P. 31 of Theorem List - Metamath Proof Explorer

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

Type	Label	Description
Statement
Theorem	nesym 3001	Characterization of inequality in terms of reversed equality (see bicom 221). (Contributed by BJ, 7-Jul-2018.)
⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐵 = 𝐴)
Theorem	nesymi 3002	Inference associated with nesym 3001. (Contributed by BJ, 7-Jul-2018.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
⊢ 𝐴 ≠ 𝐵    ⇒   ⊢ ¬ 𝐵 = 𝐴
Theorem	nesymir 3003	Inference associated with nesym 3001. (Contributed by BJ, 7-Jul-2018.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
⊢ ¬ 𝐴 = 𝐵    ⇒   ⊢ 𝐵 ≠ 𝐴
Theorem	neeq1d 3004	Deduction for inequality. (Contributed by NM, 25-Oct-1999.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶))
Theorem	neeq2d 3005	Deduction for inequality. (Contributed by NM, 25-Oct-1999.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵))
Theorem	neeq12d 3006	Deduction for inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐷))
Theorem	neeq1 3007	Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.) (Proof shortened by Wolf Lammen, 18-Nov-2019.)
⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶))
Theorem	neeq2 3008	Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.) (Proof shortened by Wolf Lammen, 18-Nov-2019.)
⊢ (𝐴 = 𝐵 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵))
Theorem	neeq1i 3009	Inference for inequality. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)
Theorem	neeq2i 3010	Inference for inequality. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵)
Theorem	neeq12i 3011	Inference for inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐷)
Theorem	eqnetrd 3012	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐶)
Theorem	eqnetrrd 3013	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐶)    ⇒   ⊢ (𝜑 → 𝐵 ≠ 𝐶)
Theorem	neeqtrd 3014	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐶)
Theorem	eqnetri 3015	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵 ≠ 𝐶    ⇒   ⊢ 𝐴 ≠ 𝐶
Theorem	eqnetrri 3016	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐴 ≠ 𝐶    ⇒   ⊢ 𝐵 ≠ 𝐶
Theorem	neeqtri 3017	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ 𝐴 ≠ 𝐵    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ 𝐴 ≠ 𝐶
Theorem	neeqtrri 3018	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ 𝐴 ≠ 𝐵    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ 𝐴 ≠ 𝐶
Theorem	neeqtrrd 3019	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐶)
Theorem	eqnetrrid 3020	A chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
⊢ 𝐵 = 𝐴    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐶)
Theorem	3netr3d 3021	Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐶 ≠ 𝐷)
Theorem	3netr4d 3022	Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 21-Nov-2019.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐴)    &   ⊢ (𝜑 → 𝐷 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 ≠ 𝐷)
Theorem	3netr3g 3023	Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ (𝜑 → 𝐶 ≠ 𝐷)
Theorem	3netr4g 3024	Substitution of equality into both sides of an inequality. (Contributed by NM, 14-Jun-2012.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ 𝐶 = 𝐴    &   ⊢ 𝐷 = 𝐵    ⇒   ⊢ (𝜑 → 𝐶 ≠ 𝐷)
Theorem	nebi 3025	Contraposition law for inequality. (Contributed by NM, 28-Dec-2008.)
⊢ ((𝐴 = 𝐵 ↔ 𝐶 = 𝐷) ↔ (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))
Theorem	pm13.18 3026	Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.) (Proof shortened by Wolf Lammen, 29-Oct-2024.)
⊢ ((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐶) → 𝐵 ≠ 𝐶)
Theorem	pm13.181 3027	Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Oct-2024.)
⊢ ((𝐴 = 𝐵 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶)
Theorem	pm13.181OLD 3028	Obsolete version of pm13.181 3027 as of 30-Oct-2024. (Contributed by Andrew Salmon, 3-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ((𝐴 = 𝐵 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶)
Theorem	pm2.61ine 3029	Inference eliminating an inequality in an antecedent. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (𝐴 = 𝐵 → 𝜑)    &   ⊢ (𝐴 ≠ 𝐵 → 𝜑)    ⇒   ⊢ 𝜑
Theorem	pm2.21ddne 3030	A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	pm2.61ne 3031	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 → (𝜓 ↔ 𝜒))    &   ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝜓)    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	pm2.61dne 3032	Deduction eliminating an inequality in an antecedent. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (𝜑 → (𝐴 = 𝐵 → 𝜓))    &   ⊢ (𝜑 → (𝐴 ≠ 𝐵 → 𝜓))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	pm2.61dane 3033	Deduction eliminating an inequality in an antecedent. (Contributed by NM, 30-Nov-2011.)
⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝜓)    &   ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	pm2.61da2ne 3034	Deduction eliminating two inequalities in an antecedent. (Contributed by NM, 29-May-2013.)
⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝜓)    &   ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝜓)    &   ⊢ ((𝜑 ∧ (𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷)) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	pm2.61da3ne 3035	Deduction eliminating three inequalities in an antecedent. (Contributed by NM, 15-Jun-2013.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝜓)    &   ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝜓)    &   ⊢ ((𝜑 ∧ 𝐸 = 𝐹) → 𝜓)    &   ⊢ ((𝜑 ∧ (𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷 ∧ 𝐸 ≠ 𝐹)) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	pm2.61iine 3036	Equality version of pm2.61ii 183. (Contributed by Scott Fenton, 13-Jun-2013.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) → 𝜑)    &   ⊢ (𝐴 = 𝐶 → 𝜑)    &   ⊢ (𝐵 = 𝐷 → 𝜑)    ⇒   ⊢ 𝜑
Theorem	neor 3037	Logical OR with an equality. (Contributed by NM, 29-Apr-2007.)
⊢ ((𝐴 = 𝐵 ∨ 𝜓) ↔ (𝐴 ≠ 𝐵 → 𝜓))
Theorem	neanior 3038	A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐶 = 𝐷))
Theorem	ne3anior 3039	A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.)
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷 ∧ 𝐸 ≠ 𝐹) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐶 = 𝐷 ∨ 𝐸 = 𝐹))
Theorem	neorian 3040	A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
⊢ ((𝐴 ≠ 𝐵 ∨ 𝐶 ≠ 𝐷) ↔ ¬ (𝐴 = 𝐵 ∧ 𝐶 = 𝐷))
Theorem	nemtbir 3041	An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)
⊢ 𝐴 ≠ 𝐵    &   ⊢ (𝜑 ↔ 𝐴 = 𝐵)    ⇒   ⊢ ¬ 𝜑
Theorem	nelne1 3042	Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.) (Proof shortened by Wolf Lammen, 14-May-2023.)
⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → 𝐵 ≠ 𝐶)
Theorem	nelne2 3043	Two classes are different if they don't belong to the same class. (Contributed by NM, 25-Jun-2012.) (Proof shortened by Wolf Lammen, 14-May-2023.)
⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → 𝐴 ≠ 𝐵)
Theorem	nelelne 3044	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.)
⊢ (¬ 𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐵 → 𝐶 ≠ 𝐴))
Theorem	neneor 3045	If two classes are different, a third class must be different of at least one of them. (Contributed by Thierry Arnoux, 8-Aug-2020.)
⊢ (𝐴 ≠ 𝐵 → (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐶))
Theorem	nfne 3046	Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐴 ≠ 𝐵
Theorem	nfned 3047	Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥 𝐴 ≠ 𝐵)
Theorem	nabbi 3048	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) ↔ {𝑥 ∣ 𝜑} ≠ {𝑥 ∣ 𝜓})
Theorem	mteqand 3049	A modus tollens deduction for inequality. (Contributed by Steven Nguyen, 1-Jun-2023.)
⊢ (𝜑 → 𝐶 ≠ 𝐷)    &   ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐵)
2.1.4.2  Negated membership
Syntax	wnel 3050	Extend wff notation to include negated membership.
wff 𝐴 ∉ 𝐵
Definition	df-nel 3051	Define negated membership. (Contributed by NM, 7-Aug-1994.)
⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
Theorem	neli 3052	Inference associated with df-nel 3051. (Contributed by BJ, 7-Jul-2018.)
⊢ 𝐴 ∉ 𝐵    ⇒   ⊢ ¬ 𝐴 ∈ 𝐵
Theorem	nelir 3053	Inference associated with df-nel 3051. (Contributed by BJ, 7-Jul-2018.)
⊢ ¬ 𝐴 ∈ 𝐵    ⇒   ⊢ 𝐴 ∉ 𝐵
Theorem	neleq12d 3054	Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷))
Theorem	neleq1 3055	Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
⊢ (𝐴 = 𝐵 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐶))
Theorem	neleq2 3056	Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
⊢ (𝐴 = 𝐵 → (𝐶 ∉ 𝐴 ↔ 𝐶 ∉ 𝐵))
Theorem	nfnel 3057	Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐴 ∉ 𝐵
Theorem	nfneld 3058	Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∉ 𝐵)
Theorem	nnel 3059	Negation of negated membership, analogous to nne 2948. (Contributed by Alexander van der Vekens, 18-Jan-2018.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
⊢ (¬ 𝐴 ∉ 𝐵 ↔ 𝐴 ∈ 𝐵)
Theorem	elnelne1 3060	Two classes are different if they don't contain the same element. (Contributed by AV, 28-Jan-2020.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∉ 𝐶) → 𝐵 ≠ 𝐶)
Theorem	elnelne2 3061	Two classes are different if they don't belong to the same class. (Contributed by AV, 28-Jan-2020.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∉ 𝐶) → 𝐴 ≠ 𝐵)
Theorem	nelcon3d 3062	Contrapositive law deduction for negated membership. (Contributed by AV, 28-Jan-2020.)
⊢ (𝜑 → (𝐴 ∈ 𝐵 → 𝐶 ∈ 𝐷))    ⇒   ⊢ (𝜑 → (𝐶 ∉ 𝐷 → 𝐴 ∉ 𝐵))
Theorem	elnelall 3063	A contradiction concerning membership implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
⊢ (𝐴 ∈ 𝐵 → (𝐴 ∉ 𝐵 → 𝜑))
Theorem	pm2.61danel 3064	Deduction eliminating an elementhood in an antecedent. (Contributed by AV, 5-Dec-2021.)
⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → 𝜓)    &   ⊢ ((𝜑 ∧ 𝐴 ∉ 𝐵) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
2.1.5  Restricted quantification
Syntax	wral 3065	Extend wff notation to include restricted universal quantification.
wff ∀𝑥 ∈ 𝐴 𝜑
Syntax	wrex 3066	Extend wff notation to include restricted existential quantification.
wff ∃𝑥 ∈ 𝐴 𝜑
Syntax	wreu 3067	Extend wff notation to include restricted existential uniqueness.
wff ∃!𝑥 ∈ 𝐴 𝜑
Syntax	wrmo 3068	Extend wff notation to include restricted "at most one".
wff ∃*𝑥 ∈ 𝐴 𝜑
Syntax	crab 3069	Extend class notation to include the restricted class abstraction (class builder).
class {𝑥 ∈ 𝐴 ∣ 𝜑}
Definition	df-ral 3070	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 Ⅎ𝑥𝐴 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 44608 or ralndv2 44609, 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.)
⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
Definition	df-rex 3071	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 Ⅎ𝑥𝐴 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 3070). (Contributed by NM, 30-Aug-1993.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
Definition	df-rmo 3072	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 Ⅎ𝑥𝐴 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 3070). (Contributed by NM, 16-Jun-2017.)
⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
Definition	df-reu 3073	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 Ⅎ𝑥𝐴 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 3070). (Contributed by NM, 22-Nov-1994.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
Definition	df-rab 3074	Define a restricted class abstraction (class builder): {𝑥 ∈ 𝐴 ∣ 𝜑} 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 Ⅎ𝑥𝐴, 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 3070. (Contributed by NM, 22-Nov-1994.)
⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
Theorem	rgen 3075	Generalization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
⊢ (𝑥 ∈ 𝐴 → 𝜑)    ⇒   ⊢ ∀𝑥 ∈ 𝐴 𝜑
Theorem	ralel 3076	All elements of a class are elements of the class. (Contributed by AV, 30-Oct-2020.)
⊢ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐴
Theorem	rgenw 3077	Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)
⊢ 𝜑    ⇒   ⊢ ∀𝑥 ∈ 𝐴 𝜑
Theorem	rgen2w 3078	Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by NM, 18-Jun-2014.)
⊢ 𝜑    ⇒   ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑
Theorem	mprg 3079	Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.)
⊢ (∀𝑥 ∈ 𝐴 𝜑 → 𝜓)    &   ⊢ (𝑥 ∈ 𝐴 → 𝜑)    ⇒   ⊢ 𝜓
Theorem	mprgbir 3080	Modus ponens on biconditional combined with restricted generalization. (Contributed by NM, 21-Mar-2004.)
⊢ (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)    &   ⊢ (𝑥 ∈ 𝐴 → 𝜓)    ⇒   ⊢ 𝜑
Theorem	alral 3081	Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)
⊢ (∀𝑥𝜑 → ∀𝑥 ∈ 𝐴 𝜑)
Theorem	raln 3082	Restricted universally quantified negation expressed as a universally quantified negation. (Contributed by BJ, 16-Jul-2021.)
⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑))
Theorem	ral2imi 3083	Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.) Allow shortening of ralim 3084. (Revised by Wolf Lammen, 1-Dec-2019.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))
Theorem	ralim 3084	Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997.) (Proof shortened by Wolf Lammen, 1-Dec-2019.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓))
Theorem	ralimi2 3085	Inference quantifying both antecedent and consequent. (Contributed by NM, 22-Feb-2004.)
⊢ ((𝑥 ∈ 𝐴 → 𝜑) → (𝑥 ∈ 𝐵 → 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐵 𝜓)
Theorem	ralimia 3086	Inference quantifying both antecedent and consequent. (Contributed by NM, 19-Jul-1996.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
Theorem	ralimiaa 3087	Inference quantifying both antecedent and consequent. (Contributed by NM, 4-Aug-2007.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
Theorem	ralimi 3088	Inference quantifying both antecedent and consequent, with strong hypothesis. (Contributed by NM, 4-Mar-1997.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
Theorem	2ralimi 3089	Inference quantifying both antecedent and consequent two times, with strong hypothesis. (Contributed by AV, 3-Dec-2021.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)
Theorem	ralbi 3090	Distribute a restricted universal quantifier over a biconditional. Restricted quantification version of albi 1821. (Contributed by NM, 6-Oct-2003.) Reduce axiom usage. (Revised by Wolf Lammen, 17-Jun-2023.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓))
Theorem	ralbii2 3091	Inference adding different restricted universal quantifiers to each side of an equivalence. (Contributed by NM, 15-Aug-2005.)
⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐵 → 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓)
Theorem	ralbiia 3092	Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 26-Nov-2000.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)
Theorem	ralbii 3093	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.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)
Theorem	2ralbii 3094	Inference adding two restricted universal quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)
Theorem	ralanid 3095	Cancellation law for restricted universal quantification. (Contributed by Peter Mazsa, 30-Dec-2018.) (Proof shortened by Wolf Lammen, 29-Jun-2023.)
⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑)
Theorem	r19.26 3096	Restricted quantifier version of 19.26 1874. (Contributed by NM, 28-Jan-1997.) (Proof shortened by Andrew Salmon, 30-May-2011.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))
Theorem	r19.26-2 3097	Restricted quantifier version of 19.26-2 1875. Version of r19.26 3096 with two quantifiers. (Contributed by NM, 10-Aug-2004.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓))
Theorem	r19.26-3 3098	Version of r19.26 3096 with three quantifiers. (Contributed by FL, 22-Nov-2010.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 𝜒))
Theorem	r19.26m 3099	Version of 19.26 1874 and r19.26 3096 with restricted quantifiers ranging over different classes. (Contributed by NM, 22-Feb-2004.)
⊢ (∀𝑥((𝑥 ∈ 𝐴 → 𝜑) ∧ (𝑥 ∈ 𝐵 → 𝜓)) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜓))
Theorem	ralbiim 3100	Split a biconditional and distribute quantifier. Restricted quantifier version of albiim 1893. (Contributed by NM, 3-Jun-2012.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)))