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Theorem List for Metamath Proof Explorer - 1901-2000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorememptynf 1901 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 1902* 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 35925 about the logical redundancy of ax-5 1902 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 2172, we can also remove the quantifier (unconditionally).

For an explanation of disjoint variable conditions, see https://us.metamath.org/mpeuni/mmset.html#distinct 2172. (Contributed by NM, 10-Jan-1993.)

(𝜑 → ∀𝑥𝜑)
 
Theoremax5d 1903* Version of ax-5 1902 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders. (Contributed by NM, 1-Mar-2013.)
(𝜑 → (𝜓 → ∀𝑥𝜓))
 
Theoremax5e 1904* A rephrasing of ax-5 1902 using the existential quantifier. (Contributed by Wolf Lammen, 4-Dec-2017.)
(∃𝑥𝜑𝜑)
 
Theoremax5ea 1905* 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 1906* 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 1907* nfv 1906 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1886. (Contributed by Mario Carneiro, 6-Oct-2016.)
(𝜑 → Ⅎ𝑥𝜓)
 
Theoremalimdv 1908* Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1802. See alimdh 1809 and alimd 2203 for versions without a distinct variable condition. (Contributed by NM, 3-Apr-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
 
Theoremeximdv 1909* Deduction form of Theorem 19.22 of [Margaris] p. 90, see exim 1825. See eximdh 1856 and eximd 2207 for versions without a distinct variable condition. (Contributed by NM, 27-Apr-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
 
Theorem2alimdv 1910* Deduction form of Theorem 19.20 of [Margaris] p. 90 with two quantifiers, see alim 1802. (Contributed by NM, 27-Apr-2004.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝑦𝜓 → ∀𝑥𝑦𝜒))
 
Theorem2eximdv 1911* Deduction form of Theorem 19.22 of [Margaris] p. 90 with two quantifiers, see exim 1825. (Contributed by NM, 3-Aug-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝜓 → ∃𝑥𝑦𝜒))
 
Theoremalbidv 1912* Formula-building rule for universal quantifier (deduction form). See also albidh 1858 and albid 2215. (Contributed by NM, 26-May-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
 
Theoremexbidv 1913* Formula-building rule for existential quantifier (deduction form). See also exbidh 1859 and exbid 2216. (Contributed by NM, 26-May-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
 
Theoremnfbidv 1914* An equality theorem for nonfreeness. See nfbidf 2217 for a version without disjoint variable condition but requiring more axioms. (Contributed by Mario Carneiro, 4-Oct-2016.) Remove dependency on ax-6 1961, ax-7 2006, ax-12 2167 by adapting proof of nfbidf 2217. (Revised by BJ, 25-Sep-2022.)
(𝜑 → (𝜓𝜒))       (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))
 
Theorem2albidv 1915* Formula-building rule for two universal quantifiers (deduction form). (Contributed by NM, 4-Mar-1997.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝑦𝜓 ↔ ∀𝑥𝑦𝜒))
 
Theorem2exbidv 1916* Formula-building rule for two existential quantifiers (deduction form). (Contributed by NM, 1-May-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝜓 ↔ ∃𝑥𝑦𝜒))
 
Theorem3exbidv 1917* Formula-building rule for three existential quantifiers (deduction form). (Contributed by NM, 1-May-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝑧𝜓 ↔ ∃𝑥𝑦𝑧𝜒))
 
Theorem4exbidv 1918* Formula-building rule for four existential quantifiers (deduction form). (Contributed by NM, 3-Aug-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝑧𝑤𝜓 ↔ ∃𝑥𝑦𝑧𝑤𝜒))
 
Theoremalrimiv 1919* Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2198 and 19.21v 1931. (Contributed by NM, 21-Jun-1993.)
(𝜑𝜓)       (𝜑 → ∀𝑥𝜓)
 
Theoremalrimivv 1920* Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2198 and 19.21v 1931. (Contributed by NM, 31-Jul-1995.)
(𝜑𝜓)       (𝜑 → ∀𝑥𝑦𝜓)
 
Theoremalrimdv 1921* Deduction form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2198 and 19.21v 1931. (Contributed by NM, 10-Feb-1997.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))
 
Theoremexlimiv 1922* Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2202.

See exlimi 2208 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 1922 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 1923. (Contributed by NM, 21-Jun-1993.) Remove dependencies on ax-6 1961 and ax-8 2107. (Revised by Wolf Lammen, 4-Dec-2017.)

(𝜑𝜓)       (∃𝑥𝜑𝜓)
 
Theoremexlimiiv 1923* Inference (Rule C) associated with exlimiv 1922. (Contributed by BJ, 19-Dec-2020.)
(𝜑𝜓)    &   𝑥𝜑       𝜓
 
Theoremexlimivv 1924* Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2202. (Contributed by NM, 1-Aug-1995.)
(𝜑𝜓)       (∃𝑥𝑦𝜑𝜓)
 
Theoremexlimdv 1925* Deduction form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2202. (Contributed by NM, 27-Apr-1994.) Remove dependencies on ax-6 1961, ax-7 2006. (Revised by Wolf Lammen, 4-Dec-2017.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓𝜒))
 
Theoremexlimdvv 1926* Deduction form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2202. (Contributed by NM, 31-Jul-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝜓𝜒))
 
Theoremexlimddv 1927* Existential elimination rule of natural deduction (Rule C, explained in exlimiv 1922). (Contributed by Mario Carneiro, 15-Jun-2016.)
(𝜑 → ∃𝑥𝜓)    &   ((𝜑𝜓) → 𝜒)       (𝜑𝜒)
 
Theoremnexdv 1928* 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 1929* Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.)
(𝜑 → ∀𝑥𝑦𝜑)
 
Theoremstdpc5v 1930* Version of stdpc5 2199 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.) Revised to shorten 19.21v 1931. (Revised by Wolf Lammen, 12-Jul-2020.)
(∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))
 
Theorem19.21v 1931* Version of 19.21 2198 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 1776) instead of a disjoint variable condition. For instance, 19.21v 1931 versus 19.21 2198 and vtoclf 3559 versus vtocl 3560. 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 1906. 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 1932* Version of 19.32 2226 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
(∀𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))
 
Theorem19.31v 1933* Version of 19.31 2227 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
(∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
 
Theorem19.23v 1934* Version of 19.23 2202 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 1961. (Revised by Rohan Ridenour, 15-Apr-2022.)
(∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
 
Theorem19.23vv 1935* Theorem 19.23v 1934 extended to two variables. (Contributed by NM, 10-Aug-2004.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝑦𝜑𝜓))
 
Theorempm11.53v 1936* Version of pm11.53 2359 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))
 
Theorem19.36imv 1937* One direction of 19.36v 1985 that can be proven without ax-6 1961. (Contributed by Rohan Ridenour, 16-Apr-2022.)
(∃𝑥(𝜑𝜓) → (∀𝑥𝜑𝜓))
 
Theorem19.36iv 1938* Inference associated with 19.36v 1985. Version of 19.36i 2224 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 1961. (Revised by Rohan Ridenour, 15-Apr-2022.)
𝑥(𝜑𝜓)       (∀𝑥𝜑𝜓)
 
Theorem19.37imv 1939* One direction of 19.37v 1989 that can be proven without ax-6 1961. (Contributed by Rohan Ridenour, 16-Apr-2022.)
(∃𝑥(𝜑𝜓) → (𝜑 → ∃𝑥𝜓))
 
Theorem19.37iv 1940* Inference associated with 19.37v 1989. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-6 1961. (Revised by Rohan Ridenour, 15-Apr-2022.)
𝑥(𝜑𝜓)       (𝜑 → ∃𝑥𝜓)
 
Theorem19.41v 1941* Version of 19.41 2228 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.) Remove dependency on ax-6 1961. (Revised by Rohan Ridenour, 15-Apr-2022.)
(∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
 
Theorem19.41vv 1942* Version of 19.41 2228 with two quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995.)
(∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝑦𝜑𝜓))
 
Theorem19.41vvv 1943* Version of 19.41 2228 with three quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995.)
(∃𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑥𝑦𝑧𝜑𝜓))
 
Theorem19.41vvvv 1944* Version of 19.41 2228 with four quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by FL, 14-Jul-2007.)
(∃𝑤𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑤𝑥𝑦𝑧𝜑𝜓))
 
Theorem19.42v 1945* Version of 19.42 2229 with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
(∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
 
Theoremexdistr 1946* Distribution of existential quantifiers. See also exdistrv 1947. (Contributed by NM, 9-Mar-1995.)
(∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
 
Theoremexdistrv 1947* Distribute a pair of existential quantifiers (over disjoint variables) over a conjunction. Combination of 19.41v 1941 and 19.42v 1945. For a version with fewer disjoint variable conditions but requiring more axioms, see eeanv 2362. (Contributed by BJ, 30-Sep-2022.)
(∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
 
Theorem4exdistrv 1948* 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 2364. (Contributed by BJ, 5-Jan-2023.)
(∃𝑥𝑧𝑦𝑤(𝜑𝜓) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝜓))
 
Theorem19.42vv 1949* Version of 19.42 2229 with two quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 16-Mar-1995.)
(∃𝑥𝑦(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝑦𝜓))
 
Theoremexdistr2 1950* Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.)
(∃𝑥𝑦𝑧(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝑧𝜓))
 
Theorem19.42vvv 1951* Version of 19.42 2229 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 1952* Obsolete version of 19.42vvv 1951 as of 27-Aug-2023. (Contributed by NM, 21-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥𝑦𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝑦𝑧𝜓))
 
Theorem3exdistr 1953* 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 1954* 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 1528 for use by df-tru 1531. See the comments in that section. In this section, we continue with its first "real" use.

 
Theoremweq 1955 Extend wff definition to include atomic formulas using the equality predicate.

(Instead of introducing weq 1955 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 1528. This lets us avoid overloading the = connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1955 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1528. 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 1956 Obsolete as of 12-Aug-2023. Use alinexa 1834 or sbn 2279 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 1957 Specialization, with additional weakening (compared to 19.2 1972) 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 1958 Alternate proof of speimfw 1957 (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 1959 Specialization, with additional weakening (compared to sp 2172) 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 1960 Inference that has ax-12 2167 (without 𝑦) as its conclusion. Uses only Tarski's FOL axiom schemes. The hypotheses may be eliminable without using ax-12 2167 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 1961 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 2396 and ax6fromc10 35914. A more convenient form of this axiom is ax6e 2394, 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 2394.

ax-6 1961 can be proved from the weaker version ax6v 1962 requiring that the variables be distinct; see theorem ax6 2395.

ax-6 1961 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 1962, this axiom should not be referenced directly. Instead, use theorem ax6 2395. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

¬ ∀𝑥 ¬ 𝑥 = 𝑦
 
Theoremax6v 1962* 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 1961 by adding a distinct variable restriction ($d). From here on, ax-6 1961 should not be referenced directly by any other proof, so that theorem ax6 2395 will show that we can recover ax-6 1961 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 1962 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 1961.

When possible, use of this theorem rather than ax6 2395 is preferred since its derivation is much shorter and requires fewer axioms. (Contributed by NM, 7-Aug-2015.)

¬ ∀𝑥 ¬ 𝑥 = 𝑦
 
Theoremax6ev 1963* At least one individual exists. Weaker version of ax6e 2394. When possible, use of this theorem rather than ax6e 2394 is preferred since its derivation is much shorter and requires fewer axioms. (Contributed by NM, 3-Aug-2017.)
𝑥 𝑥 = 𝑦
 
Theoremspimw 1964* 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 1965* 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 1966* Obsolete version of spimew 1965 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 1967* Inference from existential specialization. (Contributed by Wolf Lammen, 22-Oct-2023.)
(𝑥 = 𝑦 → (𝜓𝜑))    &   𝜓       𝑥𝜑
 
Theoremspeivw 1968* Version of spei 2406 with a disjoint variable condition, which does not require ax-13 2383 (neither ax-7 2006 nor ax-12 2167). (Contributed by BJ, 31-May-2019.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝜓       𝑥𝜑
 
Theoremexgen 1969 Rule of existential generalization, similar to universal generalization ax-gen 1787, but valid only if an individual exists. Its proof requires ax-6 1961 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 1787, ax-4 1801 and this theorem alone, not requiring ax-7 2006 or excessive distinct variable conditions. (Contributed by Wolf Lammen, 12-Nov-2017.) (Proof shortened by Wolf Lammen, 20-Oct-2023.)
𝜑       𝑥𝜑
 
TheoremexgenOLD 1970 Obsolete version of exgen 1969 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 1971 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 1972. (Contributed by Anthony Hart, 13-Sep-2011.) (Proof shortened by BJ, 12-May-2019.)
𝑥
 
Theorem19.2 1972 Theorem 19.2 of [Margaris] p. 89. This corresponds to the axiom (D) of modal logic (the other standard formulation being extru 1971). 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 2177 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 1776). (Contributed by NM, 2-Aug-2017.) Remove dependency on ax-7 2006. (Revised by Wolf Lammen, 4-Dec-2017.)
(∀𝑥𝜑 → ∃𝑥𝜑)
 
Theorem19.2d 1973 Deduction associated with 19.2 1972. (Contributed by BJ, 12-May-2019.)
(𝜑 → ∀𝑥𝜓)       (𝜑 → ∃𝑥𝜓)
 
Theorem19.8w 1974 Weak version of 19.8a 2170 and instance of 19.2d 1973. (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 1975 Weak version of sp 2172. 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 1976* Version of sp 2172 when 𝑥 does not occur in 𝜑. Converse of ax-5 1902. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.) Shorten 19.3v 1977. (Revised by Wolf Lammen, 20-Oct-2023.)
(∀𝑥𝜑𝜑)
 
Theorem19.3v 1977* Version of 19.3 2193 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 1979. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 2006. (Revised by Wolf Lammen, 4-Dec-2017.) (Proof shortened by Wolf Lammen, 20-Oct-2023.)
(∀𝑥𝜑𝜑)
 
Theorem19.8v 1978* Version of 19.8a 2170 with a disjoint variable condition, requiring fewer axioms. Converse of ax5e 1904. (Contributed by BJ, 12-Mar-2020.)
(𝜑 → ∃𝑥𝜑)
 
Theorem19.9v 1979* Version of 19.9 2196 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 1977. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 2006. (Revised by Wolf Lammen, 4-Dec-2017.)
(∃𝑥𝜑𝜑)
 
Theorem19.3vOLD 1980* Obsolete version of 19.3v 1977 as of 20-Oct-2023. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 2006. (Revised by Wolf Lammen, 4-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝜑𝜑)
 
TheoremspvwOLD 1981* Obsolete version of spvw 1976 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 1982 Theorem 19.39 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
((∃𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
 
Theorem19.24 1983 Theorem 19.24 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
((∀𝑥𝜑 → ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))
 
Theorem19.34 1984 Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
((∀𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
 
Theorem19.36v 1985* Version of 19.36 2223 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 1986* Version of 19.12vv 2360 with a disjoint variable condition, requiring fewer axioms. See also 19.12 2338. (Contributed by BJ, 18-Mar-2020.)
(∃𝑥𝑦(𝜑𝜓) ↔ ∀𝑦𝑥(𝜑𝜓))
 
Theorem19.27v 1987* Version of 19.27 2220 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 3-Jun-2004.)
(∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
 
Theorem19.28v 1988* Version of 19.28 2221 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 25-Mar-2004.)
(∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
 
Theorem19.37v 1989* Version of 19.37 2225 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
(∃𝑥(𝜑𝜓) ↔ (𝜑 → ∃𝑥𝜓))
 
Theorem19.44v 1990* Version of 19.44 2230 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
(∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
 
Theorem19.45v 1991* Version of 19.45 2231 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
(∃𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
 
Theoremspimevw 1992* Existential introduction, using implicit substitution. This is to spimew 1965 what spimvw 1993 is to spimw 1964. Version of spimev 2404 and spimefv 2189 with an additional disjoint variable condition, using only Tarski's FOL axiom schemes. (Contributed by BJ, 17-Mar-2020.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)
 
Theoremspimvw 1993* 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 2402 and spimfv 2232. (Contributed by NM, 9-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)
 
Theoremspvv 1994* Version of spv 2405 with a disjoint variable condition, which does not require ax-7 2006, ax-12 2167, ax-13 2383. (Contributed by BJ, 31-May-2019.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)
 
Theoremspfalw 1995 Version of sp 2172 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)
 
Theoremchvarvv 1996* Version of chvarv 2408 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by BJ, 31-May-2019.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝜑       𝜓
 
Theoremequs4v 1997* Version of equs4 2432 with a disjoint variable condition, which requires fewer axioms. (Contributed by BJ, 31-May-2019.)
(∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
 
Theoremalequexv 1998* Version of equs4v 1997 with its consequence simplified by exsimpr 1861. (Contributed by BJ, 9-Nov-2021.)
(∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
 
Theoremexsbim 1999* One direction of the equivalence in exsb 2371 is based on fewer axioms. (Contributed by Wolf Lammen, 2-Mar-2023.)
(∃𝑦𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
 
Theoremequsv 2000* If a formula does not contain a variable 𝑥, then it is equivalent to the corresponding prototype of substitution with a fresh variable (see sb6 2084). (Contributed by BJ, 23-Jul-2023.)
(∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑)
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