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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)

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