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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nfnan 1901 | If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑 ⊼ 𝜓). (Contributed by Scott Fenton, 2-Jan-2018.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ Ⅎ𝑥(𝜑 ⊼ 𝜓) | ||
Theorem | nf3an 1902 | If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, then it is not free in (𝜑 ∧ 𝜓 ∧ 𝜒). (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝜒 ⇒ ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓 ∧ 𝜒) | ||
Theorem | nfbid 1903 | If in a context 𝑥 is not free in 𝜓 and 𝜒, then it is not free in (𝜓 ↔ 𝜒). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 29-Dec-2017.) |
⊢ (𝜑 → Ⅎ𝑥𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝜒) ⇒ ⊢ (𝜑 → Ⅎ𝑥(𝜓 ↔ 𝜒)) | ||
Theorem | nfbi 1904 | If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑 ↔ 𝜓). (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ Ⅎ𝑥(𝜑 ↔ 𝜓) | ||
Theorem | nfor 1905 | If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑 ∨ 𝜓). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ Ⅎ𝑥(𝜑 ∨ 𝜓) | ||
Theorem | nf3or 1906 | If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, then it is not free in (𝜑 ∨ 𝜓 ∨ 𝜒). (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝜒 ⇒ ⊢ Ⅎ𝑥(𝜑 ∨ 𝜓 ∨ 𝜒) | ||
This database develops mathematics from first-order logic, which has only nonempty models. Before stating axioms excluding the empty model (typically, ax-6 1970 in logic and ax-nul 5210 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 1909) and every existential formula is false (emptyex 1908), and every variable is effectively nonfree in any formula (emptynf 1910). | ||
Theorem | empty 1907 | Two characterizations of the empty domain. (Contributed by Gérard Lang, 5-Feb-2024.) |
⊢ (¬ ∃𝑥⊤ ↔ ∀𝑥⊥) | ||
Theorem | emptyex 1908 | On the empty domain, any existentially quantified formula is false. (Contributed by Wolf Lammen, 21-Jan-2024.) |
⊢ (¬ ∃𝑥⊤ → ¬ ∃𝑥𝜑) | ||
Theorem | emptyal 1909 | On the empty domain, any universally quantified formula is true. (Contributed by Wolf Lammen, 12-Mar-2023.) |
⊢ (¬ ∃𝑥⊤ → ∀𝑥𝜑) | ||
Theorem | emptynf 1910 | On the empty domain, any variable is effectively nonfree in any formula. (Contributed by Wolf Lammen, 12-Mar-2023.) |
⊢ (¬ ∃𝑥⊤ → Ⅎ𝑥𝜑) | ||
Axiom | ax-5 1911* |
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 36058 about the logical redundancy of ax-5 1911 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 2182, we can also remove the quantifier (unconditionally). For an explanation of disjoint variable conditions, see https://us.metamath.org/mpeuni/mmset.html#distinct 2182. (Contributed by NM, 10-Jan-1993.) |
⊢ (𝜑 → ∀𝑥𝜑) | ||
Theorem | ax5d 1912* | Version of ax-5 1911 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders. (Contributed by NM, 1-Mar-2013.) |
⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) | ||
Theorem | ax5e 1913* | A rephrasing of ax-5 1911 using the existential quantifier. (Contributed by Wolf Lammen, 4-Dec-2017.) |
⊢ (∃𝑥𝜑 → 𝜑) | ||
Theorem | ax5ea 1914* | 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) | ||
Theorem | nfv 1915* | 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.) |
⊢ Ⅎ𝑥𝜑 | ||
Theorem | nfvd 1916* | nfv 1915 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1895. (Contributed by Mario Carneiro, 6-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝜓) | ||
Theorem | alimdv 1917* | Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1811. See alimdh 1818 and alimd 2212 for versions without a distinct variable condition. (Contributed by NM, 3-Apr-1994.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)) | ||
Theorem | eximdv 1918* | Deduction form of Theorem 19.22 of [Margaris] p. 90, see exim 1834. See eximdh 1865 and eximd 2216 for versions without a distinct variable condition. (Contributed by NM, 27-Apr-1994.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)) | ||
Theorem | 2alimdv 1919* | Deduction form of Theorem 19.20 of [Margaris] p. 90 with two quantifiers, see alim 1811. (Contributed by NM, 27-Apr-2004.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥∀𝑦𝜓 → ∀𝑥∀𝑦𝜒)) | ||
Theorem | 2eximdv 1920* | Deduction form of Theorem 19.22 of [Margaris] p. 90 with two quantifiers, see exim 1834. (Contributed by NM, 3-Aug-1995.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 → ∃𝑥∃𝑦𝜒)) | ||
Theorem | albidv 1921* | Formula-building rule for universal quantifier (deduction form). See also albidh 1867 and albid 2224. (Contributed by NM, 26-May-1993.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒)) | ||
Theorem | exbidv 1922* | Formula-building rule for existential quantifier (deduction form). See also exbidh 1868 and exbid 2225. (Contributed by NM, 26-May-1993.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒)) | ||
Theorem | nfbidv 1923* | An equality theorem for nonfreeness. See nfbidf 2226 for a version without disjoint variable condition but requiring more axioms. (Contributed by Mario Carneiro, 4-Oct-2016.) Remove dependency on ax-6 1970, ax-7 2015, ax-12 2177 by adapting proof of nfbidf 2226. (Revised by BJ, 25-Sep-2022.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒)) | ||
Theorem | 2albidv 1924* | Formula-building rule for two universal quantifiers (deduction form). (Contributed by NM, 4-Mar-1997.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥∀𝑦𝜓 ↔ ∀𝑥∀𝑦𝜒)) | ||
Theorem | 2exbidv 1925* | Formula-building rule for two existential quantifiers (deduction form). (Contributed by NM, 1-May-1995.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 ↔ ∃𝑥∃𝑦𝜒)) | ||
Theorem | 3exbidv 1926* | Formula-building rule for three existential quantifiers (deduction form). (Contributed by NM, 1-May-1995.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧𝜓 ↔ ∃𝑥∃𝑦∃𝑧𝜒)) | ||
Theorem | 4exbidv 1927* | Formula-building rule for four existential quantifiers (deduction form). (Contributed by NM, 3-Aug-1995.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧∃𝑤𝜓 ↔ ∃𝑥∃𝑦∃𝑧∃𝑤𝜒)) | ||
Theorem | alrimiv 1928* | Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2207 and 19.21v 1940. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → ∀𝑥𝜓) | ||
Theorem | alrimivv 1929* | Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2207 and 19.21v 1940. (Contributed by NM, 31-Jul-1995.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → ∀𝑥∀𝑦𝜓) | ||
Theorem | alrimdv 1930* | Deduction form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2207 and 19.21v 1940. (Contributed by NM, 10-Feb-1997.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒)) | ||
Theorem | exlimiv 1931* |
Inference form of Theorem 19.23 of [Margaris]
p. 90, see 19.23 2211.
See exlimi 2217 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 (𝜑 → 𝜓) where 𝜓 is the theorem to be proved and does not contain 𝑥. Then we apply exlimiv 1931 to arrive at (∃𝑥𝜑 → 𝜓). Finally, we separately prove ∃𝑥𝜑 and detach it with modus ponens ax-mp 5 to arrive at the final theorem 𝜓, see exlimiiv 1932. (Contributed by NM, 21-Jun-1993.) Remove dependencies on ax-6 1970 and ax-8 2116. (Revised by Wolf Lammen, 4-Dec-2017.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (∃𝑥𝜑 → 𝜓) | ||
Theorem | exlimiiv 1932* | Inference (Rule C) associated with exlimiv 1931. (Contributed by BJ, 19-Dec-2020.) |
⊢ (𝜑 → 𝜓) & ⊢ ∃𝑥𝜑 ⇒ ⊢ 𝜓 | ||
Theorem | exlimivv 1933* | Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2211. (Contributed by NM, 1-Aug-1995.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (∃𝑥∃𝑦𝜑 → 𝜓) | ||
Theorem | exlimdv 1934* | Deduction form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2211. (Contributed by NM, 27-Apr-1994.) Remove dependencies on ax-6 1970, ax-7 2015. (Revised by Wolf Lammen, 4-Dec-2017.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒)) | ||
Theorem | exlimdvv 1935* | Deduction form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2211. (Contributed by NM, 31-Jul-1995.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 → 𝜒)) | ||
Theorem | exlimddv 1936* | Existential elimination rule of natural deduction (Rule C, explained in exlimiv 1931). (Contributed by Mario Carneiro, 15-Jun-2016.) |
⊢ (𝜑 → ∃𝑥𝜓) & ⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | nexdv 1937* | 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) | ||
Theorem | 2ax5 1938* | Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.) |
⊢ (𝜑 → ∀𝑥∀𝑦𝜑) | ||
Theorem | stdpc5v 1939* | Version of stdpc5 2208 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.) Revised to shorten 19.21v 1940. (Revised by Wolf Lammen, 12-Jul-2020.) |
⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)) | ||
Theorem | 19.21v 1940* |
Version of 19.21 2207 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 Ⅎ𝑥𝜑. 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 1785) instead of a disjoint variable condition. For instance, 19.21v 1940 versus 19.21 2207 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 1915. 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(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) | ||
Theorem | 19.32v 1941* | Version of 19.32 2235 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.) |
⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓)) | ||
Theorem | 19.31v 1942* | Version of 19.31 2236 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.) |
⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ 𝜓)) | ||
Theorem | 19.23v 1943* | Version of 19.23 2211 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 1970. (Revised by Rohan Ridenour, 15-Apr-2022.) |
⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)) | ||
Theorem | 19.23vv 1944* | Theorem 19.23v 1943 extended to two variables. (Contributed by NM, 10-Aug-2004.) |
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥∃𝑦𝜑 → 𝜓)) | ||
Theorem | pm11.53v 1945* | Version of pm11.53 2367 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.) |
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)) | ||
Theorem | 19.36imv 1946* | One direction of 19.36v 1994 that can be proven without ax-6 1970. (Contributed by Rohan Ridenour, 16-Apr-2022.) |
⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)) | ||
Theorem | 19.36iv 1947* | Inference associated with 19.36v 1994. Version of 19.36i 2233 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 1970. (Revised by Rohan Ridenour, 15-Apr-2022.) |
⊢ ∃𝑥(𝜑 → 𝜓) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) | ||
Theorem | 19.37imv 1948* | One direction of 19.37v 1998 that can be proven without ax-6 1970. (Contributed by Rohan Ridenour, 16-Apr-2022.) |
⊢ (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)) | ||
Theorem | 19.37iv 1949* | Inference associated with 19.37v 1998. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-6 1970. (Revised by Rohan Ridenour, 15-Apr-2022.) |
⊢ ∃𝑥(𝜑 → 𝜓) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
Theorem | 19.41v 1950* | Version of 19.41 2237 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.) Remove dependency on ax-6 1970. (Revised by Rohan Ridenour, 15-Apr-2022.) |
⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓)) | ||
Theorem | 19.41vv 1951* | Version of 19.41 2237 with two quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995.) |
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ 𝜓)) | ||
Theorem | 19.41vvv 1952* | Version of 19.41 2237 with three quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995.) |
⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦∃𝑧𝜑 ∧ 𝜓)) | ||
Theorem | 19.41vvvv 1953* | Version of 19.41 2237 with four quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by FL, 14-Jul-2007.) |
⊢ (∃𝑤∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (∃𝑤∃𝑥∃𝑦∃𝑧𝜑 ∧ 𝜓)) | ||
Theorem | 19.42v 1954* | Version of 19.42 2238 with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 21-Jun-1993.) |
⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓)) | ||
Theorem | exdistr 1955* | Distribution of existential quantifiers. See also exdistrv 1956. (Contributed by NM, 9-Mar-1995.) |
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓)) | ||
Theorem | exdistrv 1956* | Distribute a pair of existential quantifiers (over disjoint variables) over a conjunction. Combination of 19.41v 1950 and 19.42v 1954. For a version with fewer disjoint variable conditions but requiring more axioms, see eeanv 2370. (Contributed by BJ, 30-Sep-2022.) |
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) | ||
Theorem | 4exdistrv 1957* | 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 2372. (Contributed by BJ, 5-Jan-2023.) |
⊢ (∃𝑥∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓)) | ||
Theorem | 19.42vv 1958* | Version of 19.42 2238 with two quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 16-Mar-1995.) |
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦𝜓)) | ||
Theorem | exdistr2 1959* | Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.) |
⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦∃𝑧𝜓)) | ||
Theorem | 19.42vvv 1960* | Version of 19.42 2238 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.) |
⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦∃𝑧𝜓)) | ||
Theorem | 19.42vvvOLD 1961* | Obsolete version of 19.42vvv 1960 as of 27-Aug-2023. (Contributed by NM, 21-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦∃𝑧𝜓)) | ||
Theorem | 3exdistr 1962* | Distribution of existential quantifiers in a triple conjunction. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒))) | ||
Theorem | 4exdistr 1963* | Distribution of existential quantifiers in a quadruple conjunction. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Wolf Lammen, 20-Jan-2018.) |
⊢ (∃𝑥∃𝑦∃𝑧∃𝑤((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧(𝜒 ∧ ∃𝑤𝜃)))) | ||
The equality predicate was introduced above in wceq 1537 for use by df-tru 1540. See the comments in that section. In this section, we continue with its first "real" use. | ||
Theorem | weq 1964 |
Extend wff definition to include atomic formulas using the equality
predicate.
(Instead of introducing weq 1964 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 1537. This lets us avoid overloading the = connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1964 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1537. 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 | ||
Theorem | equs3OLD 1965 | Obsolete as of 12-Aug-2023. Use alinexa 1843 or sbn 2287 instead. Lemma used in proofs of substitution properties. (Contributed by NM, 10-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ¬ ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) | ||
Theorem | speimfw 1966 | Specialization, with additional weakening (compared to 19.2 1981) 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 → (𝜑 → 𝜓)) ⇒ ⊢ (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑 → ∃𝑥𝜓)) | ||
Theorem | speimfwALT 1967 | Alternate proof of speimfw 1966 (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 → (𝜑 → 𝜓)) ⇒ ⊢ (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑 → ∃𝑥𝜓)) | ||
Theorem | spimfw 1968 | Specialization, with additional weakening (compared to sp 2182) 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 → (𝜑 → 𝜓)) ⇒ ⊢ (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑 → 𝜓)) | ||
Theorem | ax12i 1969 | Inference that has ax-12 2177 (without ∀𝑦) as its conclusion. Uses only Tarski's FOL axiom schemes. The hypotheses may be eliminable without using ax-12 2177 in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝜓 → ∀𝑥𝜓) ⇒ ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Axiom | ax-6 1970 |
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 2403 and ax6fromc10 36047. A more convenient form of this
axiom is ax6e 2401, 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 2401. ax-6 1970 can be proved from the weaker version ax6v 1971 requiring that the variables be distinct; see theorem ax6 2402. ax-6 1970 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 5207. Except by ax6v 1971, this axiom should not be referenced directly. Instead, use theorem ax6 2402. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.) |
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 | ||
Theorem | ax6v 1971* |
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 1970
by adding a distinct variable restriction ($d). From here on, ax-6 1970
should not be referenced directly by any other proof, so that theorem
ax6 2402 will show that we can recover ax-6 1970
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 1971 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 1970. When possible, use of this theorem rather than ax6 2402 is preferred since its derivation is much shorter and requires fewer axioms. (Contributed by NM, 7-Aug-2015.) |
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 | ||
Theorem | ax6ev 1972* | At least one individual exists. Weaker version of ax6e 2401. When possible, use of this theorem rather than ax6e 2401 is preferred since its derivation is much shorter and requires fewer axioms. (Contributed by NM, 3-Aug-2017.) |
⊢ ∃𝑥 𝑥 = 𝑦 | ||
Theorem | spimw 1973* | 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 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) | ||
Theorem | spimew 1974* | 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 → (𝜑 → 𝜓)) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
Theorem | spimehOLD 1975* | Obsolete version of spimew 1974 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 → (𝜑 → 𝜓)) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
Theorem | speiv 1976* | Inference from existential specialization. (Contributed by NM, 19-Aug-1993.) (Revised by Wolf Lammen, 22-Oct-2023.) |
⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑)) & ⊢ 𝜓 ⇒ ⊢ ∃𝑥𝜑 | ||
Theorem | speivw 1977* | Version of spei 2412 with a disjoint variable condition, which does not require ax-13 2390 (neither ax-7 2015 nor ax-12 2177). (Contributed by BJ, 31-May-2019.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ 𝜓 ⇒ ⊢ ∃𝑥𝜑 | ||
Theorem | exgen 1978 | Rule of existential generalization, similar to universal generalization ax-gen 1796, but valid only if an individual exists. Its proof requires ax-6 1970 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 1796, ax-4 1810 and this theorem alone, not requiring ax-7 2015 or excessive distinct variable conditions. (Contributed by Wolf Lammen, 12-Nov-2017.) (Proof shortened by Wolf Lammen, 20-Oct-2023.) |
⊢ 𝜑 ⇒ ⊢ ∃𝑥𝜑 | ||
Theorem | exgenOLD 1979 | Obsolete version of exgen 1978 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 | ||
Theorem | extru 1980 | 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 1981. (Contributed by Anthony Hart, 13-Sep-2011.) (Proof shortened by BJ, 12-May-2019.) |
⊢ ∃𝑥⊤ | ||
Theorem | 19.2 1981 | Theorem 19.2 of [Margaris] p. 89. This corresponds to the axiom (D) of modal logic (the other standard formulation being extru 1980). 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 2187 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 1785). (Contributed by NM, 2-Aug-2017.) Remove dependency on ax-7 2015. (Revised by Wolf Lammen, 4-Dec-2017.) |
⊢ (∀𝑥𝜑 → ∃𝑥𝜑) | ||
Theorem | 19.2d 1982 | Deduction associated with 19.2 1981. (Contributed by BJ, 12-May-2019.) |
⊢ (𝜑 → ∀𝑥𝜓) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
Theorem | 19.8w 1983 | Weak version of 19.8a 2180 and instance of 19.2d 1982. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.) (Revised by BJ, 31-Mar-2021.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (𝜑 → ∃𝑥𝜑) | ||
Theorem | spnfw 1984 | Weak version of sp 2182. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 13-Aug-2017.) |
⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑) ⇒ ⊢ (∀𝑥𝜑 → 𝜑) | ||
Theorem | spvw 1985* | Version of sp 2182 when 𝑥 does not occur in 𝜑. Converse of ax-5 1911. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.) Shorten 19.3v 1986. (Revised by Wolf Lammen, 20-Oct-2023.) |
⊢ (∀𝑥𝜑 → 𝜑) | ||
Theorem | 19.3v 1986* | Version of 19.3 2202 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 1988. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 2015. (Revised by Wolf Lammen, 4-Dec-2017.) (Proof shortened by Wolf Lammen, 20-Oct-2023.) |
⊢ (∀𝑥𝜑 ↔ 𝜑) | ||
Theorem | 19.8v 1987* | Version of 19.8a 2180 with a disjoint variable condition, requiring fewer axioms. Converse of ax5e 1913. (Contributed by BJ, 12-Mar-2020.) |
⊢ (𝜑 → ∃𝑥𝜑) | ||
Theorem | 19.9v 1988* | Version of 19.9 2205 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 1986. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 2015. (Revised by Wolf Lammen, 4-Dec-2017.) |
⊢ (∃𝑥𝜑 ↔ 𝜑) | ||
Theorem | 19.3vOLD 1989* | Obsolete version of 19.3v 1986 as of 20-Oct-2023. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 2015. (Revised by Wolf Lammen, 4-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥𝜑 ↔ 𝜑) | ||
Theorem | spvwOLD 1990* | Obsolete version of spvw 1985 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.) |
⊢ (∀𝑥𝜑 → 𝜑) | ||
Theorem | 19.39 1991 | Theorem 19.39 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
⊢ ((∃𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓)) | ||
Theorem | 19.24 1992 | Theorem 19.24 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∃𝑥(𝜑 → 𝜓)) | ||
Theorem | 19.34 1993 | Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
⊢ ((∀𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑 ∨ 𝜓)) | ||
Theorem | 19.36v 1994* | Version of 19.36 2232 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(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓)) | ||
Theorem | 19.12vvv 1995* | Version of 19.12vv 2368 with a disjoint variable condition, requiring fewer axioms. See also 19.12 2346. (Contributed by BJ, 18-Mar-2020.) |
⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓)) | ||
Theorem | 19.27v 1996* | Version of 19.27 2229 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 3-Jun-2004.) |
⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓)) | ||
Theorem | 19.28v 1997* | Version of 19.28 2230 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 25-Mar-2004.) |
⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓)) | ||
Theorem | 19.37v 1998* | Version of 19.37 2234 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.) |
⊢ (∃𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥𝜓)) | ||
Theorem | 19.44v 1999* | Version of 19.44 2239 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993.) |
⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ 𝜓)) | ||
Theorem | 19.45v 2000* | Version of 19.45 2240 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993.) |
⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓)) |
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