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Theorem List for Metamath Proof Explorer - 3101-3200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnfnel 3101 Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥 𝐴𝐵
 
Theoremnfneld 3102 Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴𝐵)
 
Theoremnnel 3103 Negation of negated membership, analogous to nne 2994. (Contributed by Alexander van der Vekens, 18-Jan-2018.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
𝐴𝐵𝐴𝐵)
 
Theoremelnelne1 3104 Two classes are different if they don't contain the same element. (Contributed by AV, 28-Jan-2020.)
((𝐴𝐵𝐴𝐶) → 𝐵𝐶)
 
Theoremelnelne2 3105 Two classes are different if they don't belong to the same class. (Contributed by AV, 28-Jan-2020.)
((𝐴𝐶𝐵𝐶) → 𝐴𝐵)
 
Theoremnelcon3d 3106 Contrapositive law deduction for negated membership. (Contributed by AV, 28-Jan-2020.)
(𝜑 → (𝐴𝐵𝐶𝐷))       (𝜑 → (𝐶𝐷𝐴𝐵))
 
Theoremelnelall 3107 A contradiction concerning membership implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
(𝐴𝐵 → (𝐴𝐵𝜑))
 
Theorempm2.61danel 3108 Deduction eliminating an elementhood in an antecedent. (Contributed by AV, 5-Dec-2021.)
((𝜑𝐴𝐵) → 𝜓)    &   ((𝜑𝐴𝐵) → 𝜓)       (𝜑𝜓)
 
2.1.5  Restricted quantification
 
Syntaxwral 3109 Extend wff notation to include restricted universal quantification.
wff 𝑥𝐴 𝜑
 
Syntaxwrex 3110 Extend wff notation to include restricted existential quantification.
wff 𝑥𝐴 𝜑
 
Syntaxwreu 3111 Extend wff notation to include restricted existential uniqueness.
wff ∃!𝑥𝐴 𝜑
 
Syntaxwrmo 3112 Extend wff notation to include restricted "at most one."
wff ∃*𝑥𝐴 𝜑
 
Syntaxcrab 3113 Extend class notation to include the restricted class abstraction (class builder).
class {𝑥𝐴𝜑}
 
Definitiondf-ral 3114 Define restricted universal quantification. Special case of Definition 4.15(3) of [TakeutiZaring] p. 22.

Note: This notation is most often used to express that 𝜑 holds for all elements of a given class 𝐴. For this reading 𝑥𝐴 is required, though, for example, asserted when 𝑥 and 𝐴 are disjoint.

Should instead 𝐴 depend on 𝑥, you rather focus on those 𝑥 that happen to be contained in the corresponding 𝐴(𝑥). This hardly used interpretation could still occur naturally. For some examples, look at ralndv1 43648 or ralndv2 43649, courtesy of AV.

So be careful to either keep 𝐴 independent of 𝑥, or adjust your comments to include such exotic cases. (Contributed by NM, 19-Aug-1993.)

(∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
 
Definitiondf-rex 3115 Define restricted existential quantification. Special case of Definition 4.15(4) of [TakeutiZaring] p. 22.

Note: This notation is most often used to express that 𝜑 holds for at least one element of a given class 𝐴. For this reading 𝑥𝐴 is required, though, for example, asserted when 𝑥 and 𝐴 are disjoint.

Should instead 𝐴 depend on 𝑥, you rather assert at least one 𝑥 fulfilling 𝜑 happens to be contained in the corresponding 𝐴(𝑥). This interpretation is rarely needed (see also df-ral 3114). (Contributed by NM, 30-Aug-1993.)

(∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
 
Definitiondf-reu 3116 Define restricted existential uniqueness.

Note: This notation is most often used to express that 𝜑 holds for exactly one element of a given class 𝐴. For this reading 𝑥𝐴 is required, though, for example, asserted when 𝑥 and 𝐴 are disjoint.

Should instead 𝐴 depend on 𝑥, you rather assert exactly one 𝑥 fulfilling 𝜑 happens to be contained in the corresponding 𝐴(𝑥). This interpretation is rarely needed (see also df-ral 3114). (Contributed by NM, 22-Nov-1994.)

(∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
 
Definitiondf-rmo 3117 Define restricted "at most one".

Note: This notation is most often used to express that 𝜑 holds for at most one element of a given class 𝐴. For this reading 𝑥𝐴 is required, though, for example, asserted when 𝑥 and 𝐴 are disjoint.

Should instead 𝐴 depend on 𝑥, you rather assert at most one 𝑥 fulfilling 𝜑 happens to be contained in the corresponding 𝐴(𝑥). This interpretation is rarely needed (see also df-ral 3114). (Contributed by NM, 16-Jun-2017.)

(∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
 
Definitiondf-rab 3118 Define a restricted class abstraction (class builder): {𝑥𝐴𝜑} is the class of all sets 𝑥 in 𝐴 such that 𝜑(𝑥) is true. Definition of [TakeutiZaring] p. 20.

For the interpretation given in the previous paragraph to be correct, we need to assume 𝑥𝐴, which is the case as soon as 𝑥 and 𝐴 are disjoint, which is generally the case. If 𝐴 were to depend on 𝑥, then the interpretation would be less obvious (think of the two extreme cases 𝐴 = {𝑥} and 𝐴 = 𝑥, for instance). See also df-ral 3114. (Contributed by NM, 22-Nov-1994.)

{𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
 
Theoremrgen 3119 Generalization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
(𝑥𝐴𝜑)       𝑥𝐴 𝜑
 
Theoremralel 3120 All elements of a class are elements of the class. (Contributed by AV, 30-Oct-2020.)
𝑥𝐴 𝑥𝐴
 
Theoremrgenw 3121 Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)
𝜑       𝑥𝐴 𝜑
 
Theoremrgen2w 3122 Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by NM, 18-Jun-2014.)
𝜑       𝑥𝐴𝑦𝐵 𝜑
 
Theoremmprg 3123 Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.)
(∀𝑥𝐴 𝜑𝜓)    &   (𝑥𝐴𝜑)       𝜓
 
Theoremmprgbir 3124 Modus ponens on biconditional combined with restricted generalization. (Contributed by NM, 21-Mar-2004.)
(𝜑 ↔ ∀𝑥𝐴 𝜓)    &   (𝑥𝐴𝜓)       𝜑
 
Theoremalral 3125 Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)
(∀𝑥𝜑 → ∀𝑥𝐴 𝜑)
 
Theoremraln 3126 Restricted universally quantified negation expressed as a universally quantified negation. (Contributed by BJ, 16-Jul-2021.)
(∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑥 ¬ (𝑥𝐴𝜑))
 
Theoremral2imi 3127 Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.) Allow shortening of ralim 3133. (Revised by Wolf Lammen, 1-Dec-2019.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝐴 𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))
 
Theoremralimi2 3128 Inference quantifying both antecedent and consequent. (Contributed by NM, 22-Feb-2004.)
((𝑥𝐴𝜑) → (𝑥𝐵𝜓))       (∀𝑥𝐴 𝜑 → ∀𝑥𝐵 𝜓)
 
Theoremralimia 3129 Inference quantifying both antecedent and consequent. (Contributed by NM, 19-Jul-1996.)
(𝑥𝐴 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓)
 
Theoremralimiaa 3130 Inference quantifying both antecedent and consequent. (Contributed by NM, 4-Aug-2007.)
((𝑥𝐴𝜑) → 𝜓)       (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓)
 
Theoremralimi 3131 Inference quantifying both antecedent and consequent, with strong hypothesis. (Contributed by NM, 4-Mar-1997.)
(𝜑𝜓)       (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓)
 
Theorem2ralimi 3132 Inference quantifying both antecedent and consequent two times, with strong hypothesis. (Contributed by AV, 3-Dec-2021.)
(𝜑𝜓)       (∀𝑥𝐴𝑦𝐵 𝜑 → ∀𝑥𝐴𝑦𝐵 𝜓)
 
Theoremralim 3133 Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997.) (Proof shortened by Wolf Lammen, 1-Dec-2019.)
(∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓))
 
Theoremralbii2 3134 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 3135 Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 26-Nov-2000.)
(𝑥𝐴 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜓)
 
Theoremralbii 3136 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 3137 Inference adding two restricted universal quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
(𝜑𝜓)       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝐴𝑦𝐵 𝜓)
 
Theoremralbi 3138 Distribute a restricted universal quantifier over a biconditional. Restricted quantification version of albi 1820. (Contributed by NM, 6-Oct-2003.) Reduce axiom usage. (Revised by Wolf Lammen, 17-Jun-2023.)
(∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜓))
 
Theoremralanid 3139 Cancellation law for restricted universal quantification. (Contributed by Peter Mazsa, 30-Dec-2018.) (Proof shortened by Wolf Lammen, 29-Jun-2023.)
(∀𝑥𝐴 (𝑥𝐴𝜑) ↔ ∀𝑥𝐴 𝜑)
 
Theoremr19.26 3140 Restricted quantifier version of 19.26 1871. (Contributed by NM, 28-Jan-1997.) (Proof shortened by Andrew Salmon, 30-May-2011.)
(∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓))
 
Theoremr19.26-2 3141 Restricted quantifier version of 19.26-2 1872. Version of r19.26 3140 with two quantifiers. (Contributed by NM, 10-Aug-2004.)
(∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∀𝑥𝐴𝑦𝐵 𝜓))
 
Theoremr19.26-3 3142 Version of r19.26 3140 with three quantifiers. (Contributed by FL, 22-Nov-2010.)
(∀𝑥𝐴 (𝜑𝜓𝜒) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓 ∧ ∀𝑥𝐴 𝜒))
 
Theoremr19.26m 3143 Version of 19.26 1871 and r19.26 3140 with restricted quantifiers ranging over different classes. (Contributed by NM, 22-Feb-2004.)
(∀𝑥((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓)) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜓))
 
Theoremralbiim 3144 Split a biconditional and distribute quantifier. Restricted quantifier version of albiim 1890. (Contributed by NM, 3-Jun-2012.)
(∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 (𝜑𝜓) ∧ ∀𝑥𝐴 (𝜓𝜑)))
 
Theoremr19.21v 3145* Restricted quantifier version of 19.21v 1940. (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 3146* Inference quantifying both antecedent and consequent. (Contributed by NM, 1-Feb-2005.)
(𝜑 → ((𝑥𝐴𝜓) → (𝑥𝐵𝜒)))       (𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐵 𝜒))
 
Theoremralimdva 3147* 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 3148* Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90 (alim 1812). (Contributed by NM, 8-Oct-2003.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))
 
Theoremralimdvva 3149* Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90 (alim 1812). (Contributed by AV, 27-Nov-2019.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 → ∀𝑥𝐴𝑦𝐵 𝜒))
 
Theoremhbralrimi 3150 Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). This theorem contains the common proof steps for ralrimi 3183 and ralrimiv 3151. 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)
 
Theoremralrimiv 3151* 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 3152* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Jan-2006.)
((𝜑𝑥𝐴) → 𝜓)       (𝜑 → ∀𝑥𝐴 𝜓)
 
Theoremralrimivw 3153* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)
(𝜑𝜓)       (𝜑 → ∀𝑥𝐴 𝜓)
 
Theoremr19.27v 3154* Restricted quantitifer version of one direction of 19.27 2228. (The other direction holds when 𝐴 is nonempty, see r19.27zv 4412.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof shortened by Wolf Lammen, 17-Jun-2023.)
((∀𝑥𝐴 𝜑𝜓) → ∀𝑥𝐴 (𝜑𝜓))
 
Theoremr19.28v 3155* Restricted quantifier version of one direction of 19.28 2229. (The other direction holds when 𝐴 is nonempty, see r19.28zv 4407.) (Contributed by NM, 2-Apr-2004.) (Proof shortened by Wolf Lammen, 17-Jun-2023.)
((𝜑 ∧ ∀𝑥𝐴 𝜓) → ∀𝑥𝐴 (𝜑𝜓))
 
Theoremralrimdv 3156* 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 3157* 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 3158* 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 3159* 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 3160* 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 3161* 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 3162* 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))
 
Theoremralbidv2 3163* Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Apr-1997.)
(𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
 
Theoremralbidva 3164* Formula-building rule for restricted universal quantifier (deduction form). (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 3165* Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 20-Nov-1994.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 5-Dec-2019.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))
 
Theorem2ralbidva 3166* Formula-building rule for restricted universal quantifiers (deduction form). (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 3167* Formula-building rule for restricted universal quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.) (Revised by Szymon Jaroszewicz, 16-Mar-2007.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 ↔ ∀𝑥𝐴𝑦𝐵 𝜒))
 
Theoremr2allem 3168 Lemma factoring out common proof steps of r2alf 3189 and r2al 3169. Introduced to reduce dependencies on axioms. (Contributed by Wolf Lammen, 9-Jan-2020.)
(∀𝑦(𝑥𝐴 → (𝑦𝐵𝜑)) ↔ (𝑥𝐴 → ∀𝑦(𝑦𝐵𝜑)))       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
 
Theoremr2al 3169* Double restricted universal quantification. (Contributed by NM, 19-Nov-1995.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 9-Jan-2020.)
(∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
 
Theoremr3al 3170* Triple restricted universal quantification. (Contributed by NM, 19-Nov-1995.) (Proof shortened by Wolf Lammen, 30-Dec-2019.)
(∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑))
 
Theoremrgen2 3171* Generalization rule for restricted quantification, with two quantifiers. This theorem should be used in place of rgen2a 3196 since it depends on a smaller set of axioms. (Contributed by NM, 30-May-1999.)
((𝑥𝐴𝑦𝐵) → 𝜑)       𝑥𝐴𝑦𝐵 𝜑
 
Theoremrgen3 3172* Generalization rule for restricted quantification, with three quantifiers. (Contributed by NM, 12-Jan-2008.)
((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑)       𝑥𝐴𝑦𝐵𝑧𝐶 𝜑
 
Theoremrsp 3173 Restricted specialization. (Contributed by NM, 17-Oct-1996.)
(∀𝑥𝐴 𝜑 → (𝑥𝐴𝜑))
 
Theoremrspa 3174 Restricted specialization. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((∀𝑥𝐴 𝜑𝑥𝐴) → 𝜑)
 
Theoremrspec 3175 Specialization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
𝑥𝐴 𝜑       (𝑥𝐴𝜑)
 
Theoremr19.21bi 3176 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.) (Proof shortened by Wolf Lammen, 11-Jun-2023.)
(𝜑 → ∀𝑥𝐴 𝜓)       ((𝜑𝑥𝐴) → 𝜓)
 
Theoremr19.21be 3177 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 21-Nov-1994.)
(𝜑 → ∀𝑥𝐴 𝜓)       𝑥𝐴 (𝜑𝜓)
 
Theoremrspec2 3178 Specialization rule for restricted quantification, with two quantifiers. (Contributed by NM, 20-Nov-1994.)
𝑥𝐴𝑦𝐵 𝜑       ((𝑥𝐴𝑦𝐵) → 𝜑)
 
Theoremrspec3 3179 Specialization rule for restricted quantification, with three quantifiers. (Contributed by NM, 20-Nov-1994.)
𝑥𝐴𝑦𝐵𝑧𝐶 𝜑       ((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑)
 
Theoremrsp2 3180 Restricted specialization, with two quantifiers. (Contributed by NM, 11-Feb-1997.)
(∀𝑥𝐴𝑦𝐵 𝜑 → ((𝑥𝐴𝑦𝐵) → 𝜑))
 
Theoremr19.21t 3181 Restricted quantifier version of 19.21t 2205; closed form of r19.21 3182. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Wolf Lammen, 2-Jan-2020.)
(Ⅎ𝑥𝜑 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓)))
 
Theoremr19.21 3182 Restricted quantifier version of 19.21 2206. (Contributed by Scott Fenton, 30-Mar-2011.)
𝑥𝜑       (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓))
 
Theoremralrimi 3183 Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 10-Oct-1999.) Shortened after introduction of hbralrimi 3150. (Revised by Wolf Lammen, 4-Dec-2019.)
𝑥𝜑    &   (𝜑 → (𝑥𝐴𝜓))       (𝜑 → ∀𝑥𝐴 𝜓)
 
Theoremralimdaa 3184 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 3185 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))
 
Theoremnfra1 3186 The setvar 𝑥 is not free in 𝑥𝐴𝜑. (Contributed by NM, 18-Oct-1996.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝑥𝐴 𝜑
 
Theoremhbra1 3187 The setvar 𝑥 is not free in 𝑥𝐴𝜑. (Contributed by NM, 18-Oct-1996.) (Proof shortened by Wolf Lammen, 7-Dec-2019.)
(∀𝑥𝐴 𝜑 → ∀𝑥𝑥𝐴 𝜑)
 
Theoremhbral 3188 Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by David Abernethy, 13-Dec-2009.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)    &   (𝜑 → ∀𝑥𝜑)       (∀𝑦𝐴 𝜑 → ∀𝑥𝑦𝐴 𝜑)
 
Theoremr2alf 3189* Double restricted universal quantification. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2allem 3168. (Revised by Wolf Lammen, 9-Jan-2020.)
𝑦𝐴       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
 
Theoremnfraldw 3190* Deduction version of nfralw 3192. Version of nfrald 3191 with a disjoint variable condition, which does not require ax-13 2382. (Contributed by NM, 15-Feb-2013.) (Revised by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
 
Theoremnfrald 3191 Deduction version of nfral 3193. Usage of this theorem is discouraged because it depends on ax-13 2382. Use the weaker nfraldw 3190 when possible. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
 
Theoremnfralw 3192* Bound-variable hypothesis builder for restricted quantification. Version of nfral 3193 with a disjoint variable condition, which does not require ax-13 2382. (Contributed by NM, 1-Sep-1999.) (Revised by Gino Giotto, 10-Jan-2024.)
𝑥𝐴    &   𝑥𝜑       𝑥𝑦𝐴 𝜑
 
Theoremnfral 3193 Bound-variable hypothesis builder for restricted quantification. Usage of this theorem is discouraged because it depends on ax-13 2382. Use the weaker nfralw 3192 when possible. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.)
𝑥𝐴    &   𝑥𝜑       𝑥𝑦𝐴 𝜑
 
Theoremnfra2w 3194* Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD 41553. Version of nfra2 3195 with a disjoint variable condition, which does not require ax-13 2382. (Contributed by Alan Sare, 31-Dec-2011.) (Revised by Gino Giotto, 10-Jan-2024.)
𝑦𝑥𝐴𝑦𝐵 𝜑
 
Theoremnfra2 3195* Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD 41553. Usage of this theorem is discouraged because it depends on ax-13 2382. Use the weaker nfra2w 3194 when possible. (Contributed by Alan Sare, 31-Dec-2011.) (New usage is discouraged.)
𝑦𝑥𝐴𝑦𝐵 𝜑
 
Theoremrgen2a 3196* Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 are not required to be disjoint. This proof illustrates the use of dvelim 2465. This theorem relies on the full set of axioms up to ax-ext 2773 and it should no longer be used. Usage of rgen2 3171 is highly encouraged. (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.) (New usage is discouraged.)
((𝑥𝐴𝑦𝐴) → 𝜑)       𝑥𝐴𝑦𝐴 𝜑
 
Theoremralbida 3197 Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Oct-2003.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))
 
Theoremralbid 3198 Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 27-Jun-1998.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))
 
Theorem2ralbida 3199* Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 24-Feb-2004.)
𝑥𝜑    &   𝑦𝜑    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 ↔ ∀𝑥𝐴𝑦𝐵 𝜒))
 
Theoremraleqbii 3200 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)
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