P. 20 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 1901-2000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nfand 1901	If in a context 𝑥 is not free in 𝜓 and 𝜒, then it is not free in (𝜓 ∧ 𝜒). (Contributed by Mario Carneiro, 7-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(𝜓 ∧ 𝜒))
Theorem	nf3and 1902	Deduction form of bound-variable hypothesis builder nf3an 1905. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → Ⅎ𝑥𝜃)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(𝜓 ∧ 𝜒 ∧ 𝜃))
Theorem	nfan 1903	If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑 ∧ 𝜓). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) (Proof shortened by Wolf Lammen, 9-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)
Theorem	nfnan 1904	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 1905	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 1906	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 1907	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 1908	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 1909	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)
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 1972 in logic and ax-nul 5307 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 1912) and every existential formula is false (emptyex 1911), and every variable is effectively nonfree in any formula (emptynf 1913).
Theorem	empty 1910	Two characterizations of the empty domain. (Contributed by Gérard Lang, 5-Feb-2024.)
⊢ (¬ ∃𝑥⊤ ↔ ∀𝑥⊥)
Theorem	emptyex 1911	On the empty domain, any existentially quantified formula is false. (Contributed by Wolf Lammen, 21-Jan-2024.)
⊢ (¬ ∃𝑥⊤ → ¬ ∃𝑥𝜑)
Theorem	emptyal 1912	On the empty domain, any universally quantified formula is true. (Contributed by Wolf Lammen, 12-Mar-2023.)
⊢ (¬ ∃𝑥⊤ → ∀𝑥𝜑)
Theorem	emptynf 1913	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
Axiom	ax-5 1914*	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 37777 about the logical redundancy of ax-5 1914 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 2177, we can also remove the quantifier (unconditionally). For an explanation of disjoint variable conditions, see https://us.metamath.org/mpeuni/mmset.html#distinct 2177. (Contributed by NM, 10-Jan-1993.)
⊢ (𝜑 → ∀𝑥𝜑)
Theorem	ax5d 1915*	Version of ax-5 1914 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders. (Contributed by NM, 1-Mar-2013.)
⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
Theorem	ax5e 1916*	A rephrasing of ax-5 1914 using the existential quantifier. (Contributed by Wolf Lammen, 4-Dec-2017.)
⊢ (∃𝑥𝜑 → 𝜑)
Theorem	ax5ea 1917*	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 1918*	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 1919*	nfv 1918 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1898. (Contributed by Mario Carneiro, 6-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝜓)
Theorem	alimdv 1920*	Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1813. See alimdh 1820 and alimd 2206 for versions without a distinct variable condition. (Contributed by NM, 3-Apr-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
Theorem	eximdv 1921*	Deduction form of Theorem 19.22 of [Margaris] p. 90, see exim 1837. See eximdh 1868 and eximd 2210 for versions without a distinct variable condition. (Contributed by NM, 27-Apr-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
Theorem	2alimdv 1922*	Deduction form of Theorem 19.20 of [Margaris] p. 90 with two quantifiers, see alim 1813. (Contributed by NM, 27-Apr-2004.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥∀𝑦𝜓 → ∀𝑥∀𝑦𝜒))
Theorem	2eximdv 1923*	Deduction form of Theorem 19.22 of [Margaris] p. 90 with two quantifiers, see exim 1837. (Contributed by NM, 3-Aug-1995.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 → ∃𝑥∃𝑦𝜒))
Theorem	albidv 1924*	Formula-building rule for universal quantifier (deduction form). See also albidh 1870 and albid 2216. (Contributed by NM, 26-May-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
Theorem	exbidv 1925*	Formula-building rule for existential quantifier (deduction form). See also exbidh 1871 and exbid 2217. (Contributed by NM, 26-May-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
Theorem	nfbidv 1926*	An equality theorem for nonfreeness. See nfbidf 2218 for a version without disjoint variable condition but requiring more axioms. (Contributed by Mario Carneiro, 4-Oct-2016.) Remove dependency on ax-6 1972, ax-7 2012, ax-12 2172 by adapting proof of nfbidf 2218. (Revised by BJ, 25-Sep-2022.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))
Theorem	2albidv 1927*	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 1928*	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 1929*	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 1930*	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 1931*	Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2201 and 19.21v 1943. (Contributed by NM, 21-Jun-1993.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥𝜓)
Theorem	alrimivv 1932*	Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2201 and 19.21v 1943. (Contributed by NM, 31-Jul-1995.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥∀𝑦𝜓)
Theorem	alrimdv 1933*	Deduction form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2201 and 19.21v 1943. (Contributed by NM, 10-Feb-1997.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))
Theorem	exlimiv 1934*	Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2205. See exlimi 2211 for a more general version requiring more axioms. This inference, along with its many variants such as rexlimdv 3154, 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 3154. 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 1934 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 1935. (Contributed by NM, 21-Jun-1993.) Remove dependencies on ax-6 1972 and ax-8 2109. (Revised by Wolf Lammen, 4-Dec-2017.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥𝜑 → 𝜓)
Theorem	exlimiiv 1935*	Inference (Rule C) associated with exlimiv 1934. (Contributed by BJ, 19-Dec-2020.)
⊢ (𝜑 → 𝜓)    &   ⊢ ∃𝑥𝜑    ⇒   ⊢ 𝜓
Theorem	exlimivv 1936*	Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2205. (Contributed by NM, 1-Aug-1995.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥∃𝑦𝜑 → 𝜓)
Theorem	exlimdv 1937*	Deduction form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2205. (Contributed by NM, 27-Apr-1994.) Remove dependencies on ax-6 1972, ax-7 2012. (Revised by Wolf Lammen, 4-Dec-2017.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))
Theorem	exlimdvv 1938*	Deduction form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2205. (Contributed by NM, 31-Jul-1995.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 → 𝜒))
Theorem	exlimddv 1939*	Existential elimination rule of natural deduction (Rule C, explained in exlimiv 1934). (Contributed by Mario Carneiro, 15-Jun-2016.)
⊢ (𝜑 → ∃𝑥𝜓)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	nexdv 1940*	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 1941*	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 1942*	Version of stdpc5 2202 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.) Revised to shorten 19.21v 1943. (Revised by Wolf Lammen, 12-Jul-2020.)
⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))
Theorem	19.21v 1943*	Version of 19.21 2201 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 nonfreeness hypothesis such as Ⅎ𝑥𝜑. Conversely, we sometimes suffix with "f" the label of a theorem introducing such a nonfreeness hypothesis ("f" stands for "not free in", see df-nf 1787) instead of a disjoint variable condition. For instance, 19.21v 1943 versus 19.21 2201 and vtoclf 3548 versus vtocl 3550. 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 nonfreeness hypothesis, by using nfv 1918. 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 1944*	Version of 19.32 2227 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))
Theorem	19.31v 1945*	Version of 19.31 2228 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ 𝜓))
Theorem	19.23v 1946*	Version of 19.23 2205 with a disjoint variable condition instead of a nonfreeness hypothesis. (Contributed by NM, 28-Jun-1998.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 11-Jan-2020.) Remove dependency on ax-6 1972. (Revised by Rohan Ridenour, 15-Apr-2022.)
⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))
Theorem	19.23vv 1947*	Theorem 19.23v 1946 extended to two variables. (Contributed by NM, 10-Aug-2004.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥∃𝑦𝜑 → 𝜓))
Theorem	pm11.53v 1948*	Version of pm11.53 2343 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))
Theorem	19.36imv 1949*	One direction of 19.36v 1992 that can be proven without ax-6 1972. (Contributed by Rohan Ridenour, 16-Apr-2022.) (Proof shortened by Wolf Lammen, 22-Sep-2024.)
⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
Theorem	19.36imvOLD 1950*	Obsolete version of 19.36imv 1949 as of 22-Sep-2024. (Contributed by Rohan Ridenour, 16-Apr-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
Theorem	19.36iv 1951*	Inference associated with 19.36v 1992. Version of 19.36i 2225 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 1972. (Revised by Rohan Ridenour, 15-Apr-2022.)
⊢ ∃𝑥(𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	19.37imv 1952*	One direction of 19.37v 1996 that can be proven without ax-6 1972. (Contributed by Rohan Ridenour, 16-Apr-2022.)
⊢ (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))
Theorem	19.37iv 1953*	Inference associated with 19.37v 1996. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-6 1972. (Revised by Rohan Ridenour, 15-Apr-2022.)
⊢ ∃𝑥(𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	19.41v 1954*	Version of 19.41 2229 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.) Remove dependency on ax-6 1972. (Revised by Rohan Ridenour, 15-Apr-2022.)
⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))
Theorem	19.41vv 1955*	Version of 19.41 2229 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 1956*	Version of 19.41 2229 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 1957*	Version of 19.41 2229 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 1958*	Version of 19.42 2230 with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
Theorem	exdistr 1959*	Distribution of existential quantifiers. See also exdistrv 1960. (Contributed by NM, 9-Mar-1995.)
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
Theorem	exdistrv 1960*	Distribute a pair of existential quantifiers (over disjoint variables) over a conjunction. Combination of 19.41v 1954 and 19.42v 1958. For a version with fewer disjoint variable conditions but requiring more axioms, see eeanv 2346. (Contributed by BJ, 30-Sep-2022.)
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
Theorem	4exdistrv 1961*	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 2348. (Contributed by BJ, 5-Jan-2023.)
⊢ (∃𝑥∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓))
Theorem	19.42vv 1962*	Version of 19.42 2230 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 1963*	Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.)
⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦∃𝑧𝜓))
Theorem	19.42vvv 1964*	Version of 19.42 2230 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	3exdistr 1965*	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 1966*	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))))
1.4.5  Equality predicate (continued) The equality predicate was introduced above in wceq 1542 for use by df-tru 1545. See the comments in that section. In this section, we continue with its first "real" use.
Theorem	weq 1967	Extend wff definition to include atomic formulas using the equality predicate. (Instead of introducing weq 1967 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 1542. This lets us avoid overloading the = connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1967 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1542. 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	speimfw 1968	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 1969	Alternate proof of speimfw 1968 (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 1970	Specialization, with additional weakening (compared to sp 2177) 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 1971	Inference that has ax-12 2172 (without ∀𝑦) as its conclusion. Uses only Tarski's FOL axiom schemes. The hypotheses may be eliminable without using ax-12 2172 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)))
1.4.6  Axiom scheme ax-6 (Existence)
Axiom	ax-6 1972	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 2385 and ax6fromc10 37766. A more convenient form of this axiom is ax6e 2383, 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 2383. ax-6 1972 can be proved from the weaker version ax6v 1973 requiring that the variables be distinct; see Theorem ax6 2384. ax-6 1972 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 5304. Except by ax6v 1973, this axiom should not be referenced directly. Instead, use Theorem ax6 2384. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Theorem	ax6v 1973*	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 1972 by adding a distinct variable restriction ($d). From here on, ax-6 1972 should not be referenced directly by any other proof, so that Theorem ax6 2384 will show that we can recover ax-6 1972 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 1973 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 1972. When possible, use of this theorem rather than ax6 2384 is preferred since its derivation is much shorter and requires fewer axioms. (Contributed by NM, 7-Aug-2015.)
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Theorem	ax6ev 1974*	At least one individual exists. Weaker version of ax6e 2383. When possible, use of this theorem rather than ax6e 2383 is preferred since its derivation is much shorter and requires fewer axioms. (Contributed by NM, 3-Aug-2017.)
⊢ ∃𝑥 𝑥 = 𝑦
Theorem	spimw 1975*	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 1976*	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	speiv 1977*	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 1978*	Version of spei 2394 with a disjoint variable condition, which does not require ax-13 2372 (neither ax-7 2012 nor ax-12 2172). (Contributed by BJ, 31-May-2019.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜓    ⇒   ⊢ ∃𝑥𝜑
Theorem	exgen 1979	Rule of existential generalization, similar to universal generalization ax-gen 1798, but valid only if an individual exists. Its proof requires ax-6 1972 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 1798, ax-4 1812 and this theorem alone, not requiring ax-7 2012 or excessive distinct variable conditions. (Contributed by Wolf Lammen, 12-Nov-2017.) (Proof shortened by Wolf Lammen, 20-Oct-2023.)
⊢ 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 2182 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 1787). (Contributed by NM, 2-Aug-2017.) Remove dependency on ax-7 2012. (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 2175 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 2177. 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 2177 when 𝑥 does not occur in 𝜑. Converse of ax-5 1914. 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 2196 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 2012. (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 2175 with a disjoint variable condition, requiring fewer axioms. Converse of ax5e 1916. (Contributed by BJ, 12-Mar-2020.)
⊢ (𝜑 → ∃𝑥𝜑)
Theorem	19.9v 1988*	Version of 19.9 2199 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 2012. (Revised by Wolf Lammen, 4-Dec-2017.)
⊢ (∃𝑥𝜑 ↔ 𝜑)
Theorem	19.39 1989	Theorem 19.39 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ ((∃𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓))
Theorem	19.24 1990	Theorem 19.24 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∃𝑥(𝜑 → 𝜓))
Theorem	19.34 1991	Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ ((∀𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑 ∨ 𝜓))
Theorem	19.36v 1992*	Version of 19.36 2224 with a disjoint variable condition instead of a nonfreeness 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 1993*	Version of 19.12vv 2344 with a disjoint variable condition, requiring fewer axioms. See also 19.12 2321. (Contributed by BJ, 18-Mar-2020.)
⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓))
Theorem	19.27v 1994*	Version of 19.27 2221 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 3-Jun-2004.)
⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓))
Theorem	19.28v 1995*	Version of 19.28 2222 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 25-Mar-2004.)
⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
Theorem	19.37v 1996*	Version of 19.37 2226 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
⊢ (∃𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥𝜓))
Theorem	19.44v 1997*	Version of 19.44 2231 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ 𝜓))
Theorem	19.45v 1998*	Version of 19.45 2232 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
Theorem	spimevw 1999*	Existential introduction, using implicit substitution. This is to spimew 1976 what spimvw 2000 is to spimw 1975. Version of spimev 2392 and spimefv 2192 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 → (𝜑 → 𝜓))    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	spimvw 2000*	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 2390 and spimfv 2233. (Contributed by NM, 9-Apr-2017.)
⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)