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	exsimpl 1901	Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜑)
Theorem	exsimpr 1902	Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜓)
Theorem	19.26 1903	Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of [WhiteheadRussell] p. 147. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
Theorem	19.26-2 1904	Theorem 19.26 1903 with two quantifiers. (Contributed by NM, 3-Feb-2005.)
⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓))
Theorem	19.26-3an 1905	Theorem 19.26 1903 with triple conjunction. (Contributed by NM, 13-Sep-2011.)
⊢ (∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓 ∧ ∀𝑥𝜒))
Theorem	19.29 1906	Theorem 19.29 of [Margaris] p. 90. See also 19.29r 1907. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))
Theorem	19.29r 1907	Variation of 19.29 1906. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2020.)
⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))
Theorem	19.29r2 1908	Variation of 19.29r 1907 with double quantification. (Contributed by NM, 3-Feb-2005.)
⊢ ((∃𝑥∃𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜓))
Theorem	19.29x 1909	Variation of 19.29 1906 with mixed quantification. (Contributed by NM, 11-Feb-2005.)
⊢ ((∃𝑥∀𝑦𝜑 ∧ ∀𝑥∃𝑦𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜓))
Theorem	19.35 1910	Theorem 19.35 of [Margaris] p. 90. This theorem is useful for moving an implication (in the form of the right-hand side) into the scope of a single existential quantifier. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)
⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
Theorem	19.35i 1911	Inference associated with 19.35 1910. (Contributed by NM, 21-Jun-1993.)
⊢ ∃𝑥(𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → ∃𝑥𝜓)
Theorem	19.35ri 1912	Inference associated with 19.35 1910. (Contributed by NM, 12-Mar-1993.)
⊢ (∀𝑥𝜑 → ∃𝑥𝜓)    ⇒   ⊢ ∃𝑥(𝜑 → 𝜓)
Theorem	19.25 1913	Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ (∀𝑦∃𝑥(𝜑 → 𝜓) → (∃𝑦∀𝑥𝜑 → ∃𝑦∃𝑥𝜓))
Theorem	19.30 1914	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 1915	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 1916	Obsolete proof of 19.43 1915. Do not delete as it is referenced on the mmrecent.html 1915 page and in conventions-labels 30882. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
Theorem	19.33 1917	Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑 ∨ 𝜓))
Theorem	19.33b 1918	The antecedent provides a condition implying the converse of 19.33 1917. (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 1919	Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 26-May-1993.)
⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
Theorem	19.40-2 1920	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 1921	The antecedent provides a condition implying the converse of 19.40 1919. This is to 19.40 1919 what 19.33b 1918 is to 19.33 1917. (Contributed by BJ, 6-May-2019.) (Proof shortened by Wolf Lammen, 13-Nov-2020.)
⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ ∃𝑥(𝜑 ∧ 𝜓)))
Theorem	albiim 1922	Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜑)))
Theorem	2albiim 1923	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 1924	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 1925	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 1926	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 1927	If in a context 𝑥 is not free in 𝜓 and 𝜒, then it is not free in (𝜓 → 𝜒). Deduction form of nfim 1929. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) df-nf 1817 changed. (Revised by Wolf Lammen, 18-Sep-2021.) Eliminate curried form of nfimt 1928. (Revised by Wolf Lammen, 10-Jul-2022.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝜒))
Theorem	nfimt 1928	Closed form of nfim 1929 and nfimd 1927. (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 1929	If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑 → 𝜓). Inference associated with nfimt 1928. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) df-nf 1817 changed. (Revised by Wolf Lammen, 17-Sep-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ Ⅎ𝑥(𝜑 → 𝜓)
Theorem	nfand 1930	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 1931	Deduction form of bound-variable hypothesis builder nf3an 1934. (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 1932	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 1933	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 1934	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 1935	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 1936	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 1937	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 1938	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 2000 in logic and ax-nul 5267 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 1941) and every existential formula is false (emptyex 1940), and every variable is effectively nonfree in any formula (emptynf 1942).
Theorem	empty 1939	Two characterizations of the empty domain. (Contributed by Gérard Lang, 5-Feb-2024.)
⊢ (¬ ∃𝑥⊤ ↔ ∀𝑥⊥)
Theorem	emptyex 1940	On the empty domain, any existentially quantified formula is false. (Contributed by Wolf Lammen, 21-Jan-2024.)
⊢ (¬ ∃𝑥⊤ → ¬ ∃𝑥𝜑)
Theorem	emptyal 1941	On the empty domain, any universally quantified formula is true. (Contributed by Wolf Lammen, 12-Mar-2023.)
⊢ (¬ ∃𝑥⊤ → ∀𝑥𝜑)
Theorem	emptynf 1942	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 1943*	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 39782 about the logical redundancy of ax-5 1943 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 1944*	Version of ax-5 1943 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders. (Contributed by NM, 1-Mar-2013.)
⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
Theorem	ax5e 1945*	A rephrasing of ax-5 1943 using the existential quantifier. (Contributed by Wolf Lammen, 4-Dec-2017.)
⊢ (∃𝑥𝜑 → 𝜑)
Theorem	ax5ea 1946*	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 1947*	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 1948*	nfv 1947 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1927. (Contributed by Mario Carneiro, 6-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝜓)
Theorem	alimdv 1949*	Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1843. See alimdh 1850 and alimd 2250 for versions without a distinct variable condition. (Contributed by NM, 3-Apr-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
Theorem	eximdv 1950*	Deduction form of Theorem 19.22 of [Margaris] p. 90, see exim 1867. See eximdh 1897 and eximd 2254 for versions without a distinct variable condition. (Contributed by NM, 27-Apr-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
Theorem	2alimdv 1951*	Deduction form of Theorem 19.20 of [Margaris] p. 90 with two quantifiers, see alim 1843. (Contributed by NM, 27-Apr-2004.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥∀𝑦𝜓 → ∀𝑥∀𝑦𝜒))
Theorem	2eximdv 1952*	Deduction form of Theorem 19.22 of [Margaris] p. 90 with two quantifiers, see exim 1867. (Contributed by NM, 3-Aug-1995.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 → ∃𝑥∃𝑦𝜒))
Theorem	albidv 1953*	Formula-building rule for universal quantifier (deduction form). See also albidh 1899 and albid 2260. (Contributed by NM, 26-May-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
Theorem	exbidv 1954*	Formula-building rule for existential quantifier (deduction form). See also exbidh 1900 and exbid 2261. (Contributed by NM, 26-May-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
Theorem	nfbidv 1955*	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 2000, ax-7 2041, ax-12 2215 by adapting proof of nfbidf 2262. (Revised by BJ, 25-Sep-2022.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))
Theorem	2albidv 1956*	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 1957*	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 1958*	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 1959*	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 1960*	Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2245 and 19.21v 1972. (Contributed by NM, 21-Jun-1993.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥𝜓)
Theorem	alrimivv 1961*	Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2245 and 19.21v 1972. (Contributed by NM, 31-Jul-1995.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥∀𝑦𝜓)
Theorem	alrimdv 1962*	Deduction form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2245 and 19.21v 1972. (Contributed by NM, 10-Feb-1997.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))
Theorem	exlimiv 1963*	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 3163, 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 3163. 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 1963 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 1964. (Contributed by NM, 21-Jun-1993.) Remove dependencies on ax-6 2000 and ax-8 2147. (Revised by Wolf Lammen, 4-Dec-2017.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥𝜑 → 𝜓)
Theorem	exlimiiv 1964*	Inference (Rule C) associated with exlimiv 1963. (Contributed by BJ, 19-Dec-2020.)
⊢ (𝜑 → 𝜓)    &   ⊢ ∃𝑥𝜑    ⇒   ⊢ 𝜓
Theorem	exlimivv 1965*	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 1966*	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 2000, ax-7 2041. (Revised by Wolf Lammen, 4-Dec-2017.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))
Theorem	exlimdvv 1967*	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 1968*	Existential elimination rule of natural deduction (Rule C, explained in exlimiv 1963). (Contributed by Mario Carneiro, 15-Jun-2016.)
⊢ (𝜑 → ∃𝑥𝜓)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	nexdv 1969*	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 1970*	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 1971*	Version of stdpc5 2246 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.) Revised to shorten 19.21v 1972. (Revised by Wolf Lammen, 12-Jul-2020.)
⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))
Theorem	19.21v 1972*	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 1817) instead of a disjoint variable condition. For instance, 19.21v 1972 versus 19.21 2245 and vtoclf 3528 versus vtocl 3523. 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 1947. 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 1973*	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 1974*	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 1975*	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 2000. (Revised by Rohan Ridenour, 15-Apr-2022.)
⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))
Theorem	19.23vv 1976*	Theorem 19.23v 1975 extended to two variables. (Contributed by NM, 10-Aug-2004.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥∃𝑦𝜑 → 𝜓))
Theorem	pm11.53v 1977*	Version of pm11.53 2377 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))
Theorem	19.36imv 1978*	One direction of 19.36v 2026 that can be proven without ax-6 2000. (Contributed by Rohan Ridenour, 16-Apr-2022.) (Proof shortened by Wolf Lammen, 22-Sep-2024.)
⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
Theorem	19.36iv 1979*	Inference associated with 19.36v 2026. 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 2000. (Revised by Rohan Ridenour, 15-Apr-2022.)
⊢ ∃𝑥(𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	19.37imv 1980*	One direction of 19.37v 2030 that can be proven without ax-6 2000. (Contributed by Rohan Ridenour, 16-Apr-2022.)
⊢ (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))
Theorem	19.37iv 1981*	Inference associated with 19.37v 2030. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-6 2000. (Revised by Rohan Ridenour, 15-Apr-2022.)
⊢ ∃𝑥(𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	19.41v 1982*	Version of 19.41 2273 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.) Remove dependency on ax-6 2000. (Revised by Rohan Ridenour, 15-Apr-2022.)
⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))
Theorem	19.41vv 1983*	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 1984*	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 1985*	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 1986*	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 1987*	Distribution of existential quantifiers. See also exdistrv 1988. (Contributed by NM, 9-Mar-1995.)
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
Theorem	exdistrv 1988*	Distribute a pair of existential quantifiers (over disjoint variables) over a conjunction. Combination of 19.41v 1982 and 19.42v 1986. For a version with fewer disjoint variable conditions but requiring more axioms, see eeanv 2380. (Contributed by BJ, 30-Sep-2022.)
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
Theorem	4exdistrv 1989*	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 2382. (Contributed by BJ, 5-Jan-2023.)
⊢ (∃𝑥∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓))
Theorem	19.42vv 1990*	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 1991*	Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.)
⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦∃𝑧𝜓))
Theorem	19.42vvv 1992*	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 1993*	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 1994*	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 1570 for use by df-tru 1573. See the comments in that section. In this section, we continue with its first "real" use.
Theorem	weq 1995	Extend wff definition to include atomic formulas using the equality predicate. (Instead of introducing weq 1995 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 1570. This lets us avoid overloading the = connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1995 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1570. 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 1996	Specialization, with additional weakening (compared to 19.2 2009) 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 1997	Alternate proof of speimfw 1996 (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 1998	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 1999	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 2000	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 2416 and ax6fromc10 39771. A more convenient form of this axiom is ax6e 2414, 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 2414. ax-6 2000 can be proved from the weaker version ax6v 2001 requiring that the variables be distinct; see Theorem ax6 2415. ax-6 2000 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 5264. Except by ax6v 2001, this axiom should not be referenced directly. Instead, use Theorem ax6 2415. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦