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Theorem List for Metamath Proof Explorer - 3001-3100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremneeqtrd 3001 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝐶)

Theoremeqnetri 3002 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴 = 𝐵    &   𝐵𝐶       𝐴𝐶

Theoremeqnetrri 3003 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴 = 𝐵    &   𝐴𝐶       𝐵𝐶

Theoremneeqtri 3004 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴𝐵    &   𝐵 = 𝐶       𝐴𝐶

Theoremneeqtrri 3005 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴𝐵    &   𝐶 = 𝐵       𝐴𝐶

Theoremneeqtrrd 3006 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝐶)

Theoremsyl5eqner 3007 A chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
𝐵 = 𝐴    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theorem3netr3d 3008 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
(𝜑𝐴𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐶𝐷)

Theorem3netr4d 3009 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 21-Nov-2019.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐶𝐷)

Theorem3netr3g 3010 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
(𝜑𝐴𝐵)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       (𝜑𝐶𝐷)

Theorem3netr4g 3011 Substitution of equality into both sides of an inequality. (Contributed by NM, 14-Jun-2012.)
(𝜑𝐴𝐵)    &   𝐶 = 𝐴    &   𝐷 = 𝐵       (𝜑𝐶𝐷)

Theoremnebi 3012 Contraposition law for inequality. (Contributed by NM, 28-Dec-2008.)
((𝐴 = 𝐵𝐶 = 𝐷) ↔ (𝐴𝐵𝐶𝐷))

Theorempm13.18 3013 Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
((𝐴 = 𝐵𝐴𝐶) → 𝐵𝐶)

Theorempm13.181 3014 Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
((𝐴 = 𝐵𝐵𝐶) → 𝐴𝐶)

Theorempm2.61ine 3015 Inference eliminating an inequality in an antecedent. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝐴 = 𝐵𝜑)    &   (𝐴𝐵𝜑)       𝜑

Theorempm2.21ddne 3016 A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴𝐵)       (𝜑𝜓)

Theorempm2.61ne 3017 Deduction eliminating an inequality in an antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
(𝐴 = 𝐵 → (𝜓𝜒))    &   ((𝜑𝐴𝐵) → 𝜓)    &   (𝜑𝜒)       (𝜑𝜓)

Theorempm2.61dne 3018 Deduction eliminating an inequality in an antecedent. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑 → (𝐴 = 𝐵𝜓))    &   (𝜑 → (𝐴𝐵𝜓))       (𝜑𝜓)

Theorempm2.61dane 3019 Deduction eliminating an inequality in an antecedent. (Contributed by NM, 30-Nov-2011.)
((𝜑𝐴 = 𝐵) → 𝜓)    &   ((𝜑𝐴𝐵) → 𝜓)       (𝜑𝜓)

Theorempm2.61da2ne 3020 Deduction eliminating two inequalities in an antecedent. (Contributed by NM, 29-May-2013.)
((𝜑𝐴 = 𝐵) → 𝜓)    &   ((𝜑𝐶 = 𝐷) → 𝜓)    &   ((𝜑 ∧ (𝐴𝐵𝐶𝐷)) → 𝜓)       (𝜑𝜓)

Theorempm2.61da3ne 3021 Deduction eliminating three inequalities in an antecedent. (Contributed by NM, 15-Jun-2013.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
((𝜑𝐴 = 𝐵) → 𝜓)    &   ((𝜑𝐶 = 𝐷) → 𝜓)    &   ((𝜑𝐸 = 𝐹) → 𝜓)    &   ((𝜑 ∧ (𝐴𝐵𝐶𝐷𝐸𝐹)) → 𝜓)       (𝜑𝜓)

Theorempm2.61iine 3022 Equality version of pm2.61ii 177. (Contributed by Scott Fenton, 13-Jun-2013.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
((𝐴𝐶𝐵𝐷) → 𝜑)    &   (𝐴 = 𝐶𝜑)    &   (𝐵 = 𝐷𝜑)       𝜑

Theoremneor 3023 Logical OR with an equality. (Contributed by NM, 29-Apr-2007.)
((𝐴 = 𝐵𝜓) ↔ (𝐴𝐵𝜓))

Theoremneanior 3024 A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
((𝐴𝐵𝐶𝐷) ↔ ¬ (𝐴 = 𝐵𝐶 = 𝐷))

Theoremne3anior 3025 A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.)
((𝐴𝐵𝐶𝐷𝐸𝐹) ↔ ¬ (𝐴 = 𝐵𝐶 = 𝐷𝐸 = 𝐹))

Theoremneorian 3026 A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
((𝐴𝐵𝐶𝐷) ↔ ¬ (𝐴 = 𝐵𝐶 = 𝐷))

Theoremnemtbir 3027 An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)
𝐴𝐵    &   (𝜑𝐴 = 𝐵)        ¬ 𝜑

Theoremnelne1 3028 Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.)
((𝐴𝐵 ∧ ¬ 𝐴𝐶) → 𝐵𝐶)

Theoremnelne2 3029 Two classes are different if they don't belong to the same class. (Contributed by NM, 25-Jun-2012.)
((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)

Theoremnelelne 3030 Two classes are different if they don't belong to the same class. (Contributed by Rodolfo Medina, 17-Oct-2010.) (Proof shortened by AV, 10-May-2020.)
𝐴𝐵 → (𝐶𝐵𝐶𝐴))

Theoremneneor 3031 If two classes are different, a third class must be different of at least one of them. (Contributed by Thierry Arnoux, 8-Aug-2020.)
(𝐴𝐵 → (𝐴𝐶𝐵𝐶))

Theoremnfne 3032 Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥 𝐴𝐵

Theoremnfned 3033 Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴𝐵)

Theoremnabbi 3034 Not equivalent wff's correspond to not equal class abstractions. (Contributed by AV, 7-Apr-2019.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
(∃𝑥(𝜑 ↔ ¬ 𝜓) ↔ {𝑥𝜑} ≠ {𝑥𝜓})

2.1.4.2  Negated membership

Syntaxwnel 3035 Extend wff notation to include negated membership.
wff 𝐴𝐵

Definitiondf-nel 3036 Define negated membership. (Contributed by NM, 7-Aug-1994.)
(𝐴𝐵 ↔ ¬ 𝐴𝐵)

Theoremneli 3037 Inference associated with df-nel 3036. (Contributed by BJ, 7-Jul-2018.)
𝐴𝐵        ¬ 𝐴𝐵

Theoremnelir 3038 Inference associated with df-nel 3036. (Contributed by BJ, 7-Jul-2018.)
¬ 𝐴𝐵       𝐴𝐵

Theoremneleq12d 3039 Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶𝐵𝐷))

Theoremneleq1 3040 Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))

Theoremneleq2 3041 Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))

Theoremnfnel 3042 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 3043 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 3044 Negation of negated membership, analogous to nne 2936. (Contributed by Alexander van der Vekens, 18-Jan-2018.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
𝐴𝐵𝐴𝐵)

Theoremelnelne1 3045 Two classes are different if they don't contain the same element. (Contributed by AV, 28-Jan-2020.)
((𝐴𝐵𝐴𝐶) → 𝐵𝐶)

Theoremelnelne2 3046 Two classes are different if they don't belong to the same class. (Contributed by AV, 28-Jan-2020.)
((𝐴𝐶𝐵𝐶) → 𝐴𝐵)

Theoremnelcon3d 3047 Contrapositive law deduction for negated membership. (Contributed by AV, 28-Jan-2020.)
(𝜑 → (𝐴𝐵𝐶𝐷))       (𝜑 → (𝐶𝐷𝐴𝐵))

Theoremelnelall 3048 A contradiction concerning membership implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
(𝐴𝐵 → (𝐴𝐵𝜑))

Theorempm2.61danel 3049 Deduction eliminating an elementhood in an antecedent. (Contributed by AV, 5-Dec-2021.)
((𝜑𝐴𝐵) → 𝜓)    &   ((𝜑𝐴𝐵) → 𝜓)       (𝜑𝜓)

2.1.5  Restricted quantification

Syntaxwral 3050 Extend wff notation to include restricted universal quantification.
wff 𝑥𝐴 𝜑

Syntaxwrex 3051 Extend wff notation to include restricted existential quantification.
wff 𝑥𝐴 𝜑

Syntaxwreu 3052 Extend wff notation to include restricted existential uniqueness.
wff ∃!𝑥𝐴 𝜑

Syntaxwrmo 3053 Extend wff notation to include restricted "at most one."
wff ∃*𝑥𝐴 𝜑

Syntaxcrab 3054 Extend class notation to include the restricted class abstraction (class builder).
class {𝑥𝐴𝜑}

Definitiondf-ral 3055 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 3056 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 3057 Define restricted existential uniqueness. (Contributed by NM, 22-Nov-1994.)
(∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))

Definitiondf-rmo 3058 Define restricted "at most one". (Contributed by NM, 16-Jun-2017.)
(∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))

Definitiondf-rab 3059 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 3060 Generalization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
(𝑥𝐴𝜑)       𝑥𝐴 𝜑

Theoremralel 3061 All elements of a class are elements of the class. (Contributed by AV, 30-Oct-2020.)
𝑥𝐴 𝑥𝐴

Theoremrgenw 3062 Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)
𝜑       𝑥𝐴 𝜑

Theoremrgen2w 3063 Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by NM, 18-Jun-2014.)
𝜑       𝑥𝐴𝑦𝐵 𝜑

Theoremmprg 3064 Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.)
(∀𝑥𝐴 𝜑𝜓)    &   (𝑥𝐴𝜑)       𝜓

Theoremmprgbir 3065 Modus ponens on biconditional combined with restricted generalization. (Contributed by NM, 21-Mar-2004.)
(𝜑 ↔ ∀𝑥𝐴 𝜓)    &   (𝑥𝐴𝜓)       𝜑

Theoremalral 3066 Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)
(∀𝑥𝜑 → ∀𝑥𝐴 𝜑)

Theoremrsp 3067 Restricted specialization. (Contributed by NM, 17-Oct-1996.)
(∀𝑥𝐴 𝜑 → (𝑥𝐴𝜑))

Theoremrspa 3068 Restricted specialization. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((∀𝑥𝐴 𝜑𝑥𝐴) → 𝜑)

Theoremrspec 3069 Specialization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
𝑥𝐴 𝜑       (𝑥𝐴𝜑)

Theoremr19.21bi 3070 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 3071 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 21-Nov-1994.)
(𝜑 → ∀𝑥𝐴 𝜓)       𝑥𝐴 (𝜑𝜓)

Theoremrspec2 3072 Specialization rule for restricted quantification, with two quantifiers. (Contributed by NM, 20-Nov-1994.)
𝑥𝐴𝑦𝐵 𝜑       ((𝑥𝐴𝑦𝐵) → 𝜑)

Theoremrspec3 3073 Specialization rule for restricted quantification, with three quantifiers. (Contributed by NM, 20-Nov-1994.)
𝑥𝐴𝑦𝐵𝑧𝐶 𝜑       ((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑)

Theoremrsp2 3074 Restricted specialization, with two quantifiers. (Contributed by NM, 11-Feb-1997.)
(∀𝑥𝐴𝑦𝐵 𝜑 → ((𝑥𝐴𝑦𝐵) → 𝜑))

Theoremr2allem 3075 Lemma factoring out common proof steps of r2alf 3076 and r2al 3077. 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 3076* Double restricted universal quantification. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2allem 3075. (Revised by Wolf Lammen, 9-Jan-2020.)
𝑦𝐴       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))

Theoremr2al 3077* 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 3078* Triple restricted universal quantification. (Contributed by NM, 19-Nov-1995.) (Proof shortened by Wolf Lammen, 30-Dec-2019.)
(∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑))

Theoremnfra1 3079 The setvar 𝑥 is not free in 𝑥𝐴𝜑. (Contributed by NM, 18-Oct-1996.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝑥𝐴 𝜑

Theoremhbra1 3080 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 3081 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 3082 Deduction version of nfral 3083. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)

Theoremnfral 3083 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 3084* Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD 39613. Contributed by Alan Sare 31-Dec-2011. (Contributed by NM, 31-Dec-2011.)
𝑦𝑥𝐴𝑦𝐵 𝜑

Theoremral2imi 3085 Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.) Allow shortening of ralim 3086. (Revised by Wolf Lammen, 1-Dec-2019.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝐴 𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))

Theoremralim 3086 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 3087 Inference quantifying both antecedent and consequent. (Contributed by NM, 22-Feb-2004.)
((𝑥𝐴𝜑) → (𝑥𝐵𝜓))       (∀𝑥𝐴 𝜑 → ∀𝑥𝐵 𝜓)

Theoremralimia 3088 Inference quantifying both antecedent and consequent. (Contributed by NM, 19-Jul-1996.)
(𝑥𝐴 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓)

Theoremralimiaa 3089 Inference quantifying both antecedent and consequent. (Contributed by NM, 4-Aug-2007.)
((𝑥𝐴𝜑) → 𝜓)       (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓)

Theoremralimi 3090 Inference quantifying both antecedent and consequent, with strong hypothesis. (Contributed by NM, 4-Mar-1997.)
(𝜑𝜓)       (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓)

Theorem2ralimi 3091 Inference quantifying both antecedent and consequent two times, with strong hypothesis. (Contributed by AV, 3-Dec-2021.)
(𝜑𝜓)       (∀𝑥𝐴𝑦𝐵 𝜑 → ∀𝑥𝐴𝑦𝐵 𝜓)

Theoremhbralrimi 3092 Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). This theorem contains the common proof steps for ralrimi 3095 and ralrimiv 3103. 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 3093 Restricted quantifier version of 19.21t 2220; closed form of r19.21 3094. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Wolf Lammen, 2-Jan-2020.)
(Ⅎ𝑥𝜑 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓)))

Theoremr19.21 3094 Restricted quantifier version of 19.21 2222. (Contributed by Scott Fenton, 30-Mar-2011.)
𝑥𝜑       (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓))

Theoremralrimi 3095 Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 10-Oct-1999.) Shortened after introduction of hbralrimi 3092. (Revised by Wolf Lammen, 4-Dec-2019.)
𝑥𝜑    &   (𝜑 → (𝑥𝐴𝜓))       (𝜑 → ∀𝑥𝐴 𝜓)

Theoremralimdaa 3096 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 3097 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 3098* Restricted quantifier version of 19.21v 2017. (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 3099* Inference quantifying both antecedent and consequent. (Contributed by NM, 1-Feb-2005.)
(𝜑 → ((𝑥𝐴𝜓) → (𝑥𝐵𝜒)))       (𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐵 𝜒))

Theoremralimdva 3100* 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))

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