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	neeq2 3001	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 3002	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 3003	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 3004	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 3005	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐶)
Theorem	eqnetrrd 3006	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐶)    ⇒   ⊢ (𝜑 → 𝐵 ≠ 𝐶)
Theorem	neeqtrd 3007	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐶)
Theorem	eqnetri 3008	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵 ≠ 𝐶    ⇒   ⊢ 𝐴 ≠ 𝐶
Theorem	eqnetrri 3009	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐴 ≠ 𝐶    ⇒   ⊢ 𝐵 ≠ 𝐶
Theorem	neeqtri 3010	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ 𝐴 ≠ 𝐵    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ 𝐴 ≠ 𝐶
Theorem	neeqtrri 3011	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ 𝐴 ≠ 𝐵    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ 𝐴 ≠ 𝐶
Theorem	neeqtrrd 3012	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐶)
Theorem	eqnetrrid 3013	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 3014	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 3015	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 3016	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 3017	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 3018	Contraposition law for inequality. (Contributed by NM, 28-Dec-2008.)
⊢ ((𝐴 = 𝐵 ↔ 𝐶 = 𝐷) ↔ (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))
Theorem	pm13.18 3019	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 3020	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 3021	Obsolete version of pm13.181 3020 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 3022	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 3023	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 3024	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 3025	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 3026	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 3027	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 3028	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 3029	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	mteqand 3030	A modus tollens deduction for inequality. (Contributed by Steven Nguyen, 1-Jun-2023.)
⊢ (𝜑 → 𝐶 ≠ 𝐷)    &   ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐵)
Theorem	neor 3031	Logical OR with an equality. (Contributed by NM, 29-Apr-2007.)
⊢ ((𝐴 = 𝐵 ∨ 𝜓) ↔ (𝐴 ≠ 𝐵 → 𝜓))
Theorem	neanior 3032	A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐶 = 𝐷))
Theorem	ne3anior 3033	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 3034	A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
⊢ ((𝐴 ≠ 𝐵 ∨ 𝐶 ≠ 𝐷) ↔ ¬ (𝐴 = 𝐵 ∧ 𝐶 = 𝐷))
Theorem	nemtbir 3035	An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)
⊢ 𝐴 ≠ 𝐵    &   ⊢ (𝜑 ↔ 𝐴 = 𝐵)    ⇒   ⊢ ¬ 𝜑
Theorem	nelne1 3036	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 3037	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 3038	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 3039	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 3040	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 3041	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	nabbib 3042	Not equivalent wff's correspond to not equal class abstractions. (Contributed by AV, 7-Apr-2019.) (Proof shortened by Wolf Lammen, 25-Nov-2019.) Definitial form. (Revised by Wolf Lammen, 5-Mar-2025.)
⊢ ({𝑥 ∣ 𝜑} ≠ {𝑥 ∣ 𝜓} ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
2.1.4.2  Negated membership
Syntax	wnel 3043	Extend wff notation to include negated membership.
wff 𝐴 ∉ 𝐵
Definition	df-nel 3044	Define negated membership. (Contributed by NM, 7-Aug-1994.)
⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
Theorem	neli 3045	Inference associated with df-nel 3044. (Contributed by BJ, 7-Jul-2018.)
⊢ 𝐴 ∉ 𝐵    ⇒   ⊢ ¬ 𝐴 ∈ 𝐵
Theorem	nelir 3046	Inference associated with df-nel 3044. (Contributed by BJ, 7-Jul-2018.)
⊢ ¬ 𝐴 ∈ 𝐵    ⇒   ⊢ 𝐴 ∉ 𝐵
Theorem	nelcon3d 3047	Contrapositive law deduction for negated membership. (Contributed by AV, 28-Jan-2020.)
⊢ (𝜑 → (𝐴 ∈ 𝐵 → 𝐶 ∈ 𝐷))    ⇒   ⊢ (𝜑 → (𝐶 ∉ 𝐷 → 𝐴 ∉ 𝐵))
Theorem	neleq12d 3048	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 3049	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 3050	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 3051	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 3052	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 3053	Negation of negated membership, analogous to nne 2941. (Contributed by Alexander van der Vekens, 18-Jan-2018.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
⊢ (¬ 𝐴 ∉ 𝐵 ↔ 𝐴 ∈ 𝐵)
Theorem	elnelne1 3054	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 3055	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	pm2.24nel 3056	A contradiction concerning membership implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
⊢ (𝐴 ∈ 𝐵 → (𝐴 ∉ 𝐵 → 𝜑))
Theorem	pm2.61danel 3057	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
2.1.5.1  Restricted universal and existential quantification
Syntax	wral 3058	Extend wff notation to include restricted universal quantification.
wff ∀𝑥 ∈ 𝐴 𝜑
Definition	df-ral 3059	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 47054 or ralndv2 47055, 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))
Theorem	rgen 3060	Generalization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
⊢ (𝑥 ∈ 𝐴 → 𝜑)    ⇒   ⊢ ∀𝑥 ∈ 𝐴 𝜑
Theorem	ralel 3061	All elements of a class are elements of the class. (Contributed by AV, 30-Oct-2020.)
⊢ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐴
Theorem	rgenw 3062	Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)
⊢ 𝜑    ⇒   ⊢ ∀𝑥 ∈ 𝐴 𝜑
Theorem	rgen2w 3063	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 3064	Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.)
⊢ (∀𝑥 ∈ 𝐴 𝜑 → 𝜓)    &   ⊢ (𝑥 ∈ 𝐴 → 𝜑)    ⇒   ⊢ 𝜓
Theorem	mprgbir 3065	Modus ponens on biconditional combined with restricted generalization. (Contributed by NM, 21-Mar-2004.)
⊢ (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)    &   ⊢ (𝑥 ∈ 𝐴 → 𝜓)    ⇒   ⊢ 𝜑
Theorem	raln 3066	Restricted universally quantified negation expressed as a universally quantified negation. (Contributed by BJ, 16-Jul-2021.)
⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑))
Syntax	wrex 3067	Extend wff notation to include restricted existential quantification.
wff ∃𝑥 ∈ 𝐴 𝜑
Definition	df-rex 3068	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 3059). (Contributed by NM, 30-Aug-1993.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
Theorem	ralnex 3069	Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) (Proof shortened by BJ, 16-Jul-2021.)
⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑)
Theorem	dfrex2 3070	Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Wolf Lammen, 26-Nov-2019.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
Theorem	nrex 3071	Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
⊢ (𝑥 ∈ 𝐴 → ¬ 𝜓)    ⇒   ⊢ ¬ ∃𝑥 ∈ 𝐴 𝜓
Theorem	alral 3072	Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)
⊢ (∀𝑥𝜑 → ∀𝑥 ∈ 𝐴 𝜑)
Theorem	rexex 3073	Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥𝜑)
Theorem	rextru 3074	Two ways of expressing that a class has at least one element. (Contributed by Zhi Wang, 23-Sep-2024.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ⊤)
Theorem	ralimi2 3075	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	reximi2 3076	Inference quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 8-Nov-2004.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜓)
Theorem	ralimia 3077	Inference quantifying both antecedent and consequent. (Contributed by NM, 19-Jul-1996.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
Theorem	reximia 3078	Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Wolf Lammen, 31-Oct-2024.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
Theorem	ralimiaa 3079	Inference quantifying both antecedent and consequent. (Contributed by NM, 4-Aug-2007.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
Theorem	ralimi 3080	Inference quantifying both antecedent and consequent, with strong hypothesis. (Contributed by NM, 4-Mar-1997.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
Theorem	reximi 3081	Inference quantifying both antecedent and consequent. (Contributed by NM, 18-Oct-1996.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
Theorem	ral2imi 3082	Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.) Allow shortening of ralim 3083. (Revised by Wolf Lammen, 1-Dec-2019.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))
Theorem	ralim 3083	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	rexim 3084	Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.) (Proof shortened by Andrew Salmon, 30-May-2011.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
Theorem	reximiaOLD 3085	Obsolete version of reximia 3078 as of 31-Oct-2024. (Contributed by NM, 10-Feb-1997.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
Theorem	ralbii2 3086	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	rexbii2 3087	Inference adding different restricted existential quantifiers to each side of an equivalence. (Contributed by NM, 4-Feb-2004.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜓)
Theorem	ralbiia 3088	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	rexbiia 3089	Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 26-Oct-1999.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓)
Theorem	ralbii 3090	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	rexbii 3091	Inference adding restricted existential 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, 6-Dec-2019.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓)
Theorem	ralanid 3092	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	rexanid 3093	Cancellation law for restricted existential quantification. (Contributed by Peter Mazsa, 24-May-2018.) (Proof shortened by Wolf Lammen, 8-Jul-2023.)
⊢ (∃𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐴 𝜑)
Theorem	ralcom3 3094	A commutation law for restricted universal quantifiers that swaps the domains of the restriction. (Contributed by NM, 22-Feb-2004.) (Proof shortened by Wolf Lammen, 22-Dec-2024.)
⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 → 𝜑))
Theorem	ralcom3OLD 3095	Obsolete version of ralcom3 3094 as of 22-Dec-2024. (Contributed by NM, 22-Feb-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 → 𝜑))
Theorem	dfral2 3096	Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) Allow shortening of rexnal 3097. (Revised by Wolf Lammen, 9-Dec-2019.)
⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑)
Theorem	rexnal 3097	Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Wolf Lammen, 9-Dec-2019.)
⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 𝜑)
Theorem	ralinexa 3098	A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))
Theorem	rexanali 3099	A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.) (Proof shortened by Wolf Lammen, 27-Dec-2019.)
⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓))
Theorem	ralbi 3100	Distribute a restricted universal quantifier over a biconditional. Restricted quantification version of albi 1814. (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))