HomeHome Metamath Proof Explorer
Theorem List (p. 20 of 449)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-28689)
  Hilbert Space Explorer  Hilbert Space Explorer
(28690-30212)
  Users' Mathboxes  Users' Mathboxes
(30213-44900)
 

Theorem List for Metamath Proof Explorer - 1901-2000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnfor 1901 If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑𝜓). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)
 
Theoremnf3or 1902 If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, then it is not free in (𝜑𝜓𝜒). (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑    &   𝑥𝜓    &   𝑥𝜒       𝑥(𝜑𝜓𝜒)
 
1.4.3.1  The empty domain of discourse

This database develops mathematics from first-order logic, which has only nonempty models. Before stating axioms excluding the empty model (typically, ax-6 1966 in logic and ax-nul 5202 in set theory), we state in this short subsection a few results relative to the empty domain, which we characterize by the assumption ¬ ∃𝑥. As expected, on the empty domain, every universally quantified formula is true (emptyal 1905) and every existential formula is false (emptyex 1904), and every variable is effectively nonfree in any formula (emptynf 1906).

 
Theoremempty 1903 Two characterizations of the empty domain. (Contributed by Gérard Lang, 5-Feb-2024.)
(¬ ∃𝑥⊤ ↔ ∀𝑥⊥)
 
Theorememptyex 1904 On the empty domain, any existentially quantified formula is false. (Contributed by Wolf Lammen, 21-Jan-2024.)
(¬ ∃𝑥⊤ → ¬ ∃𝑥𝜑)
 
Theorememptyal 1905 On the empty domain, any universally quantified formula is true. (Contributed by Wolf Lammen, 12-Mar-2023.)
(¬ ∃𝑥⊤ → ∀𝑥𝜑)
 
Theorememptynf 1906 On the empty domain, any variable is effectively nonfree in any formula. (Contributed by Wolf Lammen, 12-Mar-2023.)
(¬ ∃𝑥⊤ → Ⅎ𝑥𝜑)
 
1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d
 
Axiomax-5 1907* Axiom of Distinctness. This axiom quantifies 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.

(See comments in ax5ALT 36037 about the logical redundancy of ax-5 1907 in the presence of our obsolete axioms.)

This axiom essentially says that if 𝑥 does not occur in 𝜑, i.e. 𝜑 does not depend on 𝑥 in any way, then we can add the quantifier 𝑥 to 𝜑 with no further assumptions. By sp 2178, we can also remove the quantifier (unconditionally).

For an explanation of disjoint variable conditions, see https://us.metamath.org/mpeuni/mmset.html#distinct 2178. (Contributed by NM, 10-Jan-1993.)

(𝜑 → ∀𝑥𝜑)
 
Theoremax5d 1908* Version of ax-5 1907 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders. (Contributed by NM, 1-Mar-2013.)
(𝜑 → (𝜓 → ∀𝑥𝜓))
 
Theoremax5e 1909* A rephrasing of ax-5 1907 using the existential quantifier. (Contributed by Wolf Lammen, 4-Dec-2017.)
(∃𝑥𝜑𝜑)
 
Theoremax5ea 1910* If a formula holds for some value of a variable not occurring in it, then it holds for all values of that variable. (Contributed by BJ, 28-Dec-2020.)
(∃𝑥𝜑 → ∀𝑥𝜑)
 
Theoremnfv 1911* If 𝑥 is not present in 𝜑, then 𝑥 is not free in 𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Definition change. (Revised by Wolf Lammen, 12-Sep-2021.)
𝑥𝜑
 
Theoremnfvd 1912* nfv 1911 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1891. (Contributed by Mario Carneiro, 6-Oct-2016.)
(𝜑 → Ⅎ𝑥𝜓)
 
Theoremalimdv 1913* Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1807. See alimdh 1814 and alimd 2208 for versions without a distinct variable condition. (Contributed by NM, 3-Apr-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
 
Theoremeximdv 1914* Deduction form of Theorem 19.22 of [Margaris] p. 90, see exim 1830. See eximdh 1861 and eximd 2212 for versions without a distinct variable condition. (Contributed by NM, 27-Apr-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
 
Theorem2alimdv 1915* Deduction form of Theorem 19.20 of [Margaris] p. 90 with two quantifiers, see alim 1807. (Contributed by NM, 27-Apr-2004.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝑦𝜓 → ∀𝑥𝑦𝜒))
 
Theorem2eximdv 1916* Deduction form of Theorem 19.22 of [Margaris] p. 90 with two quantifiers, see exim 1830. (Contributed by NM, 3-Aug-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝜓 → ∃𝑥𝑦𝜒))
 
Theoremalbidv 1917* Formula-building rule for universal quantifier (deduction form). See also albidh 1863 and albid 2220. (Contributed by NM, 26-May-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
 
Theoremexbidv 1918* Formula-building rule for existential quantifier (deduction form). See also exbidh 1864 and exbid 2221. (Contributed by NM, 26-May-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
 
Theoremnfbidv 1919* An equality theorem for nonfreeness. See nfbidf 2222 for a version without disjoint variable condition but requiring more axioms. (Contributed by Mario Carneiro, 4-Oct-2016.) Remove dependency on ax-6 1966, ax-7 2011, ax-12 2173 by adapting proof of nfbidf 2222. (Revised by BJ, 25-Sep-2022.)
(𝜑 → (𝜓𝜒))       (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))
 
Theorem2albidv 1920* Formula-building rule for two universal quantifiers (deduction form). (Contributed by NM, 4-Mar-1997.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝑦𝜓 ↔ ∀𝑥𝑦𝜒))
 
Theorem2exbidv 1921* Formula-building rule for two existential quantifiers (deduction form). (Contributed by NM, 1-May-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝜓 ↔ ∃𝑥𝑦𝜒))
 
Theorem3exbidv 1922* Formula-building rule for three existential quantifiers (deduction form). (Contributed by NM, 1-May-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝑧𝜓 ↔ ∃𝑥𝑦𝑧𝜒))
 
Theorem4exbidv 1923* Formula-building rule for four existential quantifiers (deduction form). (Contributed by NM, 3-Aug-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝑧𝑤𝜓 ↔ ∃𝑥𝑦𝑧𝑤𝜒))
 
Theoremalrimiv 1924* Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2203 and 19.21v 1936. (Contributed by NM, 21-Jun-1993.)
(𝜑𝜓)       (𝜑 → ∀𝑥𝜓)
 
Theoremalrimivv 1925* Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2203 and 19.21v 1936. (Contributed by NM, 31-Jul-1995.)
(𝜑𝜓)       (𝜑 → ∀𝑥𝑦𝜓)
 
Theoremalrimdv 1926* Deduction form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2203 and 19.21v 1936. (Contributed by NM, 10-Feb-1997.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))
 
Theoremexlimiv 1927* Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2207.

See exlimi 2213 for a more general version requiring more axioms.

This inference, along with its many variants such as rexlimdv 3283, is used to implement a metatheorem called "Rule C" that is given in many logic textbooks. See, for example, Rule C in [Mendelson] p. 81, Rule C in [Margaris] p. 40, or Rule C in Hirst and Hirst's A Primer for Logic and Proof p. 59 (PDF p. 65) at http://www.appstate.edu/~hirstjl/primer/hirst.pdf 3283. In informal proofs, the statement "Let 𝐶 be an element such that..." almost always means an implicit application of Rule C.

In essence, Rule C states that if we can prove that some element 𝑥 exists satisfying a wff, i.e. 𝑥𝜑(𝑥) where 𝜑(𝑥) has 𝑥 free, then we can use 𝜑(𝐶) as a hypothesis for the proof where 𝐶 is a new (fictitious) constant not appearing previously in the proof, nor in any axioms used, nor in the theorem to be proved. The purpose of Rule C is to get rid of the existential quantifier.

We cannot do this in Metamath directly. Instead, we use the original 𝜑 (containing 𝑥) as an antecedent for the main part of the proof. We eventually arrive at (𝜑𝜓) where 𝜓 is the theorem to be proved and does not contain 𝑥. Then we apply exlimiv 1927 to arrive at (∃𝑥𝜑𝜓). Finally, we separately prove 𝑥𝜑 and detach it with modus ponens ax-mp 5 to arrive at the final theorem 𝜓, see exlimiiv 1928. (Contributed by NM, 21-Jun-1993.) Remove dependencies on ax-6 1966 and ax-8 2112. (Revised by Wolf Lammen, 4-Dec-2017.)

(𝜑𝜓)       (∃𝑥𝜑𝜓)
 
Theoremexlimiiv 1928* Inference (Rule C) associated with exlimiv 1927. (Contributed by BJ, 19-Dec-2020.)
(𝜑𝜓)    &   𝑥𝜑       𝜓
 
Theoremexlimivv 1929* Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2207. (Contributed by NM, 1-Aug-1995.)
(𝜑𝜓)       (∃𝑥𝑦𝜑𝜓)
 
Theoremexlimdv 1930* Deduction form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2207. (Contributed by NM, 27-Apr-1994.) Remove dependencies on ax-6 1966, ax-7 2011. (Revised by Wolf Lammen, 4-Dec-2017.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓𝜒))
 
Theoremexlimdvv 1931* Deduction form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2207. (Contributed by NM, 31-Jul-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝜓𝜒))
 
Theoremexlimddv 1932* Existential elimination rule of natural deduction (Rule C, explained in exlimiv 1927). (Contributed by Mario Carneiro, 15-Jun-2016.)
(𝜑 → ∃𝑥𝜓)    &   ((𝜑𝜓) → 𝜒)       (𝜑𝜒)
 
Theoremnexdv 1933* Deduction for generalization rule for negated wff. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 13-Jul-2020.) (Proof shortened by Wolf Lammen, 10-Oct-2021.)
(𝜑 → ¬ 𝜓)       (𝜑 → ¬ ∃𝑥𝜓)
 
Theorem2ax5 1934* Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.)
(𝜑 → ∀𝑥𝑦𝜑)
 
Theoremstdpc5v 1935* Version of stdpc5 2204 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.) Revised to shorten 19.21v 1936. (Revised by Wolf Lammen, 12-Jul-2020.)
(∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))
 
Theorem19.21v 1936* Version of 19.21 2203 with a disjoint variable condition, requiring fewer axioms.

Notational convention: We sometimes suffix with "v" the label of a theorem using a distinct variable ("dv") condition instead of a non-freeness hypothesis such as 𝑥𝜑. Conversely, we sometimes suffix with "f" the label of a theorem introducing such a non-freeness hypothesis ("f" stands for "not free in", see df-nf 1781) instead of a disjoint variable condition. For instance, 19.21v 1936 versus 19.21 2203 and vtoclf 3558 versus vtocl 3559. Note that "not free in" is less restrictive than "does not occur in". Note that the version with a disjoint variable condition is easily proved from the version with the corresponding non-freeness hypothesis, by using nfv 1911. However, the dv version can often be proved from fewer axioms. (Contributed by NM, 21-Jun-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 2-Jan-2020.) (Proof shortened by Wolf Lammen, 12-Jul-2020.)

(∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
 
Theorem19.32v 1937* Version of 19.32 2231 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
(∀𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))
 
Theorem19.31v 1938* Version of 19.31 2232 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
(∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
 
Theorem19.23v 1939* Version of 19.23 2207 with a disjoint variable condition instead of a non-freeness hypothesis. (Contributed by NM, 28-Jun-1998.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 11-Jan-2020.) Remove dependency on ax-6 1966. (Revised by Rohan Ridenour, 15-Apr-2022.)
(∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
 
Theorem19.23vv 1940* Theorem 19.23v 1939 extended to two variables. (Contributed by NM, 10-Aug-2004.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝑦𝜑𝜓))
 
Theorempm11.53v 1941* Version of pm11.53 2363 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))
 
Theorem19.36imv 1942* One direction of 19.36v 1990 that can be proven without ax-6 1966. (Contributed by Rohan Ridenour, 16-Apr-2022.)
(∃𝑥(𝜑𝜓) → (∀𝑥𝜑𝜓))
 
Theorem19.36iv 1943* Inference associated with 19.36v 1990. Version of 19.36i 2229 with a disjoint variable condition. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 17-Jan-2020.) Remove dependency on ax-6 1966. (Revised by Rohan Ridenour, 15-Apr-2022.)
𝑥(𝜑𝜓)       (∀𝑥𝜑𝜓)
 
Theorem19.37imv 1944* One direction of 19.37v 1994 that can be proven without ax-6 1966. (Contributed by Rohan Ridenour, 16-Apr-2022.)
(∃𝑥(𝜑𝜓) → (𝜑 → ∃𝑥𝜓))
 
Theorem19.37iv 1945* Inference associated with 19.37v 1994. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-6 1966. (Revised by Rohan Ridenour, 15-Apr-2022.)
𝑥(𝜑𝜓)       (𝜑 → ∃𝑥𝜓)
 
Theorem19.41v 1946* Version of 19.41 2233 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.) Remove dependency on ax-6 1966. (Revised by Rohan Ridenour, 15-Apr-2022.)
(∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
 
Theorem19.41vv 1947* Version of 19.41 2233 with two quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995.)
(∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝑦𝜑𝜓))
 
Theorem19.41vvv 1948* Version of 19.41 2233 with three quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995.)
(∃𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑥𝑦𝑧𝜑𝜓))
 
Theorem19.41vvvv 1949* Version of 19.41 2233 with four quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by FL, 14-Jul-2007.)
(∃𝑤𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑤𝑥𝑦𝑧𝜑𝜓))
 
Theorem19.42v 1950* Version of 19.42 2234 with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
(∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
 
Theoremexdistr 1951* Distribution of existential quantifiers. See also exdistrv 1952. (Contributed by NM, 9-Mar-1995.)
(∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
 
Theoremexdistrv 1952* Distribute a pair of existential quantifiers (over disjoint variables) over a conjunction. Combination of 19.41v 1946 and 19.42v 1950. For a version with fewer disjoint variable conditions but requiring more axioms, see eeanv 2366. (Contributed by BJ, 30-Sep-2022.)
(∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
 
Theorem4exdistrv 1953* Distribute two pairs of existential quantifiers (over disjoint variables) over a conjunction. For a version with fewer disjoint variable conditions but requiring more axioms, see ee4anv 2368. (Contributed by BJ, 5-Jan-2023.)
(∃𝑥𝑧𝑦𝑤(𝜑𝜓) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝜓))
 
Theorem19.42vv 1954* Version of 19.42 2234 with two quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 16-Mar-1995.)
(∃𝑥𝑦(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝑦𝜓))
 
Theoremexdistr2 1955* Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.)
(∃𝑥𝑦𝑧(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝑧𝜓))
 
Theorem19.42vvv 1956* Version of 19.42 2234 with three quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 21-Sep-2011.) (Proof shortened by Wolf Lammen, 27-Aug-2023.)
(∃𝑥𝑦𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝑦𝑧𝜓))
 
Theorem19.42vvvOLD 1957* Obsolete version of 19.42vvv 1956 as of 27-Aug-2023. (Contributed by NM, 21-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥𝑦𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝑦𝑧𝜓))
 
Theorem3exdistr 1958* Distribution of existential quantifiers in a triple conjunction. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒)))
 
Theorem4exdistr 1959* Distribution of existential quantifiers in a quadruple conjunction. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Wolf Lammen, 20-Jan-2018.)
(∃𝑥𝑦𝑧𝑤((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧(𝜒 ∧ ∃𝑤𝜃))))
 
1.4.5  Equality predicate (continued)

The equality predicate was introduced above in wceq 1533 for use by df-tru 1536. See the comments in that section. In this section, we continue with its first "real" use.

 
Theoremweq 1960 Extend wff definition to include atomic formulas using the equality predicate.

(Instead of introducing weq 1960 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wceq 1533. This lets us avoid overloading the = connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1960 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1533. Note: To see the proof steps of this syntax proof, type "MM> SHOW PROOF weq / ALL" in the Metamath program.) (Contributed by NM, 24-Jan-2006.)

wff 𝑥 = 𝑦
 
Theoremequs3OLD 1961 Obsolete as of 12-Aug-2023. Use alinexa 1839 or sbn 2283 instead. Lemma used in proofs of substitution properties. (Contributed by NM, 10-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥(𝑥 = 𝑦𝜑) ↔ ¬ ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))
 
Theoremspeimfw 1962 Specialization, with additional weakening (compared to 19.2 1977) to allow bundling of 𝑥 and 𝑦. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Dec-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑 → ∃𝑥𝜓))
 
TheoremspeimfwALT 1963 Alternate proof of speimfw 1962 (longer compressed proof, but fewer essential steps). (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Aug-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑 → ∃𝑥𝜓))
 
Theoremspimfw 1964 Specialization, with additional weakening (compared to sp 2178) to allow bundling of 𝑥 and 𝑦. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)
𝜓 → ∀𝑥 ¬ 𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑𝜓))
 
Theoremax12i 1965 Inference that has ax-12 2173 (without 𝑦) as its conclusion. Uses only Tarski's FOL axiom schemes. The hypotheses may be eliminable without using ax-12 2173 in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝜓 → ∀𝑥𝜓)       (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
1.4.6  Axiom scheme ax-6 (Existence)
 
Axiomax-6 1966 Axiom of Existence. One of the equality and substitution axioms of predicate calculus with equality. This axiom tells us that at least one thing exists. In this form (not requiring that 𝑥 and 𝑦 be distinct) it was used in an axiom system of Tarski (see Axiom B7' in footnote 1 of [KalishMontague] p. 81.) It is equivalent to axiom scheme C10' in [Megill] p. 448 (p. 16 of the preprint); the equivalence is established by axc10 2399 and ax6fromc10 36026. A more convenient form of this axiom is ax6e 2397, which has additional remarks.

Raph Levien proved the independence of this axiom from the other logical axioms on 12-Apr-2005. See item 16 at https://us.metamath.org/award2003.html 2397.

ax-6 1966 can be proved from the weaker version ax6v 1967 requiring that the variables be distinct; see theorem ax6 2398.

ax-6 1966 can also be proved from the Axiom of Separation (in the form that we use that axiom, where free variables are not universally quantified). See theorem ax6vsep 5199.

Except by ax6v 1967, this axiom should not be referenced directly. Instead, use theorem ax6 2398. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

¬ ∀𝑥 ¬ 𝑥 = 𝑦
 
Theoremax6v 1967* Axiom B7 of [Tarski] p. 75, which requires that 𝑥 and 𝑦 be distinct. This trivial proof is intended merely to weaken axiom ax-6 1966 by adding a distinct variable restriction ($d). From here on, ax-6 1966 should not be referenced directly by any other proof, so that theorem ax6 2398 will show that we can recover ax-6 1966 from this weaker version if it were an axiom (as it is in the case of Tarski).

Note: Introducing 𝑥, 𝑦 as a distinct variable group "out of the blue" with no apparent justification has puzzled some people, but it is perfectly sound. All we are doing is adding an additional prerequisite, similar to adding an unnecessary logical hypothesis, that results in a weakening of the theorem. This means that any future theorem that references ax6v 1967 must have a $d specified for the two variables that get substituted for 𝑥 and 𝑦. The $d does not propagate "backwards", i.e., it does not impose a requirement on ax-6 1966.

When possible, use of this theorem rather than ax6 2398 is preferred since its derivation is much shorter and requires fewer axioms. (Contributed by NM, 7-Aug-2015.)

¬ ∀𝑥 ¬ 𝑥 = 𝑦
 
Theoremax6ev 1968* At least one individual exists. Weaker version of ax6e 2397. When possible, use of this theorem rather than ax6e 2397 is preferred since its derivation is much shorter and requires fewer axioms. (Contributed by NM, 3-Aug-2017.)
𝑥 𝑥 = 𝑦
 
Theoremspimw 1969* Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)
𝜓 → ∀𝑥 ¬ 𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)
 
Theoremspimew 1970* Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Wolf Lammen, 22-Oct-2023.)
(𝜑 → ∀𝑥𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)
 
TheoremspimehOLD 1971* Obsolete version of spimew 1970 as of 22-Oct-2023. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Wolf Lammen, 10-Dec-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)
 
Theoremspeiv 1972* Inference from existential specialization. (Contributed by NM, 19-Aug-1993.) (Revised by Wolf Lammen, 22-Oct-2023.)
(𝑥 = 𝑦 → (𝜓𝜑))    &   𝜓       𝑥𝜑
 
Theoremspeivw 1973* Version of spei 2408 with a disjoint variable condition, which does not require ax-13 2386 (neither ax-7 2011 nor ax-12 2173). (Contributed by BJ, 31-May-2019.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝜓       𝑥𝜑
 
Theoremexgen 1974 Rule of existential generalization, similar to universal generalization ax-gen 1792, but valid only if an individual exists. Its proof requires ax-6 1966 in our axiomatization but the equality predicate does not occur in its statement. Some fundamental theorems of predicate calculus can be proven from ax-gen 1792, ax-4 1806 and this theorem alone, not requiring ax-7 2011 or excessive distinct variable conditions. (Contributed by Wolf Lammen, 12-Nov-2017.) (Proof shortened by Wolf Lammen, 20-Oct-2023.)
𝜑       𝑥𝜑
 
TheoremexgenOLD 1975 Obsolete version of exgen 1974 as of 20-Oct-2023. (Contributed by Wolf Lammen, 12-Nov-2017.) (Proof shortened by Wolf Lammen, 9-Dec-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑       𝑥𝜑
 
Theoremextru 1976 There exists a variable such that holds; that is, there exists a variable. This corresponds under the standard translation to one of the formulations of the modal axiom (D), the other being 19.2 1977. (Contributed by Anthony Hart, 13-Sep-2011.) (Proof shortened by BJ, 12-May-2019.)
𝑥
 
Theorem19.2 1977 Theorem 19.2 of [Margaris] p. 89. This corresponds to the axiom (D) of modal logic (the other standard formulation being extru 1976). Note: This proof is very different from Margaris' because we only have Tarski's FOL axiom schemes available at this point. See the later 19.2g 2183 for a more conventional proof of a more general result, which uses additional axioms. The reverse implication is the defining property of effective nonfreeness (see df-nf 1781). (Contributed by NM, 2-Aug-2017.) Remove dependency on ax-7 2011. (Revised by Wolf Lammen, 4-Dec-2017.)
(∀𝑥𝜑 → ∃𝑥𝜑)
 
Theorem19.2d 1978 Deduction associated with 19.2 1977. (Contributed by BJ, 12-May-2019.)
(𝜑 → ∀𝑥𝜓)       (𝜑 → ∃𝑥𝜓)
 
Theorem19.8w 1979 Weak version of 19.8a 2176 and instance of 19.2d 1978. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.) (Revised by BJ, 31-Mar-2021.)
(𝜑 → ∀𝑥𝜑)       (𝜑 → ∃𝑥𝜑)
 
Theoremspnfw 1980 Weak version of sp 2178. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 13-Aug-2017.)
𝜑 → ∀𝑥 ¬ 𝜑)       (∀𝑥𝜑𝜑)
 
Theoremspvw 1981* Version of sp 2178 when 𝑥 does not occur in 𝜑. Converse of ax-5 1907. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.) Shorten 19.3v 1982. (Revised by Wolf Lammen, 20-Oct-2023.)
(∀𝑥𝜑𝜑)
 
Theorem19.3v 1982* Version of 19.3 2198 with a disjoint variable condition, requiring fewer axioms. Any formula can be universally quantified using a variable which it does not contain. See also 19.9v 1984. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 2011. (Revised by Wolf Lammen, 4-Dec-2017.) (Proof shortened by Wolf Lammen, 20-Oct-2023.)
(∀𝑥𝜑𝜑)
 
Theorem19.8v 1983* Version of 19.8a 2176 with a disjoint variable condition, requiring fewer axioms. Converse of ax5e 1909. (Contributed by BJ, 12-Mar-2020.)
(𝜑 → ∃𝑥𝜑)
 
Theorem19.9v 1984* Version of 19.9 2201 with a disjoint variable condition, requiring fewer axioms. Any formula can be existentially quantified using a variable which it does not contain. See also 19.3v 1982. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 2011. (Revised by Wolf Lammen, 4-Dec-2017.)
(∃𝑥𝜑𝜑)
 
Theorem19.3vOLD 1985* Obsolete version of 19.3v 1982 as of 20-Oct-2023. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 2011. (Revised by Wolf Lammen, 4-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝜑𝜑)
 
TheoremspvwOLD 1986* Obsolete version of spvw 1981 as of 20-Oct-2023. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝜑𝜑)
 
Theorem19.39 1987 Theorem 19.39 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
((∃𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
 
Theorem19.24 1988 Theorem 19.24 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
((∀𝑥𝜑 → ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))
 
Theorem19.34 1989 Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
((∀𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
 
Theorem19.36v 1990* Version of 19.36 2228 with a disjoint variable condition instead of a non-freeness hypothesis. (Contributed by NM, 18-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 17-Jan-2020.)
(∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
 
Theorem19.12vvv 1991* Version of 19.12vv 2364 with a disjoint variable condition, requiring fewer axioms. See also 19.12 2342. (Contributed by BJ, 18-Mar-2020.)
(∃𝑥𝑦(𝜑𝜓) ↔ ∀𝑦𝑥(𝜑𝜓))
 
Theorem19.27v 1992* Version of 19.27 2225 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 3-Jun-2004.)
(∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
 
Theorem19.28v 1993* Version of 19.28 2226 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 25-Mar-2004.)
(∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
 
Theorem19.37v 1994* Version of 19.37 2230 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
(∃𝑥(𝜑𝜓) ↔ (𝜑 → ∃𝑥𝜓))
 
Theorem19.44v 1995* Version of 19.44 2235 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
(∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
 
Theorem19.45v 1996* Version of 19.45 2236 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
(∃𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
 
Theoremspimevw 1997* Existential introduction, using implicit substitution. This is to spimew 1970 what spimvw 1998 is to spimw 1969. Version of spimev 2406 and spimefv 2194 with an additional disjoint variable condition, using only Tarski's FOL axiom schemes. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 17-Mar-2020.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)
 
Theoremspimvw 1998* A weak form of specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. For stronger forms using more axioms, see spimv 2404 and spimfv 2237. (Contributed by NM, 9-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)
 
Theoremspvv 1999* Specialization, using implicit substitution. Version of spv 2407 with a disjoint variable condition, which does not require ax-7 2011, ax-12 2173, ax-13 2386. (Contributed by NM, 30-Aug-1993.) (Revised by BJ, 31-May-2019.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)
 
Theoremspfalw 2000 Version of sp 2178 when 𝜑 is false. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 25-Dec-2017.)
¬ 𝜑       (∀𝑥𝜑𝜑)
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900
  Copyright terms: Public domain < Previous  Next >