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Theorem List for Metamath Proof Explorer - 3201-3300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremreximdvva 3201* Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by AV, 5-Jan-2022.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 → ∃𝑥𝐴𝑦𝐵 𝜒))
 
Theoremreximddv2 3202* Double deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 15-Dec-2019.)
((((𝜑𝑥𝐴) ∧ 𝑦𝐵) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)       (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜒)
 
Theoremr19.23t 3203 Closed theorem form of r19.23 3204. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
(Ⅎ𝑥𝜓 → (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))
 
Theoremr19.23 3204 Restricted quantifier version of 19.23 2197. See r19.23v 3205 for a version requiring fewer axioms. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 8-Oct-2016.)
𝑥𝜓       (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))
 
Theoremr19.23v 3205* Restricted quantifier version of 19.23v 1985. Version of r19.23 3204 with a disjoint variable condition. (Contributed by NM, 31-Aug-1999.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2020.)
(∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))
 
Theoremrexlimi 3206 Restricted quantifier version of exlimi 2203. (Contributed by NM, 30-Nov-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
𝑥𝜓    &   (𝑥𝐴 → (𝜑𝜓))       (∃𝑥𝐴 𝜑𝜓)
 
Theoremrexlimd2 3207 Version of rexlimd 3208 with deduction version of second hypothesis. (Contributed by NM, 21-Jul-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥𝐴 → (𝜓𝜒)))       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremrexlimd 3208 Deduction form of rexlimd 3208. (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof shortened by Wolf Lammen, 14-Jan-2020.)
𝑥𝜑    &   𝑥𝜒    &   (𝜑 → (𝑥𝐴 → (𝜓𝜒)))       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremrexlimiv 3209* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2020.)
(𝑥𝐴 → (𝜑𝜓))       (∃𝑥𝐴 𝜑𝜓)
 
Theoremrexlimiva 3210* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Dec-2006.)
((𝑥𝐴𝜑) → 𝜓)       (∃𝑥𝐴 𝜑𝜓)
 
Theoremrexlimivw 3211* Weaker version of rexlimiv 3209. (Contributed by FL, 19-Sep-2011.)
(𝜑𝜓)       (∃𝑥𝐴 𝜑𝜓)
 
Theoremrexlimdv 3212* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 14-Nov-2002.) (Proof shortened by Eric Schmidt, 22-Dec-2006.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2020.)
(𝜑 → (𝑥𝐴 → (𝜓𝜒)))       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremrexlimdva 3213* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 20-Jan-2007.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremrexlimdvaa 3214* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Mario Carneiro, 15-Jun-2016.)
((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜒)       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremrexlimdv3a 3215* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 3212. (Contributed by NM, 7-Jun-2015.)
((𝜑𝑥𝐴𝜓) → 𝜒)       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremrexlimdva2 3216* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremrexlimdvw 3217* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Jun-2014.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremrexlimddv 3218* Restricted existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
(𝜑 → ∃𝑥𝐴 𝜓)    &   ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜒)       (𝜑𝜒)
 
Theoremrexlimivv 3219* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 17-Feb-2004.)
((𝑥𝐴𝑦𝐵) → (𝜑𝜓))       (∃𝑥𝐴𝑦𝐵 𝜑𝜓)
 
Theoremrexlimdvv 3220* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Jul-2004.)
(𝜑 → ((𝑥𝐴𝑦𝐵) → (𝜓𝜒)))       (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓𝜒))
 
Theoremrexlimdvva 3221* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓𝜒))
 
Theoremrexbii2 3222 Inference adding different restricted existential quantifiers to each side of an equivalence. (Contributed by NM, 4-Feb-2004.)
((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜓)
 
Theoremrexbiia 3223 Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 26-Oct-1999.)
(𝑥𝐴 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 𝜓)
 
Theoremrexbii 3224 Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 23-Nov-1994.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Dec-2019.)
(𝜑𝜓)       (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 𝜓)
 
Theorem2rexbii 3225 Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 11-Nov-1995.)
(𝜑𝜓)       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓)
 
Theoremrexnal2 3226 Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(∃𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵 𝜑)
 
Theoremrexnal3 3227 Relationship between three restricted universal and existential quantifiers. (Contributed by Thierry Arnoux, 12-Jul-2020.)
(∃𝑥𝐴𝑦𝐵𝑧𝐶 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑)
 
Theoremralnex2 3228 Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 18-May-2023.)
(∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴𝑦𝐵 𝜑)
 
Theoremralnex2OLD 3229 Obsolete version of ralnex2 3228 as of 18-May-2023. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴𝑦𝐵 𝜑)
 
Theoremralnex3 3230 Relationship between three restricted universal and existential quantifiers. (Contributed by Thierry Arnoux, 12-Jul-2020.) (Proof shortened by Wolf Lammen, 18-May-2023.)
(∀𝑥𝐴𝑦𝐵𝑧𝐶 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑)
 
Theoremralnex3OLD 3231 Obsolete version of ralnex3 3230 as of 18-May-2023. (Contributed by Thierry Arnoux, 12-Jul-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝐴𝑦𝐵𝑧𝐶 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑)
 
Theoremrexbida 3232 Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 6-Oct-2003.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
 
Theoremrexbidv2 3233* Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 22-May-1999.)
(𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))
 
Theoremrexbidva 3234* Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 9-Mar-1997.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 6-Dec-2019.) (Proof shortened by Wolf Lammen, 10-Dec-2019.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
 
TheoremrexbidvaALT 3235* Alternate proof of rexbida 3232, shorter but requires more axioms. (Contributed by NM, 9-Mar-1997.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
 
Theoremrexbid 3236 Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 27-Jun-1998.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
 
Theoremrexbidv 3237* Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 20-Nov-1994.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 6-Dec-2019.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
 
TheoremrexbidvALT 3238* Alternate proof of rexbidv 3237, shorter but requires more axioms. (Contributed by NM, 20-Nov-1994.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
 
Theoremrexeqbii 3239 Equality deduction for restricted existential quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)
𝐴 = 𝐵    &   (𝜓𝜒)       (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒)
 
Theorem2rexbiia 3240* Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
((𝑥𝐴𝑦𝐵) → (𝜑𝜓))       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓)
 
Theorem2rexbidva 3241* Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 15-Dec-2004.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))
 
Theorem2rexbidv 3242* Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))
 
Theoremrexralbidv 3243* Formula-building rule for restricted quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))
 
Theoremr2exlem 3244 Lemma factoring out common proof steps in r2exf 3245 an r2ex 3246. Introduced to reduce dependencies on axioms. (Contributed by Wolf Lammen, 10-Jan-2020.)
(∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑))       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
 
Theoremr2exf 3245* Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2exlem 3244. (Revised by Wolf Lammen, 10-Jan-2020.)
𝑦𝐴       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
 
Theoremr2ex 3246* Double restricted existential quantification. (Contributed by NM, 11-Nov-1995.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 10-Jan-2020.)
(∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
 
Theoremrisset 3247* Two ways to say "𝐴 belongs to 𝐵". (Contributed by NM, 22-Nov-1994.)
(𝐴𝐵 ↔ ∃𝑥𝐵 𝑥 = 𝐴)
 
Theoremr19.12 3248* Restricted quantifier version of 19.12 2303. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof shortened by Wolf Lammen, 19-May-2023.)
(∃𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴 𝜑)
 
Theoremr19.12OLD 3249* Obsolete version of r19.12 3248 as of 19-May-2023. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴 𝜑)
 
Theoremr19.26 3250 Restricted quantifier version of 19.26 1916. (Contributed by NM, 28-Jan-1997.) (Proof shortened by Andrew Salmon, 30-May-2011.)
(∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓))
 
Theoremr19.26-2 3251 Restricted quantifier version of 19.26-2 1917. Version of r19.26 3250 with two quantifiers. (Contributed by NM, 10-Aug-2004.)
(∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∀𝑥𝐴𝑦𝐵 𝜓))
 
Theoremr19.26-3 3252 Version of r19.26 3250 with three quantifiers. (Contributed by FL, 22-Nov-2010.)
(∀𝑥𝐴 (𝜑𝜓𝜒) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓 ∧ ∀𝑥𝐴 𝜒))
 
Theoremr19.26m 3253 Version of 19.26 1916 and r19.26 3250 with restricted quantifiers ranging over different classes. (Contributed by NM, 22-Feb-2004.)
(∀𝑥((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓)) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜓))
 
Theoremralbi 3254 Distribute a restricted universal quantifier over a biconditional. Restricted quantification version of albi 1862. (Contributed by NM, 6-Oct-2003.)
(∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜓))
 
Theoremralbiim 3255 Split a biconditional and distribute quantifier. Restricted quantifier version of albiim 1935. (Contributed by NM, 3-Jun-2012.)
(∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 (𝜑𝜓) ∧ ∀𝑥𝐴 (𝜓𝜑)))
 
Theoremr19.27v 3256* Restricted quantitifer version of one direction of 19.27 2213. (The other direction holds when 𝐴 is nonempty, see r19.27zv 4294.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.)
((∀𝑥𝐴 𝜑𝜓) → ∀𝑥𝐴 (𝜑𝜓))
 
Theoremr19.28v 3257* Restricted quantifier version of one direction of 19.28 2214. (The other direction holds when 𝐴 is nonempty, see r19.28zv 4289.) (Contributed by NM, 2-Apr-2004.)
((𝜑 ∧ ∀𝑥𝐴 𝜓) → ∀𝑥𝐴 (𝜑𝜓))
 
Theoremr19.29 3258 Restricted quantifier version of 19.29 1920. See also r19.29r 3259. (Contributed by NM, 31-Aug-1999.) (Proof shortened by Andrew Salmon, 30-May-2011.)
((∀𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 𝜓) → ∃𝑥𝐴 (𝜑𝜓))
 
Theoremr19.29r 3259 Restricted quantifier version of 19.29r 1921; variation of r19.29 3258. (Contributed by NM, 31-Aug-1999.)
((∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) → ∃𝑥𝐴 (𝜑𝜓))
 
Theoremr19.29imd 3260 Theorem 19.29 of [Margaris] p. 90 with an implication in the hypothesis containing the generalization, deduction version. (Contributed by AV, 19-Jan-2019.)
(𝜑 → ∃𝑥𝐴 𝜓)    &   (𝜑 → ∀𝑥𝐴 (𝜓𝜒))       (𝜑 → ∃𝑥𝐴 (𝜓𝜒))
 
Theoremr19.29af2 3261 A commonly used pattern based on r19.29 3258. (Contributed by Thierry Arnoux, 17-Dec-2017.) (Proof shortened by OpenAI, 25-Mar-2020.)
𝑥𝜑    &   𝑥𝜒    &   (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴 𝜓)       (𝜑𝜒)
 
Theoremr19.29af 3262* A commonly used pattern based on r19.29 3258. (Contributed by Thierry Arnoux, 29-Nov-2017.)
𝑥𝜑    &   (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴 𝜓)       (𝜑𝜒)
 
Theoremr19.29an 3263* A commonly used pattern based on r19.29 3258. (Contributed by Thierry Arnoux, 29-Dec-2019.)
(((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)       ((𝜑 ∧ ∃𝑥𝐴 𝜓) → 𝜒)
 
Theoremr19.29a 3264* A commonly used pattern based on r19.29 3258. (Contributed by Thierry Arnoux, 22-Nov-2017.)
(((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴 𝜓)       (𝜑𝜒)
 
Theorem2r19.29 3265 Theorem r19.29 3258 with two quantifiers. (Contributed by Rodolfo Medina, 25-Sep-2010.)
((∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑥𝐴𝑦𝐵 𝜓) → ∃𝑥𝐴𝑦𝐵 (𝜑𝜓))
 
Theoremr19.29d2r 3266 Theorem 19.29 of [Margaris] p. 90 with two restricted quantifiers, deduction version. (Contributed by Thierry Arnoux, 30-Jan-2017.)
(𝜑 → ∀𝑥𝐴𝑦𝐵 𝜓)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜒)       (𝜑 → ∃𝑥𝐴𝑦𝐵 (𝜓𝜒))
 
Theoremr19.29vva 3267* A commonly used pattern based on r19.29 3258, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.)
((((𝜑𝑥𝐴) ∧ 𝑦𝐵) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)       (𝜑𝜒)
 
Theoremr19.30 3268 Restricted quantifier version of 19.30 1928. (Contributed by Scott Fenton, 25-Feb-2011.)
(∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
 
Theoremr19.32v 3269* Restricted quantifier version of 19.32v 1983. (Contributed by NM, 25-Nov-2003.)
(∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝐴 𝜓))
 
Theoremr19.35 3270 Restricted quantifier version of 19.35 1924. (Contributed by NM, 20-Sep-2003.)
(∃𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
 
Theoremr19.36v 3271* Restricted quantifier version of one direction of 19.36 2216. (The other direction holds iff 𝐴 is nonempty, see r19.36zv 4295.) (Contributed by NM, 22-Oct-2003.)
(∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑𝜓))
 
Theoremr19.37 3272 Restricted quantifier version of one direction of 19.37 2218. (The other direction does not hold when 𝐴 is empty.) (Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)
𝑥𝜑       (∃𝑥𝐴 (𝜑𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))
 
Theoremr19.37v 3273* Restricted quantifier version of one direction of 19.37v 2040. (The other direction holds iff 𝐴 is nonempty, see r19.37zv 4290.) (Contributed by NM, 2-Apr-2004.)
(∃𝑥𝐴 (𝜑𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))
 
Theoremr19.40 3274 Restricted quantifier version of Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.)
(∃𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 𝜓))
 
Theoremr19.41v 3275* Restricted quantifier version 19.41v 1992. Version of r19.41 3276 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 17-Dec-2003.) Reduce dependencies on axioms. (Revised by BJ, 29-Mar-2020.)
(∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))
 
Theoremr19.41 3276 Restricted quantifier version of 19.41 2221. See r19.41v 3275 for a version with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 1-Nov-2010.)
𝑥𝜓       (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))
 
Theoremr19.41vv 3277* Version of r19.41v 3275 with two quantifiers. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴𝑦𝐵 𝜑𝜓))
 
Theoremr19.42v 3278* Restricted quantifier version of 19.42v 1996 (see also 19.42 2223). (Contributed by NM, 27-May-1998.)
(∃𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝐴 𝜓))
 
Theoremr19.43 3279 Restricted quantifier version of 19.43 1929. (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)
(∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
 
Theoremr19.44v 3280* One direction of a restricted quantifier version of 19.44 2224. The other direction holds when 𝐴 is nonempty, see r19.44zv 4292. (Contributed by NM, 2-Apr-2004.)
(∃𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑𝜓))
 
Theoremr19.45v 3281* Restricted quantifier version of one direction of 19.45 2225. The other direction holds when 𝐴 is nonempty, see r19.45zv 4291. (Contributed by NM, 2-Apr-2004.)
(∃𝑥𝐴 (𝜑𝜓) → (𝜑 ∨ ∃𝑥𝐴 𝜓))
 
Theoremralcomf 3282* Commutation of restricted universal quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑦𝐴    &   𝑥𝐵       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥𝐴 𝜑)
 
Theoremrexcomf 3283* Commutation of restricted existential quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑦𝐴    &   𝑥𝐵       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 𝜑)
 
Theoremralcom 3284* Commutation of restricted universal quantifiers. See ralcom2 3290 for a version without disjoint variable condition on 𝑥, 𝑦. (Contributed by NM, 13-Oct-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
(∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥𝐴 𝜑)
 
Theoremrexcom 3285* Commutation of restricted existential quantifiers. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
(∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 𝜑)
 
Theoremralcom13 3286* Swap first and third restricted universal quantifiers. (Contributed by AV, 3-Dec-2021.)
(∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑧𝐶𝑦𝐵𝑥𝐴 𝜑)
 
Theoremrexcom13 3287* Swap first and third restricted existential quantifiers. (Contributed by NM, 8-Apr-2015.)
(∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∃𝑧𝐶𝑦𝐵𝑥𝐴 𝜑)
 
Theoremralrot3 3288* Rotate three restricted universal quantifiers. (Contributed by AV, 3-Dec-2021.)
(∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑧𝐶𝑥𝐴𝑦𝐵 𝜑)
 
Theoremrexrot4 3289* Rotate four restricted existential quantifiers twice. (Contributed by NM, 8-Apr-2015.)
(∃𝑥𝐴𝑦𝐵𝑧𝐶𝑤𝐷 𝜑 ↔ ∃𝑧𝐶𝑤𝐷𝑥𝐴𝑦𝐵 𝜑)
 
Theoremralcom2 3290* Commutation of restricted universal quantifiers. Note that 𝑥 and 𝑦 need not be disjoint (this makes the proof longer). If 𝑥 and 𝑦 are disjoint, then one may use ralcom 3284. (Contributed by NM, 24-Nov-1994.) (Proof shortened by Mario Carneiro, 17-Oct-2016.)
(∀𝑥𝐴𝑦𝐴 𝜑 → ∀𝑦𝐴𝑥𝐴 𝜑)
 
Theoremralcom3 3291 A commutation law for restricted universal quantifiers that swaps the domains of the restriction. (Contributed by NM, 22-Feb-2004.)
(∀𝑥𝐴 (𝑥𝐵𝜑) ↔ ∀𝑥𝐵 (𝑥𝐴𝜑))
 
Theoremreean 3292* Rearrange restricted existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Andrew Salmon, 30-May-2011.)
𝑦𝜑    &   𝑥𝜓       (∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓))
 
Theoremreeanv 3293* Rearrange restricted existential quantifiers. (Contributed by NM, 9-May-1999.)
(∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓))
 
Theorem3reeanv 3294* Rearrange three restricted existential quantifiers. (Contributed by Jeff Madsen, 11-Jun-2010.)
(∃𝑥𝐴𝑦𝐵𝑧𝐶 (𝜑𝜓𝜒) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓 ∧ ∃𝑧𝐶 𝜒))
 
Theorem2ralor 3295* Distribute restricted universal quantification over "or". (Contributed by Jeff Madsen, 19-Jun-2010.)
(∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∨ ∀𝑦𝐵 𝜓))
 
Theoremnfreu1 3296 The setvar 𝑥 is not free in ∃!𝑥𝐴𝜑. (Contributed by NM, 19-Mar-1997.)
𝑥∃!𝑥𝐴 𝜑
 
Theoremnfrmo1 3297 The setvar 𝑥 is not free in ∃*𝑥𝐴𝜑. (Contributed by NM, 16-Jun-2017.)
𝑥∃*𝑥𝐴 𝜑
 
Theoremnfreud 3298 Deduction version of nfreu 3300. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃!𝑦𝐴 𝜓)
 
Theoremnfrmod 3299 Deduction version of nfrmo 3301. (Contributed by NM, 17-Jun-2017.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃*𝑦𝐴 𝜓)
 
Theoremnfreu 3300 Bound-variable hypothesis builder for restricted unique existence. (Contributed by NM, 30-Oct-2010.) (Revised by Mario Carneiro, 8-Oct-2016.)
𝑥𝐴    &   𝑥𝜑       𝑥∃!𝑦𝐴 𝜑
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