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Theorem List for Metamath Proof Explorer - 3101-3200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremralimi 3101 Inference quantifying both antecedent and consequent, with strong hypothesis. (Contributed by NM, 4-Mar-1997.)
(𝜑𝜓)       (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓)

Theorem2ralimi 3102 Inference quantifying both antecedent and consequent two times, with strong hypothesis. (Contributed by AV, 3-Dec-2021.)
(𝜑𝜓)       (∀𝑥𝐴𝑦𝐵 𝜑 → ∀𝑥𝐴𝑦𝐵 𝜓)

Theoremhbralrimi 3103 Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). This theorem contains the common proof steps for ralrimi 3106 and ralrimiv 3114. Its main advantage over these two is its minimal references to axioms. The proof is extracted from NM's previous work. (Contributed by Wolf Lammen, 4-Dec-2019.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝑥𝐴𝜓))       (𝜑 → ∀𝑥𝐴 𝜓)

Theoremr19.21t 3104 Restricted quantifier version of 19.21t 2229; closed form of r19.21 3105. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Wolf Lammen, 2-Jan-2020.)
(Ⅎ𝑥𝜑 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓)))

Theoremr19.21 3105 Restricted quantifier version of 19.21 2231. (Contributed by Scott Fenton, 30-Mar-2011.)
𝑥𝜑       (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓))

Theoremralrimi 3106 Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 10-Oct-1999.) Shortened after introduction of hbralrimi 3103. (Revised by Wolf Lammen, 4-Dec-2019.)
𝑥𝜑    &   (𝜑 → (𝑥𝐴𝜓))       (𝜑 → ∀𝑥𝐴 𝜓)

Theoremralimdaa 3107 Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-Sep-2003.) (Proof shortened by Wolf Lammen, 29-Dec-2019.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))

Theoremralrimd 3108 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 16-Feb-2004.)
𝑥𝜑    &   𝑥𝜓    &   (𝜑 → (𝜓 → (𝑥𝐴𝜒)))       (𝜑 → (𝜓 → ∀𝑥𝐴 𝜒))

Theoremr19.21v 3109* Restricted quantifier version of 19.21v 2020. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 2-Jan-2020.)
(∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓))

Theoremralimdv2 3110* Inference quantifying both antecedent and consequent. (Contributed by NM, 1-Feb-2005.)
(𝜑 → ((𝑥𝐴𝜓) → (𝑥𝐵𝜒)))       (𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐵 𝜒))

Theoremralimdva 3111* Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 5-Dec-2019.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))

Theoremralimdv 3112* Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90 (alim 1886). (Contributed by NM, 8-Oct-2003.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))

Theoremralimdvva 3113* Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90 (alim 1886). (Contributed by AV, 27-Nov-2019.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 → ∀𝑥𝐴𝑦𝐵 𝜒))

Theoremralrimiv 3114* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 4-Dec-2019.)
(𝜑 → (𝑥𝐴𝜓))       (𝜑 → ∀𝑥𝐴 𝜓)

Theoremralrimiva 3115* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Jan-2006.)
((𝜑𝑥𝐴) → 𝜓)       (𝜑 → ∀𝑥𝐴 𝜓)

Theoremralrimivw 3116* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)
(𝜑𝜓)       (𝜑 → ∀𝑥𝐴 𝜓)

Theoremralrimdv 3117* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 27-May-1998.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 28-Dec-2019.)
(𝜑 → (𝜓 → (𝑥𝐴𝜒)))       (𝜑 → (𝜓 → ∀𝑥𝐴 𝜒))

Theoremralrimdva 3118* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Feb-2008.) (Proof shortened by Wolf Lammen, 28-Dec-2019.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝐴 𝜒))

Theoremralrimivv 3119* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 24-Jul-2004.)
(𝜑 → ((𝑥𝐴𝑦𝐵) → 𝜓))       (𝜑 → ∀𝑥𝐴𝑦𝐵 𝜓)

Theoremralrimivva 3120* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by Jeff Madsen, 19-Jun-2011.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝜓)       (𝜑 → ∀𝑥𝐴𝑦𝐵 𝜓)

Theoremralrimivvva 3121* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with triple quantification.) (Contributed by Mario Carneiro, 9-Jul-2014.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵𝑧𝐶)) → 𝜓)       (𝜑 → ∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜓)

Theoremralrimdvv 3122* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 1-Jun-2005.)
(𝜑 → (𝜓 → ((𝑥𝐴𝑦𝐵) → 𝜒)))       (𝜑 → (𝜓 → ∀𝑥𝐴𝑦𝐵 𝜒))

Theoremralrimdvva 3123* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 2-Feb-2008.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝐴𝑦𝐵 𝜒))

Theoremrgen2 3124* Generalization rule for restricted quantification, with two quantifiers. (Contributed by NM, 30-May-1999.)
((𝑥𝐴𝑦𝐵) → 𝜑)       𝑥𝐴𝑦𝐵 𝜑

Theoremrgen3 3125* Generalization rule for restricted quantification, with three quantifiers. (Contributed by NM, 12-Jan-2008.)
((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑)       𝑥𝐴𝑦𝐵𝑧𝐶 𝜑

Theoremrgen2a 3126* Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 are not required to be disjoint. This proof illustrates the use of dvelim 2487. (Contributed by NM, 23-Nov-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 1-Jan-2020.) (Proof modification is discouraged.)
((𝑥𝐴𝑦𝐴) → 𝜑)       𝑥𝐴𝑦𝐴 𝜑

Theoremralbii2 3127 Inference adding different restricted universal quantifiers to each side of an equivalence. (Contributed by NM, 15-Aug-2005.)
((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓)

Theoremralbiia 3128 Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 26-Nov-2000.)
(𝑥𝐴 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜓)

Theoremralbii 3129 Inference adding restricted universal 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, 4-Dec-2019.)
(𝜑𝜓)       (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜓)

Theorem2ralbii 3130 Inference adding two restricted universal quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
(𝜑𝜓)       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝐴𝑦𝐵 𝜓)

Theoremralbida 3131 Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 6-Oct-2003.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))

Theoremralbid 3132 Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 27-Jun-1998.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))

Theoremralbidv2 3133* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 6-Apr-1997.)
(𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))

Theoremralbidva 3134* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 4-Mar-1997.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 29-Dec-2019.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))

Theoremralbidv 3135* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 20-Nov-1994.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 5-Dec-2019.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))

Theorem2ralbida 3136* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 24-Feb-2004.)
𝑥𝜑    &   𝑦𝜑    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 ↔ ∀𝑥𝐴𝑦𝐵 𝜒))

Theorem2ralbidva 3137* Formula-building rule for restricted universal quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 9-Dec-2019.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 ↔ ∀𝑥𝐴𝑦𝐵 𝜒))

Theorem2ralbidv 3138* Formula-building rule for restricted universal quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.) (Revised by Szymon Jaroszewicz, 16-Mar-2007.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 ↔ ∀𝑥𝐴𝑦𝐵 𝜒))

Theoremraleqbii 3139 Equality deduction for restricted universal quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)
𝐴 = 𝐵    &   (𝜓𝜒)       (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒)

Theoremraln 3140 Restricted universally quantified negation expressed as a universally quantified negation. (Contributed by BJ, 16-Jul-2021.)
(∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑥 ¬ (𝑥𝐴𝜑))

Theoremralnex 3141 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) (Proof shortened by BJ, 16-Jul-2021.)
(∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴 𝜑)

Theoremdfral2 3142 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) Allow shortening of rexnal 3143. (Revised by Wolf Lammen, 9-Dec-2019.)
(∀𝑥𝐴 𝜑 ↔ ¬ ∃𝑥𝐴 ¬ 𝜑)

Theoremrexnal 3143 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 3144 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)

Theoremralinexa 3145 A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)
(∀𝑥𝐴 (𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥𝐴 (𝜑𝜓))

Theoremrexanali 3146 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 3147 Negation of a complex predicate calculus formula. (Contributed by FL, 31-Jul-2009.)
(¬ ∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∀𝑥𝐴𝑦𝐵 (𝜑 ∧ ¬ 𝜓))

Theoremnrex 3148 Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
(𝑥𝐴 → ¬ 𝜓)        ¬ ∃𝑥𝐴 𝜓

Theoremnrexdv 3149* 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)

Theoremrexex 3150 Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)
(∃𝑥𝐴 𝜑 → ∃𝑥𝜑)

Theoremrspe 3151 Restricted specialization. (Contributed by NM, 12-Oct-1999.)
((𝑥𝐴𝜑) → ∃𝑥𝐴 𝜑)

Theoremrsp2e 3152 Restricted specialization. (Contributed by FL, 4-Jun-2012.) (Proof shortened by Wolf Lammen, 7-Jan-2020.)
((𝑥𝐴𝑦𝐵𝜑) → ∃𝑥𝐴𝑦𝐵 𝜑)

Theoremnfre1 3153 The setvar 𝑥 is not free in 𝑥𝐴𝜑. (Contributed by NM, 19-Mar-1997.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝑥𝐴 𝜑

Theoremnfrexd 3154 Deduction version of nfrex 3155. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)

Theoremnfrex 3155 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.)
𝑥𝐴    &   𝑥𝜑       𝑥𝑦𝐴 𝜑

Theoremrexim 3156 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 3157 Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.)
(𝑥𝐴 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓)

Theoremreximi2 3158 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)

Theoremreximi 3159 Inference quantifying both antecedent and consequent. (Contributed by NM, 18-Oct-1996.)
(𝜑𝜓)       (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓)

Theoremreximdai 3160 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 3161 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)

Theoremreximdv2 3162* 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 3163* 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 3164* 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 3165* 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 3166* 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)

Theoremreximdvva 3167* 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 3168* 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 3169 Closed theorem form of r19.23 3170. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
(Ⅎ𝑥𝜓 → (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))

Theoremr19.23 3170 Restricted quantifier version of 19.23 2236. See r19.23v 3171 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 3171* Restricted quantifier version of 19.23v 2023. Version of r19.23 3170 with a dv condition. (Contributed by NM, 31-Aug-1999.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2020.)
(∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))

Theoremrexlimi 3172 Restricted quantifier version of exlimi 2242. (Contributed by NM, 30-Nov-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
𝑥𝜓    &   (𝑥𝐴 → (𝜑𝜓))       (∃𝑥𝐴 𝜑𝜓)

Theoremrexlimd2 3173 Version of rexlimd 3174 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 3174 Deduction form of rexlimd 3174. (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 3175* 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 3176* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Dec-2006.)
((𝑥𝐴𝜑) → 𝜓)       (∃𝑥𝐴 𝜑𝜓)

Theoremrexlimivw 3177* Weaker version of rexlimiv 3175. (Contributed by FL, 19-Sep-2011.)
(𝜑𝜓)       (∃𝑥𝐴 𝜑𝜓)

Theoremrexlimdv 3178* 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 3179* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 20-Jan-2007.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓𝜒))

Theoremrexlimdvaa 3180* 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 3181* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 3178. (Contributed by NM, 7-Jun-2015.)
((𝜑𝑥𝐴𝜓) → 𝜒)       (𝜑 → (∃𝑥𝐴 𝜓𝜒))

Theoremrexlimdvw 3182* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Jun-2014.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓𝜒))

Theoremrexlimddv 3183* Restricted existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
(𝜑 → ∃𝑥𝐴 𝜓)    &   ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜒)       (𝜑𝜒)

Theoremrexlimivv 3184* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 17-Feb-2004.)
((𝑥𝐴𝑦𝐵) → (𝜑𝜓))       (∃𝑥𝐴𝑦𝐵 𝜑𝜓)

Theoremrexlimdvv 3185* 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 3186* 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 3187 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 3188 Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 26-Oct-1999.)
(𝑥𝐴 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 𝜓)

Theoremrexbii 3189 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 3190 Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 11-Nov-1995.)
(𝜑𝜓)       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓)

Theoremrexnal2 3191 Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(∃𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵 𝜑)

Theoremrexnal3 3192 Relationship between three restricted universal and existential quantifiers. (Contributed by Thierry Arnoux, 12-Jul-2020.)
(∃𝑥𝐴𝑦𝐵𝑧𝐶 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑)

Theoremralnex2 3193 Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴𝑦𝐵 𝜑)

Theoremralnex3 3194 Relationship between three restricted universal and existential quantifiers. (Contributed by Thierry Arnoux, 12-Jul-2020.)
(∀𝑥𝐴𝑦𝐵𝑧𝐶 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑)

Theoremrexbida 3195 Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 6-Oct-2003.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))

Theoremrexbidv2 3196* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 22-May-1999.)
(𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))

Theoremrexbidva 3197* Formula-building rule for restricted existential quantifier (deduction rule). (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 3198* Alternate proof of rexbida 3195, 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 3199 Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 27-Jun-1998.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))

Theoremrexbidv 3200* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 20-Nov-1994.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 6-Dec-2019.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))

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