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	19.35i 1901	Inference associated with 19.35 1900. (Contributed by NM, 21-Jun-1993.)
⊢ ∃𝑥(𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → ∃𝑥𝜓)
Theorem	19.35ri 1902	Inference associated with 19.35 1900. (Contributed by NM, 12-Mar-1993.)
⊢ (∀𝑥𝜑 → ∃𝑥𝜓)    ⇒   ⊢ ∃𝑥(𝜑 → 𝜓)
Theorem	19.25 1903	Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ (∀𝑦∃𝑥(𝜑 → 𝜓) → (∃𝑦∀𝑥𝜑 → ∃𝑦∃𝑥𝜓))
Theorem	19.30 1904	Theorem 19.30 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓))
Theorem	19.43 1905	Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)
⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
Theorem	19.43OLD 1906	Obsolete proof of 19.43 1905. Do not delete as it is referenced on the mmrecent.html 1905 page and in conventions-labels 30661. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
Theorem	19.33 1907	Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑 ∨ 𝜓))
Theorem	19.33b 1908	The antecedent provides a condition implying the converse of 19.33 1907. (Contributed by NM, 27-Mar-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 5-Jul-2014.)
⊢ (¬ (∃𝑥𝜑 ∧ ∃𝑥𝜓) → (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓)))
Theorem	19.40 1909	Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 26-May-1993.)
⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
Theorem	19.40-2 1910	Theorem *11.42 in [WhiteheadRussell] p. 163. Theorem 19.40 of [Margaris] p. 90 with two quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → (∃𝑥∃𝑦𝜑 ∧ ∃𝑥∃𝑦𝜓))
Theorem	19.40b 1911	The antecedent provides a condition implying the converse of 19.40 1909. This is to 19.40 1909 what 19.33b 1908 is to 19.33 1907. (Contributed by BJ, 6-May-2019.) (Proof shortened by Wolf Lammen, 13-Nov-2020.)
⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ ∃𝑥(𝜑 ∧ 𝜓)))
Theorem	albiim 1912	Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜑)))
Theorem	2albiim 1913	Split a biconditional and distribute two quantifiers. (Contributed by NM, 3-Feb-2005.)
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) ↔ (∀𝑥∀𝑦(𝜑 → 𝜓) ∧ ∀𝑥∀𝑦(𝜓 → 𝜑)))
Theorem	exintrbi 1914	Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ 𝜓)))
Theorem	exintr 1915	Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.) (Proof shortened by BJ, 16-Sep-2022.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))
Theorem	alsyl 1916	Universally quantified and uncurried (imported) form of syllogism. Theorem *10.3 in [WhiteheadRussell] p. 150. (Contributed by Andrew Salmon, 8-Jun-2011.)
⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜒)) → ∀𝑥(𝜑 → 𝜒))
Theorem	nfimd 1917	If in a context 𝑥 is not free in 𝜓 and 𝜒, then it is not free in (𝜓 → 𝜒). Deduction form of nfim 1919. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) df-nf 1807 changed. (Revised by Wolf Lammen, 18-Sep-2021.) Eliminate curried form of nfimt 1918. (Revised by Wolf Lammen, 10-Jul-2022.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝜒))
Theorem	nfimt 1918	Closed form of nfim 1919 and nfimd 1917. (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 1919	If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑 → 𝜓). Inference associated with nfimt 1918. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) df-nf 1807 changed. (Revised by Wolf Lammen, 17-Sep-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ Ⅎ𝑥(𝜑 → 𝜓)
Theorem	nfand 1920	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 1921	Deduction form of bound-variable hypothesis builder nf3an 1924. (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 1922	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 1923	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 1924	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 1925	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 1926	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 1927	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 1928	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 1990 in logic and ax-nul 5261 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 1931) and every existential formula is false (emptyex 1930), and every variable is effectively nonfree in any formula (emptynf 1932).
Theorem	empty 1929	Two characterizations of the empty domain. (Contributed by Gérard Lang, 5-Feb-2024.)
⊢ (¬ ∃𝑥⊤ ↔ ∀𝑥⊥)
Theorem	emptyex 1930	On the empty domain, any existentially quantified formula is false. (Contributed by Wolf Lammen, 21-Jan-2024.)
⊢ (¬ ∃𝑥⊤ → ¬ ∃𝑥𝜑)
Theorem	emptyal 1931	On the empty domain, any universally quantified formula is true. (Contributed by Wolf Lammen, 12-Mar-2023.)
⊢ (¬ ∃𝑥⊤ → ∀𝑥𝜑)
Theorem	emptynf 1932	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 1933*	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 39543 about the logical redundancy of ax-5 1933 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 2221, we can also remove the quantifier (unconditionally). For an explanation of disjoint variable conditions, see https://us.metamath.org/mpeuni/mmset.html#distinct 2221. (Contributed by NM, 10-Jan-1993.)
⊢ (𝜑 → ∀𝑥𝜑)
Theorem	ax5d 1934*	Version of ax-5 1933 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders. (Contributed by NM, 1-Mar-2013.)
⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
Theorem	ax5e 1935*	A rephrasing of ax-5 1933 using the existential quantifier. (Contributed by Wolf Lammen, 4-Dec-2017.)
⊢ (∃𝑥𝜑 → 𝜑)
Theorem	ax5ea 1936*	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 1937*	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 1938*	nfv 1937 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1917. (Contributed by Mario Carneiro, 6-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝜓)
Theorem	alimdv 1939*	Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1833. See alimdh 1840 and alimd 2250 for versions without a distinct variable condition. (Contributed by NM, 3-Apr-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
Theorem	eximdv 1940*	Deduction form of Theorem 19.22 of [Margaris] p. 90, see exim 1857. See eximdh 1887 and eximd 2254 for versions without a distinct variable condition. (Contributed by NM, 27-Apr-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
Theorem	2alimdv 1941*	Deduction form of Theorem 19.20 of [Margaris] p. 90 with two quantifiers, see alim 1833. (Contributed by NM, 27-Apr-2004.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥∀𝑦𝜓 → ∀𝑥∀𝑦𝜒))
Theorem	2eximdv 1942*	Deduction form of Theorem 19.22 of [Margaris] p. 90 with two quantifiers, see exim 1857. (Contributed by NM, 3-Aug-1995.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 → ∃𝑥∃𝑦𝜒))
Theorem	albidv 1943*	Formula-building rule for universal quantifier (deduction form). See also albidh 1889 and albid 2260. (Contributed by NM, 26-May-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
Theorem	exbidv 1944*	Formula-building rule for existential quantifier (deduction form). See also exbidh 1890 and exbid 2261. (Contributed by NM, 26-May-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
Theorem	nfbidv 1945*	An equality theorem for nonfreeness. See nfbidf 2262 for a version without disjoint variable condition but requiring more axioms. (Contributed by Mario Carneiro, 4-Oct-2016.) Remove dependency on ax-6 1990, ax-7 2031, ax-12 2215 by adapting proof of nfbidf 2262. (Revised by BJ, 25-Sep-2022.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))
Theorem	2albidv 1946*	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 1947*	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 1948*	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 1949*	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 1950*	Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2245 and 19.21v 1962. (Contributed by NM, 21-Jun-1993.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥𝜓)
Theorem	alrimivv 1951*	Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2245 and 19.21v 1962. (Contributed by NM, 31-Jul-1995.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥∀𝑦𝜓)
Theorem	alrimdv 1952*	Deduction form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2245 and 19.21v 1962. (Contributed by NM, 10-Feb-1997.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))
Theorem	exlimiv 1953*	Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2249. See exlimi 2255 for a more general version requiring more axioms. This inference, along with its many variants such as rexlimdv 3164, 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 3164. 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 1953 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 1954. (Contributed by NM, 21-Jun-1993.) Remove dependencies on ax-6 1990 and ax-8 2147. (Revised by Wolf Lammen, 4-Dec-2017.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥𝜑 → 𝜓)
Theorem	exlimiiv 1954*	Inference (Rule C) associated with exlimiv 1953. (Contributed by BJ, 19-Dec-2020.)
⊢ (𝜑 → 𝜓)    &   ⊢ ∃𝑥𝜑    ⇒   ⊢ 𝜓
Theorem	exlimivv 1955*	Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2249. (Contributed by NM, 1-Aug-1995.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥∃𝑦𝜑 → 𝜓)
Theorem	exlimdv 1956*	Deduction form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2249. (Contributed by NM, 27-Apr-1994.) Remove dependencies on ax-6 1990, ax-7 2031. (Revised by Wolf Lammen, 4-Dec-2017.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))
Theorem	exlimdvv 1957*	Deduction form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2249. (Contributed by NM, 31-Jul-1995.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 → 𝜒))
Theorem	exlimddv 1958*	Existential elimination rule of natural deduction (Rule C, explained in exlimiv 1953). (Contributed by Mario Carneiro, 15-Jun-2016.)
⊢ (𝜑 → ∃𝑥𝜓)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	nexdv 1959*	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 1960*	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 1961*	Version of stdpc5 2246 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.) Revised to shorten 19.21v 1962. (Revised by Wolf Lammen, 12-Jul-2020.)
⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))
Theorem	19.21v 1962*	Version of 19.21 2245 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 1807) instead of a disjoint variable condition. For instance, 19.21v 1962 versus 19.21 2245 and vtoclf 3533 versus vtocl 3528. 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 1937. 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 1963*	Version of 19.32 2271 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))
Theorem	19.31v 1964*	Version of 19.31 2272 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ 𝜓))
Theorem	19.23v 1965*	Version of 19.23 2249 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 1990. (Revised by Rohan Ridenour, 15-Apr-2022.)
⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))
Theorem	19.23vv 1966*	Theorem 19.23v 1965 extended to two variables. (Contributed by NM, 10-Aug-2004.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥∃𝑦𝜑 → 𝜓))
Theorem	pm11.53v 1967*	Version of pm11.53 2380 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))
Theorem	19.36imv 1968*	One direction of 19.36v 2016 that can be proven without ax-6 1990. (Contributed by Rohan Ridenour, 16-Apr-2022.) (Proof shortened by Wolf Lammen, 22-Sep-2024.)
⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
Theorem	19.36iv 1969*	Inference associated with 19.36v 2016. Version of 19.36i 2269 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 1990. (Revised by Rohan Ridenour, 15-Apr-2022.)
⊢ ∃𝑥(𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	19.37imv 1970*	One direction of 19.37v 2020 that can be proven without ax-6 1990. (Contributed by Rohan Ridenour, 16-Apr-2022.)
⊢ (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))
Theorem	19.37iv 1971*	Inference associated with 19.37v 2020. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-6 1990. (Revised by Rohan Ridenour, 15-Apr-2022.)
⊢ ∃𝑥(𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	19.41v 1972*	Version of 19.41 2273 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.) Remove dependency on ax-6 1990. (Revised by Rohan Ridenour, 15-Apr-2022.)
⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))
Theorem	19.41vv 1973*	Version of 19.41 2273 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 1974*	Version of 19.41 2273 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 1975*	Version of 19.41 2273 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 1976*	Version of 19.42 2274 with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
Theorem	exdistr 1977*	Distribution of existential quantifiers. See also exdistrv 1978. (Contributed by NM, 9-Mar-1995.)
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
Theorem	exdistrv 1978*	Distribute a pair of existential quantifiers (over disjoint variables) over a conjunction. Combination of 19.41v 1972 and 19.42v 1976. For a version with fewer disjoint variable conditions but requiring more axioms, see eeanv 2383. (Contributed by BJ, 30-Sep-2022.)
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
Theorem	4exdistrv 1979*	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 2385. (Contributed by BJ, 5-Jan-2023.)
⊢ (∃𝑥∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓))
Theorem	19.42vv 1980*	Version of 19.42 2274 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 1981*	Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.)
⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦∃𝑧𝜓))
Theorem	19.42vvv 1982*	Version of 19.42 2274 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 1983*	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 1984*	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 1563 for use by df-tru 1566. See the comments in that section. In this section, we continue with its first "real" use.
Theorem	weq 1985	Extend wff definition to include atomic formulas using the equality predicate. (Instead of introducing weq 1985 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 1563. This lets us avoid overloading the = connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1985 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1563. 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 1986	Specialization, with additional weakening (compared to 19.2 1999) 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 1987	Alternate proof of speimfw 1986 (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 1988	Specialization, with additional weakening (compared to sp 2221) 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 1989	Inference that has ax-12 2215 (without ∀𝑦) as its conclusion. Uses only Tarski's FOL axiom schemes. The hypotheses may be eliminable without using ax-12 2215 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 1990	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 2419 and ax6fromc10 39532. A more convenient form of this axiom is ax6e 2417, 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 2417. ax-6 1990 can be proved from the weaker version ax6v 1991 requiring that the variables be distinct; see Theorem ax6 2418. ax-6 1990 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 5258. Except by ax6v 1991, this axiom should not be referenced directly. Instead, use Theorem ax6 2418. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Theorem	ax6v 1991*	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 1990 by adding a distinct variable restriction ($d). From here on, ax-6 1990 should not be referenced directly by any other proof, so that Theorem ax6 2418 will show that we can recover ax-6 1990 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 1991 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 1990. When possible, use of this theorem rather than ax6 2418 is preferred since its derivation is much shorter and requires fewer axioms. (Contributed by NM, 7-Aug-2015.)
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Theorem	ax6ev 1992*	At least one individual exists. Weaker version of ax6e 2417. When possible, use of this theorem rather than ax6e 2417 is preferred since its derivation is much shorter and requires fewer axioms. (Contributed by NM, 3-Aug-2017.)
⊢ ∃𝑥 𝑥 = 𝑦
Theorem	spimw 1993*	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 1994*	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 1995*	Inference from existential specialization. (Contributed by NM, 19-Aug-1993.) Use spimew 1994. (Revised by Wolf Lammen, 22-Oct-2023.)
⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑))    &   ⊢ 𝜓    ⇒   ⊢ ∃𝑥𝜑
Theorem	speivw 1996*	Version of spei 2428 with a disjoint variable condition, which does not require ax-13 2406 (neither ax-7 2031 nor ax-12 2215). (Contributed by BJ, 31-May-2019.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜓    ⇒   ⊢ ∃𝑥𝜑
Theorem	exgen 1997	Rule of existential generalization, similar to universal generalization ax-gen 1818, but valid only if an individual exists. Its proof requires ax-6 1990 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 1818, ax-4 1832 and this theorem alone, not requiring ax-7 2031 or excessive distinct variable conditions. (Contributed by Wolf Lammen, 12-Nov-2017.) (Proof shortened by Wolf Lammen, 20-Oct-2023.)
⊢ 𝜑    ⇒   ⊢ ∃𝑥𝜑
Theorem	extru 1998	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 1999. (Contributed by Anthony Hart, 13-Sep-2011.) (Proof shortened by BJ, 12-May-2019.)
⊢ ∃𝑥⊤
Theorem	19.2 1999	Theorem 19.2 of [Margaris] p. 89. This corresponds to the axiom (D) of modal logic (the other standard formulation being extru 1998). 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 2226 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 1807). (Contributed by NM, 2-Aug-2017.) Remove dependency on ax-7 2031. (Revised by Wolf Lammen, 4-Dec-2017.)
⊢ (∀𝑥𝜑 → ∃𝑥𝜑)
Theorem	19.2d 2000	Deduction associated with 19.2 1999. (Contributed by BJ, 12-May-2019.)
⊢ (𝜑 → ∀𝑥𝜓)    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)