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Theorem List for Metamath Proof Explorer - 3201-3300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrexnal 3201 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Wolf Lammen, 9-Dec-2019.)
(∃𝑥𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴 𝜑)

Theoremdfrex2 3202 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Wolf Lammen, 26-Nov-2019.)
(∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜑)

Theoremrexex 3203 Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)
(∃𝑥𝐴 𝜑 → ∃𝑥𝜑)

Theoremrexim 3204 Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.) (Proof shortened by Andrew Salmon, 30-May-2011.)
(∀𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))

Theoremreximia 3205 Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.)
(𝑥𝐴 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓)

Theoremreximi 3206 Inference quantifying both antecedent and consequent. (Contributed by NM, 18-Oct-1996.)
(𝜑𝜓)       (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓)

Theoremreximi2 3207 Inference quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 8-Nov-2004.)
((𝑥𝐴𝜑) → (𝑥𝐵𝜓))       (∃𝑥𝐴 𝜑 → ∃𝑥𝐵 𝜓)

Theoremrexbii2 3208 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 3209 Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 26-Oct-1999.)
(𝑥𝐴 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 𝜓)

Theoremrexbii 3210 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 3211 Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 11-Nov-1995.)
(𝜑𝜓)       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓)

Theoremrexcom4 3212* Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) Reduce axiom dependencies. (Revised by BJ, 13-Jun-2019.)
(∃𝑥𝐴𝑦𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)

Theorem2ex2rexrot 3213* Rotate two existential quantifiers and two restricted existential quantifiers. (Contributed by AV, 9-Nov-2023.)
(∃𝑥𝑦𝑧𝐴𝑤𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵𝑥𝑦𝜑)

Theoremrexcom4a 3214* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
(∃𝑥𝑦𝐴 (𝜑𝜓) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥𝜓))

Theoremrexanid 3215 Cancellation law for restricted existential quantification. (Contributed by Peter Mazsa, 24-May-2018.) (Proof shortened by Wolf Lammen, 8-Jul-2023.)
(∃𝑥𝐴 (𝑥𝐴𝜑) ↔ ∃𝑥𝐴 𝜑)

Theoremr19.29 3216 Restricted quantifier version of 19.29 1874. See also r19.29r 3217. (Contributed by NM, 31-Aug-1999.) (Proof shortened by Andrew Salmon, 30-May-2011.)
((∀𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 𝜓) → ∃𝑥𝐴 (𝜑𝜓))

Theoremr19.29r 3217 Restricted quantifier version of 19.29r 1875; variation of r19.29 3216. (Contributed by NM, 31-Aug-1999.) (Proof shortened by Wolf Lammen, 29-Jun-2023.)
((∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) → ∃𝑥𝐴 (𝜑𝜓))

Theoremr19.29imd 3218 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))

Theoremrexnal2 3219 Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(∃𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵 𝜑)

Theoremrexnal3 3220 Relationship between three restricted universal and existential quantifiers. (Contributed by Thierry Arnoux, 12-Jul-2020.)
(∃𝑥𝐴𝑦𝐵𝑧𝐶 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑)

Theoremralnex2 3221 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)

Theoremralnex3 3222 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)

Theoremralinexa 3223 A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)
(∀𝑥𝐴 (𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥𝐴 (𝜑𝜓))

Theoremrexanali 3224 A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.) (Proof shortened by Wolf Lammen, 27-Dec-2019.)
(∃𝑥𝐴 (𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥𝐴 (𝜑𝜓))

Theoremnrexralim 3225 Negation of a complex predicate calculus formula. (Contributed by FL, 31-Jul-2009.)
(¬ ∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∀𝑥𝐴𝑦𝐵 (𝜑 ∧ ¬ 𝜓))

Theoremrisset 3226* Two ways to say "𝐴 belongs to 𝐵". (Contributed by NM, 22-Nov-1994.)
(𝐴𝐵 ↔ ∃𝑥𝐵 𝑥 = 𝐴)

Theoremnelb 3227* A definition of ¬ 𝐴𝐵. (Contributed by Thierry Arnoux, 20-Nov-2023.) (Proof shortened by SN, 23-Jan-2024.)
𝐴𝐵 ↔ ∀𝑥𝐵 𝑥𝐴)

Theoremnrex 3228 Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
(𝑥𝐴 → ¬ 𝜓)        ¬ ∃𝑥𝐴 𝜓

Theoremnrexdv 3229* Deduction adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Wolf Lammen, 5-Jan-2020.)
((𝜑𝑥𝐴) → ¬ 𝜓)       (𝜑 → ¬ ∃𝑥𝐴 𝜓)

Theoremreximdv2 3230* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 17-Sep-2003.)
(𝜑 → ((𝑥𝐴𝜓) → (𝑥𝐵𝜒)))       (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐵 𝜒))

Theoremreximdvai 3231* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 14-Nov-2002.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 8-Jan-2020.)
(𝜑 → (𝑥𝐴 → (𝜓𝜒)))       (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 𝜒))

Theoremreximdv 3232* Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version with strong hypothesis.) (Contributed by NM, 24-Jun-1998.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 𝜒))

Theoremreximdva 3233* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 𝜒))

Theoremreximddv 3234* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 7-Dec-2016.)
((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜒)    &   (𝜑 → ∃𝑥𝐴 𝜓)       (𝜑 → ∃𝑥𝐴 𝜒)

Theoremreximssdv 3235* Derivation of a restricted existential quantification over a subset (the second hypothesis implies 𝐴𝐵), deduction form. (Contributed by AV, 21-Aug-2022.)
(𝜑 → ∃𝑥𝐵 𝜓)    &   ((𝜑 ∧ (𝑥𝐵𝜓)) → 𝑥𝐴)    &   ((𝜑 ∧ (𝑥𝐵𝜓)) → 𝜒)       (𝜑 → ∃𝑥𝐴 𝜒)

Theoremreximdvva 3236* 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 3237* 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)

Theoremr19.23v 3238* Restricted quantifier version of 19.23v 1943. Version of r19.23 3273 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))

Theoremrexlimiv 3239* 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 3240* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Dec-2006.)
((𝑥𝐴𝜑) → 𝜓)       (∃𝑥𝐴 𝜑𝜓)

Theoremrexlimivw 3241* Weaker version of rexlimiv 3239. (Contributed by FL, 19-Sep-2011.)
(𝜑𝜓)       (∃𝑥𝐴 𝜑𝜓)

Theoremrexlimdv 3242* 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 3243* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 20-Jan-2007.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓𝜒))

Theoremrexlimdvaa 3244* 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 3245* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 3242. (Contributed by NM, 7-Jun-2015.)
((𝜑𝑥𝐴𝜓) → 𝜒)       (𝜑 → (∃𝑥𝐴 𝜓𝜒))

Theoremrexlimdva2 3246* 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))

Theoremr19.29an 3247* A commonly used pattern in the spirit of r19.29 3216. (Contributed by Thierry Arnoux, 29-Dec-2019.) (Proof shortened by Wolf Lammen, 17-Jun-2023.)
(((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)       ((𝜑 ∧ ∃𝑥𝐴 𝜓) → 𝜒)

Theoremr19.29a 3248* A commonly used pattern in the spirit of r19.29 3216. (Contributed by Thierry Arnoux, 22-Nov-2017.) Reduce axiom usage. (Revised by Wolf Lammen, 17-Jun-2023.)
(((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴 𝜓)       (𝜑𝜒)

Theoremrexlimdvw 3249* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Jun-2014.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓𝜒))

Theoremrexlimddv 3250* Restricted existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
(𝜑 → ∃𝑥𝐴 𝜓)    &   ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜒)       (𝜑𝜒)

Theoremrexlimivv 3251* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 17-Feb-2004.)
((𝑥𝐴𝑦𝐵) → (𝜑𝜓))       (∃𝑥𝐴𝑦𝐵 𝜑𝜓)

Theoremrexlimdvv 3252* 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 3253* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓𝜒))

Theoremrexbidv2 3254* Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 22-May-1999.)
(𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))

Theoremrexbidva 3255* 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))

Theoremrexbidv 3256* 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))

Theorem2rexbiia 3257* 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 3258* Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 15-Dec-2004.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))

Theorem2rexbidv 3259* Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))

Theoremrexralbidv 3260* Formula-building rule for restricted quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))

Theoremr2exlem 3261 Lemma factoring out common proof steps in r2exf 3284 an r2ex 3262. Introduced to reduce dependencies on axioms. (Contributed by Wolf Lammen, 10-Jan-2020.)
(∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑))       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))

Theoremr2ex 3262* 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))

Theoremrspe 3263 Restricted specialization. (Contributed by NM, 12-Oct-1999.)
((𝑥𝐴𝜑) → ∃𝑥𝐴 𝜑)

Theoremrsp2e 3264 Restricted specialization. (Contributed by FL, 4-Jun-2012.) (Proof shortened by Wolf Lammen, 7-Jan-2020.)
((𝑥𝐴𝑦𝐵𝜑) → ∃𝑥𝐴𝑦𝐵 𝜑)

Theoremnfre1 3265 The setvar 𝑥 is not free in 𝑥𝐴𝜑. (Contributed by NM, 19-Mar-1997.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝑥𝐴 𝜑

Theoremnfrexd 3266* Deduction version of nfrex 3268. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2379. See nfrexdg 3267 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)

Theoremnfrexdg 3267 Deduction version of nfrexg 3269. Usage of this theorem is discouraged because it depends on ax-13 2379. See nfrexd 3266 for a version with a disjoint variable condition, but not requiring ax-13 2379. (Contributed by Mario Carneiro, 14-Oct-2016.) (New usage is discouraged.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)

Theoremnfrex 3268* Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2019.) Add disjoint variable condition to avoid ax-13 2379. See nfrexg 3269 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
𝑥𝐴    &   𝑥𝜑       𝑥𝑦𝐴 𝜑

Theoremnfrexg 3269 Bound-variable hypothesis builder for restricted quantification. Usage of this theorem is discouraged because it depends on ax-13 2379. See nfrex 3268 for a version with a disjoint variable condition, but not requiring ax-13 2379. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2019.) (New usage is discouraged.)
𝑥𝐴    &   𝑥𝜑       𝑥𝑦𝐴 𝜑

Theoremreximdai 3270 Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 31-Aug-1999.)
𝑥𝜑    &   (𝜑 → (𝑥𝐴 → (𝜓𝜒)))       (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 𝜒))

Theoremreximd2a 3271 Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 27-Jan-2020.)
𝑥𝜑    &   (((𝜑𝑥𝐴) ∧ 𝜓) → 𝑥𝐵)    &   (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴 𝜓)       (𝜑 → ∃𝑥𝐵 𝜒)

Theoremr19.23t 3272 Closed theorem form of r19.23 3273. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
(Ⅎ𝑥𝜓 → (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))

Theoremr19.23 3273 Restricted quantifier version of 19.23 2209. See r19.23v 3238 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))

Theoremrexlimi 3274 Restricted quantifier version of exlimi 2215. (Contributed by NM, 30-Nov-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
𝑥𝜓    &   (𝑥𝐴 → (𝜑𝜓))       (∃𝑥𝐴 𝜑𝜓)

Theoremrexlimd2 3275 Version of rexlimd 3276 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 3276 Deduction form of rexlimd 3276. (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))

Theoremrexbida 3277 Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 6-Oct-2003.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))

TheoremrexbidvaALT 3278* Alternate proof of rexbidva 3255, 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 3279 Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 27-Jun-1998.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))

TheoremrexbidvALT 3280* Alternate proof of rexbidv 3256, 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))

Theoremralrexbid 3281 Formula-building rule for restricted existential quantifier, using a restricted universal quantifier to bind the quantified variable in the antecedent. (Contributed by AV, 21-Oct-2023.) Reduce axiom usage. (Revised by SN, 13-Nov-2023.)
(𝜑 → (𝜓𝜃))       (∀𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜃))

TheoremralrexbidOLD 3282 Obsolete version of ralrexbid 3281 as of 13-Nov-2023. (Contributed by AV, 21-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜃))       (∀𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜃))

Theoremr19.12 3283* Restricted quantifier version of 19.12 2335. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) Avoid ax-13 2379, ax-ext 2770. (Revised by Wolf Lammen, 17-Jun-2023.)
(∃𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴 𝜑)

Theoremr2exf 3284* Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2exlem 3261. (Revised by Wolf Lammen, 10-Jan-2020.)
𝑦𝐴       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))

Theoremrexeqbii 3285 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)

Theoremreuanid 3286 Cancellation law for restricted unique existential quantification. (Contributed by Peter Mazsa, 12-Feb-2018.)
(∃!𝑥𝐴 (𝑥𝐴𝜑) ↔ ∃!𝑥𝐴 𝜑)

Theoremrmoanid 3287 Cancellation law for restricted at-most-one quantification. (Contributed by Peter Mazsa, 24-May-2018.)
(∃*𝑥𝐴 (𝑥𝐴𝜑) ↔ ∃*𝑥𝐴 𝜑)

Theoremr19.29af2 3288 A commonly used pattern based on r19.29 3216. (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 3289* A commonly used pattern based on r19.29 3216. See r19.29a 3248, r19.29an 3247 for a variant when 𝑥 is disjoint from 𝜑. (Contributed by Thierry Arnoux, 29-Nov-2017.)
𝑥𝜑    &   (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴 𝜓)       (𝜑𝜒)

Theorem2r19.29 3290 Theorem r19.29 3216 with two quantifiers. (Contributed by Rodolfo Medina, 25-Sep-2010.)
((∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑥𝐴𝑦𝐵 𝜓) → ∃𝑥𝐴𝑦𝐵 (𝜑𝜓))

Theoremr19.29d2r 3291 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 3292* A commonly used pattern based on r19.29 3216, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.) (Proof shortened by Wolf Lammen, 29-Jun-2023.)
((((𝜑𝑥𝐴) ∧ 𝑦𝐵) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)       (𝜑𝜒)

Theoremr19.30 3293 Restricted quantifier version of 19.30 1882. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by Wolf Lammen, 18-Jun-2023.)
(∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))

Theoremr19.32v 3294* Restricted quantifier version of 19.32v 1941. (Contributed by NM, 25-Nov-2003.)
(∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝐴 𝜓))

Theoremr19.35 3295 Restricted quantifier version of 19.35 1878. (Contributed by NM, 20-Sep-2003.)
(∃𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))

Theoremr19.36v 3296* Restricted quantifier version of one direction of 19.36 2230. (The other direction holds iff 𝐴 is nonempty, see r19.36zv 4410.) (Contributed by NM, 22-Oct-2003.)
(∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑𝜓))

Theoremr19.37 3297 Restricted quantifier version of one direction of 19.37 2232. (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 3298* Restricted quantifier version of one direction of 19.37v 1998. (The other direction holds iff 𝐴 is nonempty, see r19.37zv 4405.) (Contributed by NM, 2-Apr-2004.) Reduce axiom usage. (Revised by Wolf Lammen, 18-Jun-2023.)
(∃𝑥𝐴 (𝜑𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))

Theoremr19.40 3299 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 3300* Restricted quantifier version 19.41v 1950. Version of r19.41 3301 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))

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