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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rgen3 3201* | Generalization rule for restricted quantification, with three quantifiers. (Contributed by NM, 12-Jan-2008.) |
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜑) ⇒ ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 | ||
Theorem | rsp 3202 | Restricted specialization. (Contributed by NM, 17-Oct-1996.) |
⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑)) | ||
Theorem | rspa 3203 | Restricted specialization. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜑) | ||
Theorem | rspec 3204 | Specialization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.) |
⊢ ∀𝑥 ∈ 𝐴 𝜑 ⇒ ⊢ (𝑥 ∈ 𝐴 → 𝜑) | ||
Theorem | r19.21bi 3205 | Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.) (Proof shortened by Wolf Lammen, 11-Jun-2023.) |
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓) | ||
Theorem | r19.21biOLD 3206 | Obsolete version of r19.21bi 3205 as of 11-Jun-2023. (Contributed by NM, 20-Nov-1994.) (Proof shortened by Wolf Lammen, 1-Jan-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓) | ||
Theorem | r19.21be 3207 | Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 21-Nov-1994.) |
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) ⇒ ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) | ||
Theorem | rspec2 3208 | Specialization rule for restricted quantification, with two quantifiers. (Contributed by NM, 20-Nov-1994.) |
⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ⇒ ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑) | ||
Theorem | rspec3 3209 | Specialization rule for restricted quantification, with three quantifiers. (Contributed by NM, 20-Nov-1994.) |
⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ⇒ ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜑) | ||
Theorem | rsp2 3210 | Restricted specialization, with two quantifiers. (Contributed by NM, 11-Feb-1997.) |
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) | ||
Theorem | r19.21t 3211 | Restricted quantifier version of 19.21t 2196; closed form of r19.21 3212. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Wolf Lammen, 2-Jan-2020.) |
⊢ (Ⅎ𝑥𝜑 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))) | ||
Theorem | r19.21 3212 | Restricted quantifier version of 19.21 2197. (Contributed by Scott Fenton, 30-Mar-2011.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) | ||
Theorem | ralrimi 3213 | Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 10-Oct-1999.) Shortened after introduction of hbralrimi 3177. (Revised by Wolf Lammen, 4-Dec-2019.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) | ||
Theorem | ralimdaa 3214 | 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.) |
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | ralrimd 3215 | Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 16-Feb-2004.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝜑 → (𝜓 → (𝑥 ∈ 𝐴 → 𝜒))) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | nfra1 3216 | The setvar 𝑥 is not free in ∀𝑥 ∈ 𝐴𝜑. (Contributed by NM, 18-Oct-1996.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑 | ||
Theorem | hbra1 3217 | The setvar 𝑥 is not free in ∀𝑥 ∈ 𝐴𝜑. (Contributed by NM, 18-Oct-1996.) (Proof shortened by Wolf Lammen, 7-Dec-2019.) |
⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥∀𝑥 ∈ 𝐴 𝜑) | ||
Theorem | hbral 3218 | 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 | r2alf 3219* | Double restricted universal quantification. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2allem 3197. (Revised by Wolf Lammen, 9-Jan-2020.) |
⊢ Ⅎ𝑦𝐴 ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) | ||
Theorem | nfraldw 3220* | Version of nfrald 3221 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓) | ||
Theorem | nfrald 3221 | Deduction version of nfral 3223. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓) | ||
Theorem | nfralw 3222* | Version of nfral 3223 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑 | ||
Theorem | nfral 3223 | Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑 | ||
Theorem | nfra2w 3224* | Version of nfra2 3225 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 | ||
Theorem | nfra2 3225* | Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD 41071. Contributed by Alan Sare 31-Dec-2011. (Contributed by NM, 31-Dec-2011.) |
⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 | ||
Theorem | rgen2a 3226* | Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 are not required to be disjoint. This proof illustrates the use of dvelim 2465. (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.) |
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝜑) ⇒ ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 | ||
Theorem | ralbida 3227 | Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Oct-2003.) |
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | ralbid 3228 | Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 27-Jun-1998.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | 2ralbida 3229* | Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 24-Feb-2004.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)) | ||
Theorem | ralbiOLD 3230 | Obsolete version of ralbi 3164 as of 17-Jun-2023. (Contributed by NM, 6-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)) | ||
Theorem | raleqbii 3231 | 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 | ralcom4 3232* | Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) Reduce axiom dependencies. (Revised by BJ, 13-Jun-2019.) |
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑) | ||
Theorem | ralnex 3233 | 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 | dfral2 3234 | Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) Allow shortening of rexnal 3235. (Revised by Wolf Lammen, 9-Dec-2019.) |
⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑) | ||
Theorem | rexnal 3235 | 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 | dfrex2 3236 | 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 | rexex 3237 | Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.) |
⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥𝜑) | ||
Theorem | rexim 3238 | 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 | reximia 3239 | Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.) |
⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) | ||
Theorem | reximi 3240 | Inference quantifying both antecedent and consequent. (Contributed by NM, 18-Oct-1996.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) | ||
Theorem | reximi2 3241 | 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 | rexbii2 3242 | 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 | rexbiia 3243 | 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 | rexbii 3244 | 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 | 2rexbii 3245 | Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 11-Nov-1995.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) | ||
Theorem | rexcom4 3246* | Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) Reduce axiom dependencies. (Revised by BJ, 13-Jun-2019.) |
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑) | ||
Theorem | 2ex2rexrot 3247* | Rotate two existential quantifiers and two restricted existential quantifiers. (Contributed by AV, 9-Nov-2023.) |
⊢ (∃𝑥∃𝑦∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∃𝑥∃𝑦𝜑) | ||
Theorem | rexcom4a 3248* | Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.) |
⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥𝜓)) | ||
Theorem | rexanid 3249 | 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 | rexanidOLD 3250 | Obsolete version of rexanid 3249 as of 8-Jul-2023. (Contributed by Peter Mazsa, 24-May-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (∃𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐴 𝜑) | ||
Theorem | r19.29 3251 | Restricted quantifier version of 19.29 1865. See also r19.29r 3252. (Contributed by NM, 31-Aug-1999.) (Proof shortened by Andrew Salmon, 30-May-2011.) |
⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) | ||
Theorem | r19.29r 3252 | Restricted quantifier version of 19.29r 1866; variation of r19.29 3251. (Contributed by NM, 31-Aug-1999.) (Proof shortened by Wolf Lammen, 29-Jun-2023.) |
⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) | ||
Theorem | r19.29rOLD 3253 | Obsolete version of r19.29r 3252 as of 29-Jun-2023. (Contributed by NM, 31-Aug-1999.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) | ||
Theorem | r19.29imd 3254 | Theorem 19.29 of [Margaris] p. 90 with an implication in the hypothesis containing the generalization, deduction version. (Contributed by AV, 19-Jan-2019.) |
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒)) | ||
Theorem | rexnal2 3255 | Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑) | ||
Theorem | rexnal3 3256 | Relationship between three restricted universal and existential quantifiers. (Contributed by Thierry Arnoux, 12-Jul-2020.) |
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑) | ||
Theorem | ralnex2 3257 | Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 18-May-2023.) |
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) | ||
Theorem | ralnex2OLD 3258 | Obsolete version of ralnex2 3257 as of 18-May-2023. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) | ||
Theorem | ralnex3 3259 | Relationship between three restricted universal and existential quantifiers. (Contributed by Thierry Arnoux, 12-Jul-2020.) (Proof shortened by Wolf Lammen, 18-May-2023.) |
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜑) | ||
Theorem | ralnex3OLD 3260 | Obsolete version of ralnex3 3259 as of 18-May-2023. (Contributed by Thierry Arnoux, 12-Jul-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜑) | ||
Theorem | ralinexa 3261 | A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.) |
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) | ||
Theorem | rexanali 3262 | 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 | nrexralim 3263 | Negation of a complex predicate calculus formula. (Contributed by FL, 31-Jul-2009.) |
⊢ (¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ ¬ 𝜓)) | ||
Theorem | risset 3264* | Two ways to say "𝐴 belongs to 𝐵". (Contributed by NM, 22-Nov-1994.) |
⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴) | ||
Theorem | nelb 3265* | A definition of ¬ 𝐴 ∈ 𝐵. (Contributed by Thierry Arnoux, 20-Nov-2023.) (Proof shortened by SN, 23-Jan-2024.) |
⊢ (¬ 𝐴 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴) | ||
Theorem | nrex 3266 | Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.) |
⊢ (𝑥 ∈ 𝐴 → ¬ 𝜓) ⇒ ⊢ ¬ ∃𝑥 ∈ 𝐴 𝜓 | ||
Theorem | nrexdv 3267* | Deduction adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Wolf Lammen, 5-Jan-2020.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ 𝜓) ⇒ ⊢ (𝜑 → ¬ ∃𝑥 ∈ 𝐴 𝜓) | ||
Theorem | reximdv2 3268* | Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 17-Sep-2003.) |
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) → (𝑥 ∈ 𝐵 ∧ 𝜒))) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐵 𝜒)) | ||
Theorem | reximdvai 3269* | 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 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | reximdv 3270* | Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version with strong hypothesis.) (Contributed by NM, 24-Jun-1998.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | reximdva 3271* | Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | reximddv 3272* | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 7-Dec-2016.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝜒) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒) | ||
Theorem | reximssdv 3273* | Derivation of a restricted existential quantification over a subset (the second hypothesis implies 𝐴 ⊆ 𝐵), deduction form. (Contributed by AV, 21-Aug-2022.) |
⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝜓)) → 𝑥 ∈ 𝐴) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝜓)) → 𝜒) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒) | ||
Theorem | reximdvva 3274* | Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by AV, 5-Jan-2022.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒)) | ||
Theorem | reximddv2 3275* | Double deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → 𝜒) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒) | ||
Theorem | r19.23v 3276* | Restricted quantifier version of 19.23v 1934. Version of r19.23 3311 with a disjoint variable condition. (Contributed by NM, 31-Aug-1999.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2020.) |
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)) | ||
Theorem | rexlimiv 3277* | Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2020.) |
⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓) | ||
Theorem | rexlimiva 3278* | Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Dec-2006.) |
⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓) | ||
Theorem | rexlimivw 3279* | Weaker version of rexlimiv 3277. (Contributed by FL, 19-Sep-2011.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓) | ||
Theorem | rexlimdv 3280* | Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 14-Nov-2002.) (Proof shortened by Eric Schmidt, 22-Dec-2006.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2020.) |
⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) | ||
Theorem | rexlimdva 3281* | Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 20-Jan-2007.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) | ||
Theorem | rexlimdvaa 3282* | Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Mario Carneiro, 15-Jun-2016.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝜒) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) | ||
Theorem | rexlimdv3a 3283* | Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 3280. (Contributed by NM, 7-Jun-2015.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓) → 𝜒) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) | ||
Theorem | rexlimdva2 3284* | Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) | ||
Theorem | r19.29an 3285* | A commonly used pattern in the spirit of r19.29 3251. (Contributed by Thierry Arnoux, 29-Dec-2019.) (Proof shortened by Wolf Lammen, 17-Jun-2023.) |
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) ⇒ ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) → 𝜒) | ||
Theorem | r19.29a 3286* | A commonly used pattern in the spirit of r19.29 3251. (Contributed by Thierry Arnoux, 22-Nov-2017.) Reduce axiom usage. (Revised by Wolf Lammen, 17-Jun-2023.) |
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | rexlimdvw 3287* | Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Jun-2014.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) | ||
Theorem | rexlimddv 3288* | Restricted existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.) |
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝜒) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | rexlimivv 3289* | Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 17-Feb-2004.) |
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜑 → 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → 𝜓) | ||
Theorem | rexlimdvv 3290* | Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Jul-2004.) |
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒))) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → 𝜒)) | ||
Theorem | rexlimdvva 3291* | Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → 𝜒)) | ||
Theorem | rexbidv2 3292* | Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 22-May-1999.) |
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) | ||
Theorem | rexbidva 3293* | Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 9-Mar-1997.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 6-Dec-2019.) (Proof shortened by Wolf Lammen, 10-Dec-2019.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | rexbidv 3294* | Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 20-Nov-1994.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 6-Dec-2019.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | 2rexbiia 3295* | Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.) |
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) | ||
Theorem | 2rexbidva 3296* | Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 15-Dec-2004.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒)) | ||
Theorem | 2rexbidv 3297* | Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒)) | ||
Theorem | rexralbidv 3298* | Formula-building rule for restricted quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)) | ||
Theorem | r2exlem 3299 | Lemma factoring out common proof steps in r2exf 3322 an r2ex 3300. Introduced to reduce dependencies on axioms. (Contributed by Wolf Lammen, 10-Jan-2020.) |
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ¬ 𝜑)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)) | ||
Theorem | r2ex 3300* | Double restricted existential quantification. (Contributed by NM, 11-Nov-1995.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 10-Jan-2020.) |
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)) |
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