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Theorem List for Metamath Proof Explorer - 2801-2900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxwrex 2801 Extend wff notation to include restricted existential quantification.
wff 𝑥𝐴 𝜑
 
Syntaxwreu 2802 Extend wff notation to include restricted existential uniqueness.
wff ∃!𝑥𝐴 𝜑
 
Syntaxwrmo 2803 Extend wff notation to include restricted "at most one."
wff ∃*𝑥𝐴 𝜑
 
Syntaxcrab 2804 Extend class notation to include the restricted class abstraction (class builder).
class {𝑥𝐴𝜑}
 
Definitiondf-ral 2805 Define restricted universal quantification. Special case of Definition 4.15(3) of [TakeutiZaring] p. 22. (Contributed by NM, 19-Aug-1993.)
(∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
 
Definitiondf-rex 2806 Define restricted existential quantification. Special case of Definition 4.15(4) of [TakeutiZaring] p. 22. (Contributed by NM, 30-Aug-1993.)
(∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
 
Definitiondf-reu 2807 Define restricted existential uniqueness. (Contributed by NM, 22-Nov-1994.)
(∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
 
Definitiondf-rmo 2808 Define restricted "at most one". (Contributed by NM, 16-Jun-2017.)
(∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
 
Definitiondf-rab 2809 Define a restricted class abstraction (class builder), which is the class of all 𝑥 in 𝐴 such that 𝜑 is true. Definition of [TakeutiZaring] p. 20. (Contributed by NM, 22-Nov-1994.)
{𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
 
Theoremrgen 2810 Generalization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
(𝑥𝐴𝜑)       𝑥𝐴 𝜑
 
Theoremralel 2811 All elements of a class are elements of the class. (Contributed by AV, 30-Oct-2020.)
𝑥𝐴 𝑥𝐴
 
Theoremrgenw 2812 Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)
𝜑       𝑥𝐴 𝜑
 
Theoremrgen2w 2813 Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by NM, 18-Jun-2014.)
𝜑       𝑥𝐴𝑦𝐵 𝜑
 
Theoremmprg 2814 Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.)
(∀𝑥𝐴 𝜑𝜓)    &   (𝑥𝐴𝜑)       𝜓
 
Theoremmprgbir 2815 Modus ponens on biconditional combined with restricted generalization. (Contributed by NM, 21-Mar-2004.)
(𝜑 ↔ ∀𝑥𝐴 𝜓)    &   (𝑥𝐴𝜓)       𝜑
 
Theoremalral 2816 Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)
(∀𝑥𝜑 → ∀𝑥𝐴 𝜑)
 
Theoremrsp 2817 Restricted specialization. (Contributed by NM, 17-Oct-1996.)
(∀𝑥𝐴 𝜑 → (𝑥𝐴𝜑))
 
Theoremrspa 2818 Restricted specialization. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((∀𝑥𝐴 𝜑𝑥𝐴) → 𝜑)
 
Theoremrspec 2819 Specialization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
𝑥𝐴 𝜑       (𝑥𝐴𝜑)
 
Theoremr19.21bi 2820 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.) (Proof shortened by Wolf Lammen, 1-Jan-2020.)
(𝜑 → ∀𝑥𝐴 𝜓)       ((𝜑𝑥𝐴) → 𝜓)
 
Theoremr19.21be 2821 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 21-Nov-1994.)
(𝜑 → ∀𝑥𝐴 𝜓)       𝑥𝐴 (𝜑𝜓)
 
Theoremrspec2 2822 Specialization rule for restricted quantification, with two quantifiers. (Contributed by NM, 20-Nov-1994.)
𝑥𝐴𝑦𝐵 𝜑       ((𝑥𝐴𝑦𝐵) → 𝜑)
 
Theoremrspec3 2823 Specialization rule for restricted quantification, with three quantifiers. (Contributed by NM, 20-Nov-1994.)
𝑥𝐴𝑦𝐵𝑧𝐶 𝜑       ((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑)
 
Theoremrsp2 2824 Restricted specialization, with two quantifiers. (Contributed by NM, 11-Feb-1997.)
(∀𝑥𝐴𝑦𝐵 𝜑 → ((𝑥𝐴𝑦𝐵) → 𝜑))
 
Theoremr2allem 2825 Lemma factoring out common proof steps of r2alf 2826 and r2al 2827. Introduced to reduce dependencies on axioms. (Contributed by Wolf Lammen, 9-Jan-2020.)
(∀𝑦(𝑥𝐴 → (𝑦𝐵𝜑)) ↔ (𝑥𝐴 → ∀𝑦(𝑦𝐵𝜑)))       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
 
Theoremr2alf 2826* Double restricted universal quantification. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2allem 2825. (Revised by Wolf Lammen, 9-Jan-2020.)
𝑦𝐴       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
 
Theoremr2al 2827* 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 2828* Triple restricted universal quantification. (Contributed by NM, 19-Nov-1995.) (Proof shortened by Wolf Lammen, 30-Dec-2019.)
(∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑))
 
Theoremnfra1 2829 The setvar 𝑥 is not free in 𝑥𝐴𝜑. (Contributed by NM, 18-Oct-1996.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝑥𝐴 𝜑
 
Theoremhbra1 2830 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 2831 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)
 
Theoremnfrald 2832 Deduction version of nfral 2833. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
 
Theoremnfral 2833 Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝐴    &   𝑥𝜑       𝑥𝑦𝐴 𝜑
 
Theoremnfra2 2834* Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD 38015. Contributed by Alan Sare 31-Dec-2011. (Contributed by NM, 31-Dec-2011.)
𝑦𝑥𝐴𝑦𝐵 𝜑
 
Theoremral2imi 2835 Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.) Allow shortening of ralim 2836. (Revised by Wolf Lammen, 1-Dec-2019.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝐴 𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))
 
Theoremralim 2836 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))
 
Theoremralimi2 2837 Inference quantifying both antecedent and consequent. (Contributed by NM, 22-Feb-2004.)
((𝑥𝐴𝜑) → (𝑥𝐵𝜓))       (∀𝑥𝐴 𝜑 → ∀𝑥𝐵 𝜓)
 
Theoremralimia 2838 Inference quantifying both antecedent and consequent. (Contributed by NM, 19-Jul-1996.)
(𝑥𝐴 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓)
 
Theoremralimiaa 2839 Inference quantifying both antecedent and consequent. (Contributed by NM, 4-Aug-2007.)
((𝑥𝐴𝜑) → 𝜓)       (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓)
 
Theoremralimi 2840 Inference quantifying both antecedent and consequent, with strong hypothesis. (Contributed by NM, 4-Mar-1997.)
(𝜑𝜓)       (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓)
 
Theoremhbralrimi 2841 Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). This theorem contains the common proof steps for ralrimi 2844 and ralrimiv 2852. 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 2842 Restricted quantifier version of 19.21t 2035; closed form of r19.21 2843. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Wolf Lammen, 2-Jan-2020.)
(Ⅎ𝑥𝜑 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓)))
 
Theoremr19.21 2843 Restricted quantifier version of 19.21 2036. (Contributed by Scott Fenton, 30-Mar-2011.)
𝑥𝜑       (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓))
 
Theoremralrimi 2844 Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 10-Oct-1999.) Shortened after introduction of hbralrimi 2841. (Revised by Wolf Lammen, 4-Dec-2019.)
𝑥𝜑    &   (𝜑 → (𝑥𝐴𝜓))       (𝜑 → ∀𝑥𝐴 𝜓)
 
Theoremralimdaa 2845 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 2846 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 2847* Restricted quantifier version of 19.21v 1821. (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 2848* Inference quantifying both antecedent and consequent. (Contributed by NM, 1-Feb-2005.)
(𝜑 → ((𝑥𝐴𝜓) → (𝑥𝐵𝜒)))       (𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐵 𝜒))
 
Theoremralimdva 2849* 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 2850* Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90 (alim 1714). (Contributed by NM, 8-Oct-2003.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))
 
Theoremralimdvva 2851* Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90 (alim 1714). (Contributed by AV, 27-Nov-2019.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 → ∀𝑥𝐴𝑦𝐵 𝜒))
 
Theoremralrimiv 2852* 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 2853* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Jan-2006.)
((𝜑𝑥𝐴) → 𝜓)       (𝜑 → ∀𝑥𝐴 𝜓)
 
Theoremralrimivw 2854* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)
(𝜑𝜓)       (𝜑 → ∀𝑥𝐴 𝜓)
 
Theoremralrimdv 2855* 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 2856* 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 2857* 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 2858* 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 2859* 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 2860* 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 2861* 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 2862* Generalization rule for restricted quantification, with two quantifiers. (Contributed by NM, 30-May-1999.)
((𝑥𝐴𝑦𝐵) → 𝜑)       𝑥𝐴𝑦𝐵 𝜑
 
Theoremrgen3 2863* Generalization rule for restricted quantification, with three quantifiers. (Contributed by NM, 12-Jan-2008.)
((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑)       𝑥𝐴𝑦𝐵𝑧𝐶 𝜑
 
Theoremrgen2a 2864* Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 are not required to be disjoint. This proof illustrates the use of dvelim 2229. (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 2865 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 2866 Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 26-Nov-2000.)
(𝑥𝐴 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜓)
 
Theoremralbii 2867 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 2868 Inference adding two restricted universal quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
(𝜑𝜓)       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝐴𝑦𝐵 𝜓)
 
Theoremralbida 2869 Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 6-Oct-2003.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))
 
Theoremralbid 2870 Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 27-Jun-1998.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))
 
Theoremralbidv2 2871* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 6-Apr-1997.)
(𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
 
Theoremralbidva 2872* 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 2873* 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 2874* 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 2875* 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 2876* 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 2877 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 2878 Restricted universally quantified negation expressed as a universally quantified negation. (Contributed by BJ, 16-Jul-2021.)
(∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑥 ¬ (𝑥𝐴𝜑))
 
Theoremralnex 2879 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) (Proof shortened by BJ, 16-Jul-2021.)
(∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴 𝜑)
 
TheoremralnexOLD 2880 Obsolete proof of ralnex 2879 as of 16-Jul-2021. (Contributed by NM, 21-Jan-1997.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴 𝜑)
 
Theoremdfral2 2881 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) Allow shortening of rexnal 2882. (Revised by Wolf Lammen, 9-Dec-2019.)
(∀𝑥𝐴 𝜑 ↔ ¬ ∃𝑥𝐴 ¬ 𝜑)
 
Theoremrexnal 2882 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 2883 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 2884 A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)
(∀𝑥𝐴 (𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥𝐴 (𝜑𝜓))
 
Theoremrexanali 2885 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 2886 Negation of a complex predicate calculus formula. (Contributed by FL, 31-Jul-2009.)
(¬ ∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∀𝑥𝐴𝑦𝐵 (𝜑 ∧ ¬ 𝜓))
 
Theoremnrex 2887 Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
(𝑥𝐴 → ¬ 𝜓)        ¬ ∃𝑥𝐴 𝜓
 
Theoremnrexdv 2888* 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 2889 Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)
(∃𝑥𝐴 𝜑 → ∃𝑥𝜑)
 
Theoremrspe 2890 Restricted specialization. (Contributed by NM, 12-Oct-1999.)
((𝑥𝐴𝜑) → ∃𝑥𝐴 𝜑)
 
Theoremrsp2e 2891 Restricted specialization. (Contributed by FL, 4-Jun-2012.) (Proof shortened by Wolf Lammen, 7-Jan-2020.)
((𝑥𝐴𝑦𝐵𝜑) → ∃𝑥𝐴𝑦𝐵 𝜑)
 
Theoremnfre1 2892 The setvar 𝑥 is not free in 𝑥𝐴𝜑. (Contributed by NM, 19-Mar-1997.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝑥𝐴 𝜑
 
Theoremnfrexd 2893 Deduction version of nfrex 2894. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
 
Theoremnfrex 2894 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 2895 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 2896 Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.)
(𝑥𝐴 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓)
 
Theoremreximi2 2897 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 2898 Inference quantifying both antecedent and consequent. (Contributed by NM, 18-Oct-1996.)
(𝜑𝜓)       (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓)
 
Theoremreximdai 2899 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 2900 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)
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