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Type | Label | Description |
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Statement | ||
Theorem | cbvralfw 3301* | Rule used to change bound variables, using implicit substitution. Version of cbvralf 3356 with a disjoint variable condition, which does not require ax-10 2137, ax-13 2371. For a version not dependent on ax-11 2154 and ax-12, see cbvralvw 3234. (Contributed by NM, 7-Mar-2004.) Avoid ax-10 2137, ax-13 2371. (Revised by Gino Giotto, 23-May-2024.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvrexfw 3302* | Rule used to change bound variables, using implicit substitution. Version of cbvrexf 3357 with a disjoint variable condition, which does not require ax-13 2371. For a version not dependent on ax-11 2154 and ax-12, see cbvrexvw 3235. (Contributed by FL, 27-Apr-2008.) Avoid ax-10 2137, ax-13 2371. (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvralw 3303* | Rule used to change bound variables, using implicit substitution. Version of cbvralfw 3301 with more disjoint variable conditions. (Contributed by NM, 31-Jul-2003.) Avoid ax-13 2371. (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvrexw 3304* | Rule used to change bound variables, using implicit substitution. Version of cbvrexfw 3302 with more disjoint variable conditions. (Contributed by NM, 31-Jul-2003.) Avoid ax-13 2371. (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓) | ||
Theorem | hbral 3305 | Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by David Abernethy, 13-Dec-2009.) |
⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∀𝑦 ∈ 𝐴 𝜑 → ∀𝑥∀𝑦 ∈ 𝐴 𝜑) | ||
Theorem | nfraldw 3306* | Deduction version of nfralw 3308. Version of nfrald 3368 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 15-Feb-2013.) Avoid ax-9 2116, ax-ext 2703. (Revised by Gino Giotto, 24-Sep-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓) | ||
Theorem | nfrexdw 3307* | Deduction version of nfrexw 3310. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2371. See nfrexd 3369 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓) | ||
Theorem | nfralw 3308* | Bound-variable hypothesis builder for restricted quantification. Version of nfral 3370 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 1-Sep-1999.) Avoid ax-13 2371. (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 13-Dec-2024.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑 | ||
Theorem | nfralwOLD 3309* | Obsolete version of nfralw 3308 as of 13-Dec-2024. (Contributed by NM, 1-Sep-1999.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑 | ||
Theorem | nfrexw 3310* | 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.) Add disjoint variable condition to avoid ax-13 2371. See nfrex 3371 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑 | ||
Theorem | r19.12 3311* | Restricted quantifier version of 19.12 2320. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) Avoid ax-13 2371, ax-ext 2703. (Revised by Wolf Lammen, 17-Jun-2023.) (Proof shortened by Wolf Lammen, 4-Nov-2024.) |
⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) | ||
Theorem | r19.12OLD 3312* | Obsolete version of 19.12 2320 as of 4-Nov-2024. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) Avoid ax-13 2371, ax-ext 2703. (Revised by Wolf Lammen, 17-Jun-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) | ||
Theorem | reean 3313* | Rearrange restricted existential quantifiers. For a version based on fewer axioms see reeanv 3226. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Andrew Salmon, 30-May-2011.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓)) | ||
Theorem | cbvralsvw 3314* | Change bound variable by using a substitution. Version of cbvralsv 3362 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 20-Nov-2005.) Avoid ax-13 2371. (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 8-Mar-2025.) |
⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) | ||
Theorem | cbvrexsvw 3315* | Change bound variable by using a substitution. Version of cbvrexsv 3363 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 2-Mar-2008.) Avoid ax-13 2371. (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 8-Mar-2025.) |
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) | ||
Theorem | cbvralsvwOLD 3316* | Obsolete version of cbvralsvw 3314 as of 8-Mar-2025. (Contributed by NM, 20-Nov-2005.) Avoid ax-13 2371. (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) | ||
Theorem | cbvrexsvwOLD 3317* | Obsolete version of cbvrexsvw 3315 as of 8-Mar-2025. (Contributed by NM, 20-Nov-2005.) Avoid ax-13 2371. (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) | ||
Theorem | nfraldwOLD 3318* | Obsolete version of nfraldw 3306 as of 24-Sep-2024. (Contributed by NM, 15-Feb-2013.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓) | ||
Theorem | nfra2wOLDOLD 3319* | Obsolete version of nfra2w 3296 as of 24-Sep-2024. (Contributed by Alan Sare, 31-Dec-2011.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 | ||
Theorem | cbvralfwOLD 3320* | Obsolete version of cbvralfw 3301 as of 23-May-2024. (Contributed by NM, 7-Mar-2004.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓) | ||
Theorem | rexeq 3321* | Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.) Remove usage of ax-10 2137, ax-11 2154, and ax-12 2171. (Revised by Steven Nguyen, 30-Apr-2023.) Shorten other proofs. (Revised by Wolf Lammen, 8-Mar-2025.) |
⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | raleq 3322* | Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.) Remove usage of ax-10 2137, ax-11 2154, and ax-12 2171. (Revised by Steven Nguyen, 30-Apr-2023.) Shorten other proofs. (Revised by Wolf Lammen, 8-Mar-2025.) |
⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | raleqi 3323* | Equality inference for restricted universal quantifier. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑) | ||
Theorem | rexeqi 3324* | Equality inference for restricted existential quantifier. (Contributed by Mario Carneiro, 23-Apr-2015.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑) | ||
Theorem | raleqdv 3325* | Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓)) | ||
Theorem | rexeqdv 3326* | Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓)) | ||
Theorem | raleqbidva 3327* | Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)) | ||
Theorem | rexeqbidva 3328* | Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) | ||
Theorem | raleqbidvv 3329* | Version of raleqbidv 3342 with additional disjoint variable conditions, not requiring ax-8 2108 nor df-clel 2810. (Contributed by BJ, 22-Sep-2024.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)) | ||
Theorem | raleqbidvvOLD 3330* | Version of raleqbidv 3342 with additional disjoint variable conditions, not requiring ax-8 2108 nor df-clel 2810. (Contributed by BJ, 22-Sep-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)) | ||
Theorem | rexeqbidvv 3331* | Version of rexeqbidv 3343 with additional disjoint variable conditions, not requiring ax-8 2108 nor df-clel 2810. (Contributed by Wolf Lammen, 25-Sep-2024.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) | ||
Theorem | rexeqbidvvOLD 3332* | Version of rexeqbidv 3343 with additional disjoint variable conditions, not requiring ax-8 2108 nor df-clel 2810. (Contributed by Wolf Lammen, 25-Sep-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) | ||
Theorem | raleqbi1dv 3333* | Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.) (Proof shortened by Steven Nguyen, 5-May-2023.) |
⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓)) | ||
Theorem | rexeqbi1dv 3334* | Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.) (Proof shortened by Steven Nguyen, 5-May-2023.) |
⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜓)) | ||
Theorem | raleqOLD 3335* | Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.) Remove usage of ax-10 2137, ax-11 2154, and ax-12 2171. (Revised by Steven Nguyen, 30-Apr-2023.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | rexeqOLD 3336* | Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.) Remove usage of ax-10 2137, ax-11 2154, and ax-12 2171. (Revised by Steven Nguyen, 30-Apr-2023.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | raleleq 3337* | All elements of a class are elements of a class equal to this class. (Contributed by AV, 30-Oct-2020.) Avoid ax-8 2108. (Revised by Wolf Lammen, 9-Mar-2025.) |
⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) | ||
Theorem | raleqbii 3338 | Equality deduction for restricted universal quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝜓 ↔ 𝜒) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒) | ||
Theorem | rexeqbii 3339 | Equality deduction for restricted existential quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝜓 ↔ 𝜒) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒) | ||
Theorem | raleleqOLD 3340* | Obsolete version of raleleq 3337 as of 9-Mar-2025. (Contributed by AV, 30-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) | ||
Theorem | raleleqALT 3341* | Alternate proof of raleleq 3337 using ralel 3064, being longer and using more axioms. (Contributed by AV, 30-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) | ||
Theorem | raleqbidv 3342* | Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.) Remove usage of ax-10 2137, ax-11 2154, and ax-12 2171 and reduce distinct variable conditions. (Revised by Steven Nguyen, 30-Apr-2023.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)) | ||
Theorem | rexeqbidv 3343* | Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.) Remove usage of ax-10 2137, ax-11 2154, and ax-12 2171 and reduce distinct variable conditions. (Revised by Steven Nguyen, 30-Apr-2023.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) | ||
Theorem | cbvraldva2 3344* | Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) & ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒)) | ||
Theorem | cbvrexdva2 3345* | Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.) (Proof shortened by Wolf Lammen, 8-Jan-2025.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) & ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒)) | ||
Theorem | cbvrexdva2OLD 3346* | Obsolete version of cbvrexdva2 3345 as of 8-Jan-2025. (Contributed by David Moews, 1-May-2017.) (Proof shortened by Wolf Lammen, 12-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) & ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒)) | ||
Theorem | cbvraldvaOLD 3347* | Obsolete version of cbvraldva 3236 as of 9-Mar-2025. (Contributed by David Moews, 1-May-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐴 𝜒)) | ||
Theorem | cbvrexdvaOLD 3348* | Obsolete version of cbvrexdva 3237 as of 9-Mar-2025. (Contributed by David Moews, 1-May-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐴 𝜒)) | ||
Theorem | raleqf 3349 | Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. See raleq 3322 for a version based on fewer axioms. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | rexeqf 3350 | Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. See rexeq 3321 for a version based on fewer axioms. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 9-Mar-2025.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | rexeqfOLD 3351 | Obsolete version of rexeqf 3350 as of 9-Mar-2025. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | raleqbid 3352 | Equality deduction for restricted universal quantifier. See raleqbidv 3342 for a version based on fewer axioms. (Contributed by Thierry Arnoux, 8-Mar-2017.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)) | ||
Theorem | rexeqbid 3353 | Equality deduction for restricted existential quantifier. See rexeqbidv 3343 for a version based on fewer axioms. (Contributed by Thierry Arnoux, 8-Mar-2017.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) | ||
Theorem | sbralie 3354* | Implicit to explicit substitution that swaps variables in a rectrictedly universally quantified expression. (Contributed by NM, 5-Sep-2004.) Avoid ax-ext 2703. (Revised by Wolf Lammen, 10-Mar-2025.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓) | ||
Theorem | sbralieALT 3355* | Alternative shorter proof of sbralie 3354 dependent on ax-ext 2703. (Contributed by NM, 5-Sep-2004.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓) | ||
Theorem | cbvralf 3356 | Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker cbvralfw 3301 when possible. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvrexf 3357 | Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker cbvrexfw 3302 when possible. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvral 3358* | Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker cbvralw 3303 when possible. (Contributed by NM, 31-Jul-2003.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvrex 3359* | Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker cbvrexw 3304 when possible. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvralv 3360* | Change the bound variable of a restricted universal quantifier using implicit substitution. See cbvralvw 3234 based on fewer axioms , but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker cbvralvw 3234 when possible. (Contributed by NM, 28-Jan-1997.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvrexv 3361* | Change the bound variable of a restricted existential quantifier using implicit substitution. See cbvrexvw 3235 based on fewer axioms , but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker cbvrexvw 3235 when possible. (Contributed by NM, 2-Jun-1998.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvralsv 3362* | Change bound variable by using a substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker cbvralsvw 3314 when possible. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.) (New usage is discouraged.) |
⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) | ||
Theorem | cbvrexsv 3363* | Change bound variable by using a substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker cbvrexsvw 3315 when possible. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.) (New usage is discouraged.) |
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) | ||
Theorem | cbvral2v 3364* | Change bound variables of double restricted universal quantification, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker cbvral2vw 3238 when possible. (Contributed by NM, 10-Aug-2004.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓) | ||
Theorem | cbvrex2v 3365* | Change bound variables of double restricted universal quantification, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker cbvrex2vw 3239 when possible. (Contributed by FL, 2-Jul-2012.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓) | ||
Theorem | cbvral3v 3366* | Change bound variables of triple restricted universal quantification, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker cbvral3vw 3240 when possible. (Contributed by NM, 10-May-2005.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 = 𝑣 → (𝜒 ↔ 𝜃)) & ⊢ (𝑧 = 𝑢 → (𝜃 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓) | ||
Theorem | rgen2a 3367* | Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 are not required to be disjoint. This proof illustrates the use of dvelim 2450. This theorem relies on the full set of axioms up to ax-ext 2703 and it should no longer be used. Usage of rgen2 3197 is highly encouraged. (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.) (New usage is discouraged.) |
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝜑) ⇒ ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 | ||
Theorem | nfrald 3368 | Deduction version of nfral 3370. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker nfraldw 3306 when possible. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓) | ||
Theorem | nfrexd 3369 | Deduction version of nfrex 3371. Usage of this theorem is discouraged because it depends on ax-13 2371. See nfrexdw 3307 for a version with a disjoint variable condition, but not requiring ax-13 2371. (Contributed by Mario Carneiro, 14-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓) | ||
Theorem | nfral 3370 | Bound-variable hypothesis builder for restricted quantification. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker nfralw 3308 when possible. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑 | ||
Theorem | nfrex 3371 | Bound-variable hypothesis builder for restricted quantification. Usage of this theorem is discouraged because it depends on ax-13 2371. See nfrexw 3310 for a version with a disjoint variable condition, but not requiring ax-13 2371. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2019.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑 | ||
Theorem | nfra2 3372* | Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD 43611. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker nfra2w 3296 when possible. (Contributed by Alan Sare, 31-Dec-2011.) (New usage is discouraged.) |
⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 | ||
Theorem | ralcom2 3373* | Commutation of restricted universal quantifiers. Note that 𝑥 and 𝑦 need not be disjoint (this makes the proof longer). This theorem relies on the full set of axioms up to ax-ext 2703 and it should no longer be used. Usage of ralcom 3286 is highly encouraged. (Contributed by NM, 24-Nov-1994.) (Proof shortened by Mario Carneiro, 17-Oct-2016.) (New usage is discouraged.) |
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 → ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑) | ||
Syntax | wreu 3374 | Extend wff notation to include restricted existential uniqueness. |
wff ∃!𝑥 ∈ 𝐴 𝜑 | ||
Syntax | wrmo 3375 | Extend wff notation to include restricted "at most one". |
wff ∃*𝑥 ∈ 𝐴 𝜑 | ||
Definition | df-rmo 3376 |
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 3062). (Contributed by NM, 16-Jun-2017.) |
⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | ||
Definition | df-reu 3377 |
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 3062). (Contributed by NM, 22-Nov-1994.) |
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | ||
Theorem | reu5 3378 | Restricted uniqueness in terms of "at most one". (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.) |
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | reurmo 3379 | Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.) |
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑) | ||
Theorem | reurex 3380 | Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.) |
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑) | ||
Theorem | mormo 3381 | Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.) |
⊢ (∃*𝑥𝜑 → ∃*𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rmobiia 3382 | Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 16-Jun-2017.) |
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐴 𝜓) | ||
Theorem | reubiia 3383 | Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 14-Nov-2004.) |
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐴 𝜓) | ||
Theorem | rmobii 3384 | Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 16-Jun-2017.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐴 𝜓) | ||
Theorem | reubii 3385 | Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 22-Oct-1999.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐴 𝜓) | ||
Theorem | rmoanid 3386 | Cancellation law for restricted at-most-one quantification. (Contributed by Peter Mazsa, 24-May-2018.) (Proof shortened by Wolf Lammen, 12-Jan-2025.) |
⊢ (∃*𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃*𝑥 ∈ 𝐴 𝜑) | ||
Theorem | reuanid 3387 | Cancellation law for restricted unique existential quantification. (Contributed by Peter Mazsa, 12-Feb-2018.) (Proof shortened by Wolf Lammen, 12-Jan-2025.) |
⊢ (∃!𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rmoanidOLD 3388 | Obsolete version of rmoanid 3386 as of 12-Jan-2025. (Contributed by Peter Mazsa, 24-May-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∃*𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃*𝑥 ∈ 𝐴 𝜑) | ||
Theorem | reuanidOLD 3389 | Obsolete version of reuanid 3387 as of 12-Jan-2025. (Contributed by Peter Mazsa, 12-Feb-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∃!𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑥 ∈ 𝐴 𝜑) | ||
Theorem | 2reu2rex 3390 | Double restricted existential uniqueness, analogous to 2eu2ex 2639. (Contributed by Alexander van der Vekens, 25-Jun-2017.) |
⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) | ||
Theorem | rmobidva 3391* | Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.) Avoid ax-6 1971, ax-7 2011, ax-12 2171. (Revised by Wolf Lammen, 23-Nov-2024.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | reubidva 3392* | Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 13-Nov-2004.) Reduce axiom usage. (Revised by Wolf Lammen, 14-Jan-2023.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | rmobidv 3393* | Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | reubidv 3394* | Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 17-Oct-1996.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | reueubd 3395* | Restricted existential uniqueness is equivalent to existential uniqueness if the unique element is in the restricting class. (Contributed by AV, 4-Jan-2021.) |
⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝑉) ⇒ ⊢ (𝜑 → (∃!𝑥 ∈ 𝑉 𝜓 ↔ ∃!𝑥𝜓)) | ||
Theorem | rmo5 3396 | Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.) |
⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | nrexrmo 3397 | Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.) |
⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑) | ||
Theorem | moel 3398* | "At most one" element in a set. (Contributed by Thierry Arnoux, 26-Jul-2018.) Avoid ax-11 2154. (Revised by Wolf Lammen, 23-Nov-2024.) |
⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦) | ||
Theorem | cbvrmovw 3399* | Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmov 3426 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 16-Jun-2017.) (Revised by Gino Giotto, 30-Sep-2024.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvreuvw 3400* | Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreuv 3427 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 5-Apr-2004.) (Revised by Gino Giotto, 30-Sep-2024.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) |
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