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	nfimd 1901	If in a context 𝑥 is not free in 𝜓 and 𝜒, then it is not free in (𝜓 → 𝜒). Deduction form of nfim 1903. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) df-nf 1791 changed. (Revised by Wolf Lammen, 18-Sep-2021.) Eliminate curried form of nfimt 1902. (Revised by Wolf Lammen, 10-Jul-2022.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝜒))
Theorem	nfimt 1902	Closed form of nfim 1903 and nfimd 1901. (Contributed by BJ, 20-Oct-2021.) Eliminate curried form, former name nfimt2. (Revised by Wolf Lammen, 6-Jul-2022.)
⊢ ((Ⅎ𝑥𝜑 ∧ Ⅎ𝑥𝜓) → Ⅎ𝑥(𝜑 → 𝜓))
Theorem	nfim 1903	If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑 → 𝜓). Inference associated with nfimt 1902. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) df-nf 1791 changed. (Revised by Wolf Lammen, 17-Sep-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ Ⅎ𝑥(𝜑 → 𝜓)
Theorem	nfand 1904	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 1905	Deduction form of bound-variable hypothesis builder nf3an 1908. (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 1906	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 1907	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 1908	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 1909	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 1910	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 1911	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 1912	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 1974 in logic and ax-nul 5235 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 1915) and every existential formula is false (emptyex 1914), and every variable is effectively nonfree in any formula (emptynf 1916).
Theorem	empty 1913	Two characterizations of the empty domain. (Contributed by Gérard Lang, 5-Feb-2024.)
⊢ (¬ ∃𝑥⊤ ↔ ∀𝑥⊥)
Theorem	emptyex 1914	On the empty domain, any existentially quantified formula is false. (Contributed by Wolf Lammen, 21-Jan-2024.)
⊢ (¬ ∃𝑥⊤ → ¬ ∃𝑥𝜑)
Theorem	emptyal 1915	On the empty domain, any universally quantified formula is true. (Contributed by Wolf Lammen, 12-Mar-2023.)
⊢ (¬ ∃𝑥⊤ → ∀𝑥𝜑)
Theorem	emptynf 1916	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 1917*	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 39400 about the logical redundancy of ax-5 1917 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 2195, we can also remove the quantifier (unconditionally). For an explanation of disjoint variable conditions, see https://us.metamath.org/mpeuni/mmset.html#distinct 2195. (Contributed by NM, 10-Jan-1993.)
⊢ (𝜑 → ∀𝑥𝜑)
Theorem	ax5d 1918*	Version of ax-5 1917 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders. (Contributed by NM, 1-Mar-2013.)
⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
Theorem	ax5e 1919*	A rephrasing of ax-5 1917 using the existential quantifier. (Contributed by Wolf Lammen, 4-Dec-2017.)
⊢ (∃𝑥𝜑 → 𝜑)
Theorem	ax5ea 1920*	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 1921*	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 1922*	nfv 1921 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1901. (Contributed by Mario Carneiro, 6-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝜓)
Theorem	alimdv 1923*	Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1817. See alimdh 1824 and alimd 2224 for versions without a distinct variable condition. (Contributed by NM, 3-Apr-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
Theorem	eximdv 1924*	Deduction form of Theorem 19.22 of [Margaris] p. 90, see exim 1841. See eximdh 1871 and eximd 2228 for versions without a distinct variable condition. (Contributed by NM, 27-Apr-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
Theorem	2alimdv 1925*	Deduction form of Theorem 19.20 of [Margaris] p. 90 with two quantifiers, see alim 1817. (Contributed by NM, 27-Apr-2004.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥∀𝑦𝜓 → ∀𝑥∀𝑦𝜒))
Theorem	2eximdv 1926*	Deduction form of Theorem 19.22 of [Margaris] p. 90 with two quantifiers, see exim 1841. (Contributed by NM, 3-Aug-1995.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 → ∃𝑥∃𝑦𝜒))
Theorem	albidv 1927*	Formula-building rule for universal quantifier (deduction form). See also albidh 1873 and albid 2234. (Contributed by NM, 26-May-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
Theorem	exbidv 1928*	Formula-building rule for existential quantifier (deduction form). See also exbidh 1874 and exbid 2235. (Contributed by NM, 26-May-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
Theorem	nfbidv 1929*	An equality theorem for nonfreeness. See nfbidf 2236 for a version without disjoint variable condition but requiring more axioms. (Contributed by Mario Carneiro, 4-Oct-2016.) Remove dependency on ax-6 1974, ax-7 2015, ax-12 2189 by adapting proof of nfbidf 2236. (Revised by BJ, 25-Sep-2022.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))
Theorem	2albidv 1930*	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 1931*	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 1932*	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 1933*	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 1934*	Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2219 and 19.21v 1946. (Contributed by NM, 21-Jun-1993.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥𝜓)
Theorem	alrimivv 1935*	Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2219 and 19.21v 1946. (Contributed by NM, 31-Jul-1995.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥∀𝑦𝜓)
Theorem	alrimdv 1936*	Deduction form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2219 and 19.21v 1946. (Contributed by NM, 10-Feb-1997.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))
Theorem	exlimiv 1937*	Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2223. See exlimi 2229 for a more general version requiring more axioms. This inference, along with its many variants such as rexlimdv 3139, 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 3139. 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 1937 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 1938. (Contributed by NM, 21-Jun-1993.) Remove dependencies on ax-6 1974 and ax-8 2121. (Revised by Wolf Lammen, 4-Dec-2017.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥𝜑 → 𝜓)
Theorem	exlimiiv 1938*	Inference (Rule C) associated with exlimiv 1937. (Contributed by BJ, 19-Dec-2020.)
⊢ (𝜑 → 𝜓)    &   ⊢ ∃𝑥𝜑    ⇒   ⊢ 𝜓
Theorem	exlimivv 1939*	Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2223. (Contributed by NM, 1-Aug-1995.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥∃𝑦𝜑 → 𝜓)
Theorem	exlimdv 1940*	Deduction form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2223. (Contributed by NM, 27-Apr-1994.) Remove dependencies on ax-6 1974, ax-7 2015. (Revised by Wolf Lammen, 4-Dec-2017.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))
Theorem	exlimdvv 1941*	Deduction form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2223. (Contributed by NM, 31-Jul-1995.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 → 𝜒))
Theorem	exlimddv 1942*	Existential elimination rule of natural deduction (Rule C, explained in exlimiv 1937). (Contributed by Mario Carneiro, 15-Jun-2016.)
⊢ (𝜑 → ∃𝑥𝜓)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	nexdv 1943*	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 1944*	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 1945*	Version of stdpc5 2220 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.) Revised to shorten 19.21v 1946. (Revised by Wolf Lammen, 12-Jul-2020.)
⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))
Theorem	19.21v 1946*	Version of 19.21 2219 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 1791) instead of a disjoint variable condition. For instance, 19.21v 1946 versus 19.21 2219 and vtoclf 3512 versus vtocl 3506. 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 1921. 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 1947*	Version of 19.32 2245 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))
Theorem	19.31v 1948*	Version of 19.31 2246 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ 𝜓))
Theorem	19.23v 1949*	Version of 19.23 2223 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 1974. (Revised by Rohan Ridenour, 15-Apr-2022.)
⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))
Theorem	19.23vv 1950*	Theorem 19.23v 1949 extended to two variables. (Contributed by NM, 10-Aug-2004.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥∃𝑦𝜑 → 𝜓))
Theorem	pm11.53v 1951*	Version of pm11.53 2354 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))
Theorem	19.36imv 1952*	One direction of 19.36v 2000 that can be proven without ax-6 1974. (Contributed by Rohan Ridenour, 16-Apr-2022.) (Proof shortened by Wolf Lammen, 22-Sep-2024.)
⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
Theorem	19.36iv 1953*	Inference associated with 19.36v 2000. Version of 19.36i 2243 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 1974. (Revised by Rohan Ridenour, 15-Apr-2022.)
⊢ ∃𝑥(𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	19.37imv 1954*	One direction of 19.37v 2004 that can be proven without ax-6 1974. (Contributed by Rohan Ridenour, 16-Apr-2022.)
⊢ (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))
Theorem	19.37iv 1955*	Inference associated with 19.37v 2004. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-6 1974. (Revised by Rohan Ridenour, 15-Apr-2022.)
⊢ ∃𝑥(𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	19.41v 1956*	Version of 19.41 2247 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.) Remove dependency on ax-6 1974. (Revised by Rohan Ridenour, 15-Apr-2022.)
⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))
Theorem	19.41vv 1957*	Version of 19.41 2247 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 1958*	Version of 19.41 2247 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 1959*	Version of 19.41 2247 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 1960*	Version of 19.42 2248 with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
Theorem	exdistr 1961*	Distribution of existential quantifiers. See also exdistrv 1962. (Contributed by NM, 9-Mar-1995.)
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
Theorem	exdistrv 1962*	Distribute a pair of existential quantifiers (over disjoint variables) over a conjunction. Combination of 19.41v 1956 and 19.42v 1960. For a version with fewer disjoint variable conditions but requiring more axioms, see eeanv 2357. (Contributed by BJ, 30-Sep-2022.)
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
Theorem	4exdistrv 1963*	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 2359. (Contributed by BJ, 5-Jan-2023.)
⊢ (∃𝑥∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓))
Theorem	19.42vv 1964*	Version of 19.42 2248 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 1965*	Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.)
⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦∃𝑧𝜓))
Theorem	19.42vvv 1966*	Version of 19.42 2248 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 1967*	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 1968*	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 1547 for use by df-tru 1550. See the comments in that section. In this section, we continue with its first "real" use.
Theorem	weq 1969	Extend wff definition to include atomic formulas using the equality predicate. (Instead of introducing weq 1969 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 1547. This lets us avoid overloading the = connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1969 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1547. 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 1970	Specialization, with additional weakening (compared to 19.2 1983) 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 1971	Alternate proof of speimfw 1970 (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 1972	Specialization, with additional weakening (compared to sp 2195) 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 1973	Inference that has ax-12 2189 (without ∀𝑦) as its conclusion. Uses only Tarski's FOL axiom schemes. The hypotheses may be eliminable without using ax-12 2189 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 1974	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 2393 and ax6fromc10 39389. A more convenient form of this axiom is ax6e 2391, 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 2391. ax-6 1974 can be proved from the weaker version ax6v 1975 requiring that the variables be distinct; see Theorem ax6 2392. ax-6 1974 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 5232. Except by ax6v 1975, this axiom should not be referenced directly. Instead, use Theorem ax6 2392. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Theorem	ax6v 1975*	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 1974 by adding a distinct variable restriction ($d). From here on, ax-6 1974 should not be referenced directly by any other proof, so that Theorem ax6 2392 will show that we can recover ax-6 1974 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 1975 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 1974. When possible, use of this theorem rather than ax6 2392 is preferred since its derivation is much shorter and requires fewer axioms. (Contributed by NM, 7-Aug-2015.)
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Theorem	ax6ev 1976*	At least one individual exists. Weaker version of ax6e 2391. When possible, use of this theorem rather than ax6e 2391 is preferred since its derivation is much shorter and requires fewer axioms. (Contributed by NM, 3-Aug-2017.)
⊢ ∃𝑥 𝑥 = 𝑦
Theorem	spimw 1977*	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 1978*	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 1979*	Inference from existential specialization. (Contributed by NM, 19-Aug-1993.) Use spimew 1978. (Revised by Wolf Lammen, 22-Oct-2023.)
⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑))    &   ⊢ 𝜓    ⇒   ⊢ ∃𝑥𝜑
Theorem	speivw 1980*	Version of spei 2402 with a disjoint variable condition, which does not require ax-13 2380 (neither ax-7 2015 nor ax-12 2189). (Contributed by BJ, 31-May-2019.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜓    ⇒   ⊢ ∃𝑥𝜑
Theorem	exgen 1981	Rule of existential generalization, similar to universal generalization ax-gen 1802, but valid only if an individual exists. Its proof requires ax-6 1974 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 1802, ax-4 1816 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	extru 1982	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 1983. (Contributed by Anthony Hart, 13-Sep-2011.) (Proof shortened by BJ, 12-May-2019.)
⊢ ∃𝑥⊤
Theorem	19.2 1983	Theorem 19.2 of [Margaris] p. 89. This corresponds to the axiom (D) of modal logic (the other standard formulation being extru 1982). 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 2200 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 1791). (Contributed by NM, 2-Aug-2017.) Remove dependency on ax-7 2015. (Revised by Wolf Lammen, 4-Dec-2017.)
⊢ (∀𝑥𝜑 → ∃𝑥𝜑)
Theorem	19.2d 1984	Deduction associated with 19.2 1983. (Contributed by BJ, 12-May-2019.)
⊢ (𝜑 → ∀𝑥𝜓)    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	19.8w 1985	Weak version of 19.8a 2193 and instance of 19.2d 1984. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (𝜑 → ∃𝑥𝜑)
Theorem	spnfw 1986	Weak version of sp 2195. 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	spfalw 1987	Version of sp 2195 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    ⇒   ⊢ (∀𝑥𝜑 → 𝜑)
Theorem	spvw 1988*	Version of sp 2195 when 𝑥 does not occur in 𝜑. Converse of ax-5 1917. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.) Shorten 19.3v 1989. (Revised by Wolf Lammen, 20-Oct-2023.)
⊢ (∀𝑥𝜑 → 𝜑)
Theorem	19.3v 1989*	Version of 19.3 2214 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 1991. (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 1990*	Version of 19.8a 2193 with a disjoint variable condition, requiring fewer axioms. Converse of ax5e 1919. (Contributed by BJ, 12-Mar-2020.)
⊢ (𝜑 → ∃𝑥𝜑)
Theorem	19.9v 1991*	Version of 19.9 2217 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 1989. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 2015. (Revised by Wolf Lammen, 4-Dec-2017.)
⊢ (∃𝑥𝜑 ↔ 𝜑)
Theorem	spimevw 1992*	Existential introduction, using implicit substitution. This is to spimew 1978 what spimvw 1993 is to spimw 1977. Version of spimev 2400 and spimefv 2210 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 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 2398 and spimfv 2251. (Contributed by NM, 9-Apr-2017.)
⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	spsv 1994*	Generalization of antecedent. A trivial weak version of sps 2197 avoiding ax-12 2189. (Contributed by SN, 13-Nov-2025.) (Proof shortened by WL, 19-Nov-2025.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	spvv 1995*	Specialization, using implicit substitution. Version of spv 2401 with a disjoint variable condition, which does not require ax-7 2015, ax-12 2189, ax-13 2380. (Contributed by NM, 30-Aug-1993.) (Revised by BJ, 31-May-2019.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	chvarvv 1996*	Implicit substitution of 𝑦 for 𝑥 into a theorem. Version of chvarv 2404 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by NM, 20-Apr-1994.) (Revised by BJ, 31-May-2019.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
Theorem	19.39 1997	Theorem 19.39 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ ((∃𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓))
Theorem	19.24 1998	Theorem 19.24 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∃𝑥(𝜑 → 𝜓))
Theorem	19.34 1999	Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ ((∀𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑 ∨ 𝜓))
Theorem	19.36v 2000*	Version of 19.36 2242 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))