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Theorem List for Metamath Proof Explorer - 2901-3000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-ral 2901 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 2902 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 2903 Define restricted existential uniqueness. (Contributed by NM, 22-Nov-1994.)
(∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))

Definitiondf-rmo 2904 Define restricted "at most one". (Contributed by NM, 16-Jun-2017.)
(∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))

Definitiondf-rab 2905 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 2906 Generalization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
(𝑥𝐴𝜑)       𝑥𝐴 𝜑

Theoremralel 2907 All elements of a class are elements of the class. (Contributed by AV, 30-Oct-2020.)
𝑥𝐴 𝑥𝐴

Theoremrgenw 2908 Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)
𝜑       𝑥𝐴 𝜑

Theoremrgen2w 2909 Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by NM, 18-Jun-2014.)
𝜑       𝑥𝐴𝑦𝐵 𝜑

Theoremmprg 2910 Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.)
(∀𝑥𝐴 𝜑𝜓)    &   (𝑥𝐴𝜑)       𝜓

Theoremmprgbir 2911 Modus ponens on biconditional combined with restricted generalization. (Contributed by NM, 21-Mar-2004.)
(𝜑 ↔ ∀𝑥𝐴 𝜓)    &   (𝑥𝐴𝜓)       𝜑

Theoremalral 2912 Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)
(∀𝑥𝜑 → ∀𝑥𝐴 𝜑)

Theoremrsp 2913 Restricted specialization. (Contributed by NM, 17-Oct-1996.)
(∀𝑥𝐴 𝜑 → (𝑥𝐴𝜑))

Theoremrspa 2914 Restricted specialization. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((∀𝑥𝐴 𝜑𝑥𝐴) → 𝜑)

Theoremrspec 2915 Specialization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
𝑥𝐴 𝜑       (𝑥𝐴𝜑)

Theoremr19.21bi 2916 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 2917 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 21-Nov-1994.)
(𝜑 → ∀𝑥𝐴 𝜓)       𝑥𝐴 (𝜑𝜓)

Theoremrspec2 2918 Specialization rule for restricted quantification, with two quantifiers. (Contributed by NM, 20-Nov-1994.)
𝑥𝐴𝑦𝐵 𝜑       ((𝑥𝐴𝑦𝐵) → 𝜑)

Theoremrspec3 2919 Specialization rule for restricted quantification, with three quantifiers. (Contributed by NM, 20-Nov-1994.)
𝑥𝐴𝑦𝐵𝑧𝐶 𝜑       ((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑)

Theoremrsp2 2920 Restricted specialization, with two quantifiers. (Contributed by NM, 11-Feb-1997.)
(∀𝑥𝐴𝑦𝐵 𝜑 → ((𝑥𝐴𝑦𝐵) → 𝜑))

Theoremr2allem 2921 Lemma factoring out common proof steps of r2alf 2922 and r2al 2923. 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 2922* Double restricted universal quantification. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2allem 2921. (Revised by Wolf Lammen, 9-Jan-2020.)
𝑦𝐴       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))

Theoremr2al 2923* 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 2924* Triple restricted universal quantification. (Contributed by NM, 19-Nov-1995.) (Proof shortened by Wolf Lammen, 30-Dec-2019.)
(∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑))

Theoremnfra1 2925 The setvar 𝑥 is not free in 𝑥𝐴𝜑. (Contributed by NM, 18-Oct-1996.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝑥𝐴 𝜑

Theoremhbra1 2926 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 2927 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 2928 Deduction version of nfral 2929. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)

Theoremnfral 2929 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 2930* Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD 38118. Contributed by Alan Sare 31-Dec-2011. (Contributed by NM, 31-Dec-2011.)
𝑦𝑥𝐴𝑦𝐵 𝜑

Theoremral2imi 2931 Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.) Allow shortening of ralim 2932. (Revised by Wolf Lammen, 1-Dec-2019.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝐴 𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))

Theoremralim 2932 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 2933 Inference quantifying both antecedent and consequent. (Contributed by NM, 22-Feb-2004.)
((𝑥𝐴𝜑) → (𝑥𝐵𝜓))       (∀𝑥𝐴 𝜑 → ∀𝑥𝐵 𝜓)

Theoremralimia 2934 Inference quantifying both antecedent and consequent. (Contributed by NM, 19-Jul-1996.)
(𝑥𝐴 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓)

Theoremralimiaa 2935 Inference quantifying both antecedent and consequent. (Contributed by NM, 4-Aug-2007.)
((𝑥𝐴𝜑) → 𝜓)       (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓)

Theoremralimi 2936 Inference quantifying both antecedent and consequent, with strong hypothesis. (Contributed by NM, 4-Mar-1997.)
(𝜑𝜓)       (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓)

Theoremhbralrimi 2937 Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). This theorem contains the common proof steps for ralrimi 2940 and ralrimiv 2948. 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 2938 Restricted quantifier version of 19.21t 2061; closed form of r19.21 2939. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Wolf Lammen, 2-Jan-2020.)
(Ⅎ𝑥𝜑 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓)))

Theoremr19.21 2939 Restricted quantifier version of 19.21 2062. (Contributed by Scott Fenton, 30-Mar-2011.)
𝑥𝜑       (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓))

Theoremralrimi 2940 Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 10-Oct-1999.) Shortened after introduction of hbralrimi 2937. (Revised by Wolf Lammen, 4-Dec-2019.)
𝑥𝜑    &   (𝜑 → (𝑥𝐴𝜓))       (𝜑 → ∀𝑥𝐴 𝜓)

Theoremralimdaa 2941 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 2942 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 2943* Restricted quantifier version of 19.21v 1855. (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 2944* Inference quantifying both antecedent and consequent. (Contributed by NM, 1-Feb-2005.)
(𝜑 → ((𝑥𝐴𝜓) → (𝑥𝐵𝜒)))       (𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐵 𝜒))

Theoremralimdva 2945* 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 2946* Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90 (alim 1729). (Contributed by NM, 8-Oct-2003.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))

Theoremralimdvva 2947* Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90 (alim 1729). (Contributed by AV, 27-Nov-2019.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 → ∀𝑥𝐴𝑦𝐵 𝜒))

Theoremralrimiv 2948* 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 2949* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Jan-2006.)
((𝜑𝑥𝐴) → 𝜓)       (𝜑 → ∀𝑥𝐴 𝜓)

Theoremralrimivw 2950* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)
(𝜑𝜓)       (𝜑 → ∀𝑥𝐴 𝜓)

Theoremralrimdv 2951* 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 2952* 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 2953* 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 2954* 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 2955* 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 2956* 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 2957* 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 2958* Generalization rule for restricted quantification, with two quantifiers. (Contributed by NM, 30-May-1999.)
((𝑥𝐴𝑦𝐵) → 𝜑)       𝑥𝐴𝑦𝐵 𝜑

Theoremrgen3 2959* Generalization rule for restricted quantification, with three quantifiers. (Contributed by NM, 12-Jan-2008.)
((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑)       𝑥𝐴𝑦𝐵𝑧𝐶 𝜑

Theoremrgen2a 2960* Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 are not required to be disjoint. This proof illustrates the use of dvelim 2325. (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 2961 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 2962 Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 26-Nov-2000.)
(𝑥𝐴 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜓)

Theoremralbii 2963 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 2964 Inference adding two restricted universal quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
(𝜑𝜓)       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝐴𝑦𝐵 𝜓)

Theoremralbida 2965 Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 6-Oct-2003.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))

Theoremralbid 2966 Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 27-Jun-1998.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))

Theoremralbidv2 2967* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 6-Apr-1997.)
(𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))

Theoremralbidva 2968* 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 2969* 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 2970* 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 2971* 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 2972* 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 2973 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 2974 Restricted universally quantified negation expressed as a universally quantified negation. (Contributed by BJ, 16-Jul-2021.)
(∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑥 ¬ (𝑥𝐴𝜑))

Theoremralnex 2975 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 2976 Obsolete proof of ralnex 2975 as of 16-Jul-2021. (Contributed by NM, 21-Jan-1997.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴 𝜑)

Theoremdfral2 2977 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) Allow shortening of rexnal 2978. (Revised by Wolf Lammen, 9-Dec-2019.)
(∀𝑥𝐴 𝜑 ↔ ¬ ∃𝑥𝐴 ¬ 𝜑)

Theoremrexnal 2978 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 2979 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 2980 A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)
(∀𝑥𝐴 (𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥𝐴 (𝜑𝜓))

Theoremrexanali 2981 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 2982 Negation of a complex predicate calculus formula. (Contributed by FL, 31-Jul-2009.)
(¬ ∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∀𝑥𝐴𝑦𝐵 (𝜑 ∧ ¬ 𝜓))

Theoremnrex 2983 Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
(𝑥𝐴 → ¬ 𝜓)        ¬ ∃𝑥𝐴 𝜓

Theoremnrexdv 2984* 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 2985 Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)
(∃𝑥𝐴 𝜑 → ∃𝑥𝜑)

Theoremrspe 2986 Restricted specialization. (Contributed by NM, 12-Oct-1999.)
((𝑥𝐴𝜑) → ∃𝑥𝐴 𝜑)

Theoremrsp2e 2987 Restricted specialization. (Contributed by FL, 4-Jun-2012.) (Proof shortened by Wolf Lammen, 7-Jan-2020.)
((𝑥𝐴𝑦𝐵𝜑) → ∃𝑥𝐴𝑦𝐵 𝜑)

Theoremnfre1 2988 The setvar 𝑥 is not free in 𝑥𝐴𝜑. (Contributed by NM, 19-Mar-1997.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝑥𝐴 𝜑

Theoremnfrexd 2989 Deduction version of nfrex 2990. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)

Theoremnfrex 2990 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 2991 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 2992 Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.)
(𝑥𝐴 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓)

Theoremreximi2 2993 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 2994 Inference quantifying both antecedent and consequent. (Contributed by NM, 18-Oct-1996.)
(𝜑𝜓)       (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓)

Theoremreximdai 2995 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 2996 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 2997* 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 2998* 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 2999* 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 3000* 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))

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