theorem | hba1-o 2149 | x is not free in ∀xφ. 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.) |
⊢ (∀xφ → ∀x∀xφ) | ||
Theorem | a5i-o 2150 | Inference version of ax-5o 2136. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
⊢ (∀xφ → ψ) ⇒ ⊢ (∀xφ → ∀xψ) | ||
Theorem | aecom-o 2151 | Commutation law for identical variable specifiers. The antecedent and consequent are true when x and y are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). Version of aecom 1946 using ax-10o 2139. Unlike ax10from10o 2177, this version does not require ax-17 1616. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
⊢ (∀x x = y → ∀y y = x) | ||
Theorem | aecoms-o 2152 | A commutation rule for identical variable specifiers. Version of aecoms 1947 using ax-10o . (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
⊢ (∀x x = y → φ) ⇒ ⊢ (∀y y = x → φ) | ||
Theorem | hbae-o 2153 | All variables are effectively bound in an identical variable specifier. Version of hbae 1953 using ax-10o 2139. (Contributed by NM, 5-Aug-1993.) (Proof modification is disccouraged.) (New usage is discouraged.) |
⊢ (∀x x = y → ∀z∀x x = y) | ||
Theorem | dral1-o 2154 | 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 1965 using ax-10o 2139. (Contributed by NM, 24-Nov-1994.) (New usage is discouraged.) |
⊢ (∀x x = y → (φ ↔ ψ)) ⇒ ⊢ (∀x x = y → (∀xφ ↔ ∀yψ)) | ||
Theorem | ax11 2155 |
Rederivation of axiom ax-11 1746 from ax-11o 2141, ax-10o 2139, and other older
axioms. The proof does not require ax-16 2144 or ax-17 1616. See theorem
ax11o 1994 for the derivation of ax-11o 2141 from ax-11 1746.
An open problem is whether we can prove this using ax-10 2140 instead of ax-10o 2139. This proof uses newer axioms ax-5 1557 and ax-9 1654, 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 2136 and ax-9o 2138. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (x = y → (∀yφ → ∀x(x = y → φ))) | ||
Theorem | ax12from12o 2156 |
Derive ax-12 1925 from ax-12o 2142 and other older axioms.
This proof uses newer axioms ax-5 1557 and ax-9 1654, 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 2136 and ax-9o 2138. (Contributed by NM, 21-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ x = y → (y = z → ∀x y = z)) | ||
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 2157* |
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 1616 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 1546, ax-5o 2136, ax-4 2135, ax-7 1734, ax-6o 2137, ax-8 1675, ax-12o 2142, ax-9o 2138, ax-10o 2139, ax-13 1712, ax-14 1714, ax-15 2143, ax-11o 2141, and ax-16 2144: in that system, we can derive any instance of ax-17 1616 not containing wff variables by induction on formula length, using ax17eq 2183 and ax17el 2189 for the basis together hbn 1776, hbal 1736, and hbim 1817. 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 setvar variable not occurring in a wff (as opposed to just two setvar variables being distinct). (Contributed by NM, 19-Aug-2017.) (New usage is discouraged.) (Proof modification discouraged.) |
⊢ (φ → ∀xφ) | ||
Theorem | equid1 2158 | 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 1616; see the proof of equid 1676. See equid1ALT 2176 for an alternate proof. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ x = x | ||
Theorem | sps-o 2159 | Generalization of antecedent. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (φ → ψ) ⇒ ⊢ (∀xφ → ψ) | ||
Theorem | hbequid 2160 | Bound-variable hypothesis builder for x = x. This theorem tells us that any variable, including x, is effectively not free in x = x, even though x is technically free according to the traditional definition of free variable. (The proof does not use ax-9o 2138.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (x = x → ∀y x = x) | ||
Theorem | nfequid-o 2161 | Bound-variable hypothesis builder for x = x. This theorem tells us that any variable, including x, is effectively not free in x = x, even though x is technically free according to the traditional definition of free variable. (The proof uses only ax-5 1557, ax-8 1675, ax-12o 2142, and ax-gen 1546. This shows that this can be proved without ax9 1949, even though the theorem equid 1676 cannot be. A shorter proof using ax9 1949 is obtainable from equid 1676 and hbth 1552.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax9v 1655, which is used for the derivation of ax12o 1934, unless we consider ax-12o 2142 the starting axiom rather than ax-12 1925. (Contributed by NM, 13-Jan-2011.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎy x = x | ||
Theorem | ax46 2162 | Proof of a single axiom that can replace ax-4 2135 and ax-6o 2137. See ax46to4 2163 and ax46to6 2164 for the re-derivation of those axioms. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((∀x ¬ ∀xφ → ∀xφ) → φ) | ||
Theorem | ax46to4 2163 | Re-derivation of ax-4 2135 from ax46 2162. Only propositional calculus is used for the re-derivation. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀xφ → φ) | ||
Theorem | ax46to6 2164 | Re-derivation of ax-6o 2137 from ax46 2162. Only propositional calculus is used for the re-derivation. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀x ¬ ∀xφ → φ) | ||
Theorem | ax67 2165 | Proof of a single axiom that can replace both ax-6o 2137 and ax-7 1734. See ax67to6 2167 and ax67to7 2168 for the re-derivation of those axioms. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀x ¬ ∀y∀xφ → ∀yφ) | ||
Theorem | nfa1-o 2166 | x is not free in ∀xφ. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎx∀xφ | ||
Theorem | ax67to6 2167 | Re-derivation of ax-6o 2137 from ax67 2165. Note that ax-6o 2137 and ax-7 1734 are not used by the re-derivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀x ¬ ∀xφ → φ) | ||
Theorem | ax67to7 2168 | Re-derivation of ax-7 1734 from ax67 2165. Note that ax-6o 2137 and ax-7 1734 are not used by the re-derivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀x∀yφ → ∀y∀xφ) | ||
Theorem | ax467 2169 | Proof of a single axiom that can replace ax-4 2135, ax-6o 2137, and ax-7 1734 in a subsystem that includes these axioms plus ax-5o 2136 and ax-gen 1546 (and propositional calculus). See ax467to4 2170, ax467to6 2171, and ax467to7 2172 for the re-derivation of those axioms. This theorem extends the idea in Scott Fenton's ax46 2162. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((∀x∀y ¬ ∀x∀yφ → ∀xφ) → φ) | ||
Theorem | ax467to4 2170 | Re-derivation of ax-4 2135 from ax467 2169. Only propositional calculus is used by the re-derivation. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀xφ → φ) | ||
Theorem | ax467to6 2171 | Re-derivation of ax-6o 2137 from ax467 2169. Note that ax-6o 2137 and ax-7 1734 are not used by the re-derivation. The use of alimi 1559 (which uses ax-4 2135) is allowed since we have already proved ax467to4 2170. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀x ¬ ∀xφ → φ) | ||
Theorem | ax467to7 2172 | Re-derivation of ax-7 1734 from ax467 2169. Note that ax-6o 2137 and ax-7 1734 are not used by the re-derivation. The use of alimi 1559 (which uses ax-4 2135) is allowed since we have already proved ax467to4 2170. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀x∀yφ → ∀y∀xφ) | ||
Theorem | equidqe 2173 | equid 1676 with existential quantifier without using ax-4 2135 or ax-17 1616. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) (Proof modification is discouraged.) |
⊢ ¬ ∀y ¬ x = x | ||
Theorem | ax4sp1 2174 | A special case of ax-4 2135 without using ax-4 2135 or ax-17 1616. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) |
⊢ (∀y ¬ x = x → ¬ x = x) | ||
Theorem | equidq 2175 | equid 1676 with universal quantifier without using ax-4 2135 or ax-17 1616. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ∀y x = x | ||
Theorem | equid1ALT 2176 | Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. Alternate proof of equid1 2158 from older axioms ax-6o 2137 and ax-9o 2138. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ x = x | ||
Theorem | ax10from10o 2177 |
Rederivation of ax-10 2140 from original version ax-10o 2139. See theorem
ax10o 1952 for the derivation of ax-10o 2139 from ax-10 2140.
This theorem should not be referenced in any proof. Instead, use ax-10 2140 above so that uses of ax-10 2140 can be more easily identified, or use aecom-o 2151 when this form is needed for studies involving ax-10o 2139 and omitting ax-17 1616. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀x x = y → ∀y y = x) | ||
Theorem | naecoms-o 2178 | A commutation rule for distinct variable specifiers. Version of naecoms 1948 using ax-10o 2139. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀x x = y → φ) ⇒ ⊢ (¬ ∀y y = x → φ) | ||
Theorem | hbnae-o 2179 | All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). Version of hbnae 1955 using ax-10o 2139. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀x x = y → ∀z ¬ ∀x x = y) | ||
Theorem | dvelimf-o 2180 | Proof of dvelimh 1964 that uses ax-10o 2139 but not ax-11o 2141, ax-10 2140, or ax-11 1746. Version of dvelimh 1964 using ax-10o 2139 instead of ax10o 1952. (Contributed by NM, 12-Nov-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (φ → ∀xφ) & ⊢ (ψ → ∀zψ) & ⊢ (z = y → (φ ↔ ψ)) ⇒ ⊢ (¬ ∀x x = y → (ψ → ∀xψ)) | ||
Theorem | dral2-o 2181 | 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 1966 using ax-10o 2139. (Contributed by NM, 27-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀x x = y → (φ ↔ ψ)) ⇒ ⊢ (∀x x = y → (∀zφ ↔ ∀zψ)) | ||
Theorem | aev-o 2182* | A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 2144. Version of aev 1991 using ax-10o 2139. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀x x = y → ∀z w = v) | ||
Theorem | ax17eq 2183* | Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-17 1616 considered as a metatheorem. Do not use it for later proofs - use ax-17 1616 instead, to avoid reference to the redundant axiom ax-16 2144.) (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (x = y → ∀z x = y) | ||
Theorem | dveeq2-o 2184* | Quantifier introduction when one pair of variables is distinct. Version of dveeq2 1940 using ax-11o 2141. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀x x = y → (z = y → ∀x z = y)) | ||
Theorem | dveeq2-o16 2185* | Version of dveeq2 1940 using ax-16 2144 instead of ax-17 1616. TO DO: Recover proof from older set.mm to remove use of ax-17 1616. (Contributed by NM, 29-Apr-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀x x = y → (z = y → ∀x z = y)) | ||
Theorem | a16g-o 2186* | A generalization of axiom ax-16 2144. Version of a16g 1945 using ax-10o 2139. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀x x = y → (φ → ∀zφ)) | ||
Theorem | dveeq1-o 2187* | Quantifier introduction when one pair of variables is distinct. Version of dveeq1 2018 using ax-10o . (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀x x = y → (y = z → ∀x y = z)) | ||
Theorem | dveeq1-o16 2188* | Version of dveeq1 2018 using ax-16 2144 instead of ax-17 1616. (Contributed by NM, 29-Apr-2008.) TO DO: Recover proof from older set.mm to remove use of ax-17 1616. (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀x x = y → (y = z → ∀x y = z)) | ||
Theorem | ax17el 2189* | Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-17 1616 considered as a metatheorem.) (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (x ∈ y → ∀z x ∈ y) | ||
Theorem | ax10-16 2190* | This theorem shows that, given ax-16 2144, we can derive a version of ax-10 2140. However, it is weaker than ax-10 2140 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 27-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀x x = z → ∀z z = x) | ||
Theorem | dveel2ALT 2191* | Version of dveel2 2020 using ax-16 2144 instead of ax-17 1616. (Contributed by NM, 10-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀x x = y → (z ∈ y → ∀x z ∈ y)) | ||
Theorem | ax11f 2192 | Basis step for constructing a substitution instance of ax-11o 2141 without using ax-11o 2141. We can start with any formula φ in which x is not free. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (φ → ∀xφ) ⇒ ⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ)))) | ||
Theorem | ax11eq 2193 | Basis step for constructing a substitution instance of ax-11o 2141 without using ax-11o 2141. Atomic formula for equality predicate. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀x x = y → (x = y → (z = w → ∀x(x = y → z = w)))) | ||
Theorem | ax11el 2194 | Basis step for constructing a substitution instance of ax-11o 2141 without using ax-11o 2141. Atomic formula for membership predicate. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀x x = y → (x = y → (z ∈ w → ∀x(x = y → z ∈ w)))) | ||
Theorem | ax11indn 2195 | Induction step for constructing a substitution instance of ax-11o 2141 without using ax-11o 2141. Negation case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ)))) ⇒ ⊢ (¬ ∀x x = y → (x = y → (¬ φ → ∀x(x = y → ¬ φ)))) | ||
Theorem | ax11indi 2196 | Induction step for constructing a substitution instance of ax-11o 2141 without using ax-11o 2141. Implication case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ)))) & ⊢ (¬ ∀x x = y → (x = y → (ψ → ∀x(x = y → ψ)))) ⇒ ⊢ (¬ ∀x x = y → (x = y → ((φ → ψ) → ∀x(x = y → (φ → ψ))))) | ||
Theorem | ax11indalem 2197 | Lemma for ax11inda2 2199 and ax11inda 2200. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ)))) ⇒ ⊢ (¬ ∀y y = z → (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ))))) | ||
Theorem | ax11inda2ALT 2198* | A proof of ax11inda2 2199 that is slightly more direct. (Contributed by NM, 4-May-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ)))) ⇒ ⊢ (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ)))) | ||
Theorem | ax11inda2 2199* | Induction step for constructing a substitution instance of ax-11o 2141 without using ax-11o 2141. Quantification case. When z and y are distinct, this theorem avoids the dummy variables needed by the more general ax11inda 2200. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ)))) ⇒ ⊢ (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ)))) | ||
Theorem | ax11inda 2200* | Induction step for constructing a substitution instance of ax-11o 2141 without using ax-11o 2141. Quantification case. (When z and y are distinct, ax11inda2 2199 may be used instead to avoid the dummy variable w in the proof.) (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀x x = w → (x = w → (φ → ∀x(x = w → φ)))) ⇒ ⊢ (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ)))) |
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