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Theorem List for Intuitionistic Logic Explorer - 2501-2600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrexim 2501 Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.) (Proof shortened by Andrew Salmon, 30-May-2011.)
(∀𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
 
Theoremreximia 2502 Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.)
(𝑥𝐴 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓)
 
Theoremreximi2 2503 Inference quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 8-Nov-2004.)
((𝑥𝐴𝜑) → (𝑥𝐵𝜓))       (∃𝑥𝐴 𝜑 → ∃𝑥𝐵 𝜓)
 
Theoremreximi 2504 Inference quantifying both antecedent and consequent. (Contributed by NM, 18-Oct-1996.)
(𝜑𝜓)       (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓)
 
Theoremreximdai 2505 Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 31-Aug-1999.)
𝑥𝜑    &   (𝜑 → (𝑥𝐴 → (𝜓𝜒)))       (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 𝜒))
 
Theoremreximdv2 2506* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 17-Sep-2003.)
(𝜑 → ((𝑥𝐴𝜓) → (𝑥𝐵𝜒)))       (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐵 𝜒))
 
Theoremreximdvai 2507* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 14-Nov-2002.)
(𝜑 → (𝑥𝐴 → (𝜓𝜒)))       (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 𝜒))
 
Theoremreximdv 2508* Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version with strong hypothesis.) (Contributed by NM, 24-Jun-1998.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 𝜒))
 
Theoremreximdva 2509* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 𝜒))
 
Theoremreximddv 2510* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 7-Dec-2016.)
((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜒)    &   (𝜑 → ∃𝑥𝐴 𝜓)       (𝜑 → ∃𝑥𝐴 𝜒)
 
Theoremreximssdv 2511* Derivation of a restricted existential quantification over a subset (the second hypothesis implies 𝐴𝐵), deduction form. (Contributed by AV, 21-Aug-2022.)
(𝜑 → ∃𝑥𝐵 𝜓)    &   ((𝜑 ∧ (𝑥𝐵𝜓)) → 𝑥𝐴)    &   ((𝜑 ∧ (𝑥𝐵𝜓)) → 𝜒)       (𝜑 → ∃𝑥𝐴 𝜒)
 
Theoremreximddv2 2512* Double deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 15-Dec-2019.)
((((𝜑𝑥𝐴) ∧ 𝑦𝐵) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)       (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜒)
 
Theoremr19.12 2513* Theorem 19.12 of [Margaris] p. 89 with restricted quantifiers. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
(∃𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴 𝜑)
 
Theoremr19.23t 2514 Closed theorem form of r19.23 2515. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
(Ⅎ𝑥𝜓 → (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))
 
Theoremr19.23 2515 Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 8-Oct-2016.)
𝑥𝜓       (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))
 
Theoremr19.23v 2516* Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)
(∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))
 
Theoremrexlimi 2517 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 30-Nov-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
𝑥𝜓    &   (𝑥𝐴 → (𝜑𝜓))       (∃𝑥𝐴 𝜑𝜓)
 
Theoremrexlimiv 2518* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.)
(𝑥𝐴 → (𝜑𝜓))       (∃𝑥𝐴 𝜑𝜓)
 
Theoremrexlimiva 2519* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Dec-2006.)
((𝑥𝐴𝜑) → 𝜓)       (∃𝑥𝐴 𝜑𝜓)
 
Theoremrexlimivw 2520* Weaker version of rexlimiv 2518. (Contributed by FL, 19-Sep-2011.)
(𝜑𝜓)       (∃𝑥𝐴 𝜑𝜓)
 
Theoremrexlimd 2521 Deduction from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)
𝑥𝜑    &   𝑥𝜒    &   (𝜑 → (𝑥𝐴 → (𝜓𝜒)))       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremrexlimd2 2522 Version of rexlimd 2521 with deduction version of second hypothesis. (Contributed by NM, 21-Jul-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥𝐴 → (𝜓𝜒)))       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremrexlimdv 2523* 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.)
(𝜑 → (𝑥𝐴 → (𝜓𝜒)))       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremrexlimdva 2524* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 20-Jan-2007.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremrexlimdvaa 2525* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Mario Carneiro, 15-Jun-2016.)
((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜒)       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremrexlimdv3a 2526* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 2523. (Contributed by NM, 7-Jun-2015.)
((𝜑𝑥𝐴𝜓) → 𝜒)       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremrexlimdva2 2527* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremrexlimdvw 2528* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Jun-2014.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremrexlimddv 2529* Restricted existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
(𝜑 → ∃𝑥𝐴 𝜓)    &   ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜒)       (𝜑𝜒)
 
Theoremrexlimivv 2530* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 17-Feb-2004.)
((𝑥𝐴𝑦𝐵) → (𝜑𝜓))       (∃𝑥𝐴𝑦𝐵 𝜑𝜓)
 
Theoremrexlimdvv 2531* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Jul-2004.)
(𝜑 → ((𝑥𝐴𝑦𝐵) → (𝜓𝜒)))       (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓𝜒))
 
Theoremrexlimdvva 2532* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓𝜒))
 
Theoremr19.26 2533 Theorem 19.26 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 28-Jan-1997.) (Proof shortened by Andrew Salmon, 30-May-2011.)
(∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓))
 
Theoremr19.27v 2534* Restricted quantitifer version of one direction of 19.27 1523. (The other direction holds when 𝐴 is inhabited, see r19.27mv 3427.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof shortened by Wolf Lammen, 17-Jun-2023.)
((∀𝑥𝐴 𝜑𝜓) → ∀𝑥𝐴 (𝜑𝜓))
 
Theoremr19.28v 2535* Restricted quantifier version of one direction of 19.28 1525. (The other direction holds when 𝐴 is inhabited, see r19.28mv 3423.) (Contributed by NM, 2-Apr-2004.) (Proof shortened by Wolf Lammen, 17-Jun-2023.)
((𝜑 ∧ ∀𝑥𝐴 𝜓) → ∀𝑥𝐴 (𝜑𝜓))
 
Theoremr19.26-2 2536 Theorem 19.26 of [Margaris] p. 90 with 2 restricted quantifiers. (Contributed by NM, 10-Aug-2004.)
(∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∀𝑥𝐴𝑦𝐵 𝜓))
 
Theoremr19.26-3 2537 Theorem 19.26 of [Margaris] p. 90 with 3 restricted quantifiers. (Contributed by FL, 22-Nov-2010.)
(∀𝑥𝐴 (𝜑𝜓𝜒) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓 ∧ ∀𝑥𝐴 𝜒))
 
Theoremr19.26m 2538 Theorem 19.26 of [Margaris] p. 90 with mixed quantifiers. (Contributed by NM, 22-Feb-2004.)
(∀𝑥((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓)) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜓))
 
Theoremralbi 2539 Distribute a restricted universal quantifier over a biconditional. Theorem 19.15 of [Margaris] p. 90 with restricted quantification. (Contributed by NM, 6-Oct-2003.)
(∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜓))
 
Theoremrexbi 2540 Distribute a restricted existential quantifier over a biconditional. Theorem 19.18 of [Margaris] p. 90 with restricted quantification. (Contributed by Jim Kingdon, 21-Jan-2019.)
(∀𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 𝜓))
 
Theoremralbiim 2541 Split a biconditional and distribute quantifier. (Contributed by NM, 3-Jun-2012.)
(∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 (𝜑𝜓) ∧ ∀𝑥𝐴 (𝜓𝜑)))
 
Theoremr19.27av 2542* Restricted version of one direction of Theorem 19.27 of [Margaris] p. 90. (The other direction doesn't hold when 𝐴 is empty.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.)
((∀𝑥𝐴 𝜑𝜓) → ∀𝑥𝐴 (𝜑𝜓))
 
Theoremr19.28av 2543* Restricted version of one direction of Theorem 19.28 of [Margaris] p. 90. (The other direction doesn't hold when 𝐴 is empty.) (Contributed by NM, 2-Apr-2004.)
((𝜑 ∧ ∀𝑥𝐴 𝜓) → ∀𝑥𝐴 (𝜑𝜓))
 
Theoremr19.29 2544 Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.) (Proof shortened by Andrew Salmon, 30-May-2011.)
((∀𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 𝜓) → ∃𝑥𝐴 (𝜑𝜓))
 
Theoremr19.29r 2545 Variation of Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)
((∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) → ∃𝑥𝐴 (𝜑𝜓))
 
Theoremralnex2 2546 Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 18-May-2023.)
(∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴𝑦𝐵 𝜑)
 
Theoremr19.29af2 2547 A commonly used pattern based on r19.29 2544 (Contributed by Thierry Arnoux, 17-Dec-2017.)
𝑥𝜑    &   𝑥𝜒    &   (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴 𝜓)       (𝜑𝜒)
 
Theoremr19.29af 2548* A commonly used pattern based on r19.29 2544 (Contributed by Thierry Arnoux, 29-Nov-2017.)
𝑥𝜑    &   (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴 𝜓)       (𝜑𝜒)
 
Theoremr19.29an 2549* A commonly used pattern based on r19.29 2544. (Contributed by Thierry Arnoux, 29-Dec-2019.)
(((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)       ((𝜑 ∧ ∃𝑥𝐴 𝜓) → 𝜒)
 
Theoremr19.29a 2550* A commonly used pattern based on r19.29 2544 (Contributed by Thierry Arnoux, 22-Nov-2017.)
(((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴 𝜓)       (𝜑𝜒)
 
Theoremr19.29d2r 2551 Theorem 19.29 of [Margaris] p. 90 with two restricted quantifiers, deduction version (Contributed by Thierry Arnoux, 30-Jan-2017.)
(𝜑 → ∀𝑥𝐴𝑦𝐵 𝜓)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜒)       (𝜑 → ∃𝑥𝐴𝑦𝐵 (𝜓𝜒))
 
Theoremr19.29vva 2552* A commonly used pattern based on r19.29 2544, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.)
((((𝜑𝑥𝐴) ∧ 𝑦𝐵) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)       (𝜑𝜒)
 
Theoremr19.32r 2553 One direction of Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. For decidable propositions this is an equivalence. (Contributed by Jim Kingdon, 19-Aug-2018.)
𝑥𝜑       ((𝜑 ∨ ∀𝑥𝐴 𝜓) → ∀𝑥𝐴 (𝜑𝜓))
 
Theoremr19.32vr 2554* One direction of Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. For decidable propositions this is an equivalence, as seen at r19.32vdc 2555. (Contributed by Jim Kingdon, 19-Aug-2018.)
((𝜑 ∨ ∀𝑥𝐴 𝜓) → ∀𝑥𝐴 (𝜑𝜓))
 
Theoremr19.32vdc 2555* Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, where 𝜑 is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)
(DECID 𝜑 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝐴 𝜓)))
 
Theoremr19.35-1 2556 Restricted quantifier version of 19.35-1 1586. (Contributed by Jim Kingdon, 4-Jun-2018.)
(∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
 
Theoremr19.36av 2557* One direction of a restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. In classical logic, the converse would hold if 𝐴 has at least one element, but in intuitionistic logic, that is not a sufficient condition. (Contributed by NM, 22-Oct-2003.)
(∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑𝜓))
 
Theoremr19.37 2558 Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. In classical logic the converse would hold if 𝐴 has at least one element, but that is not sufficient in intuitionistic logic. (Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)
𝑥𝜑       (∃𝑥𝐴 (𝜑𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))
 
Theoremr19.37av 2559* Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.)
(∃𝑥𝐴 (𝜑𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))
 
Theoremr19.40 2560 Restricted quantifier version of Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.)
(∃𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 𝜓))
 
Theoremr19.41 2561 Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 1-Nov-2010.)
𝑥𝜓       (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))
 
Theoremr19.41v 2562* Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 17-Dec-2003.)
(∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))
 
Theoremr19.42v 2563* Restricted version of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
(∃𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝐴 𝜓))
 
Theoremr19.43 2564 Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) (Proof rewritten by Jim Kingdon, 5-Jun-2018.)
(∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
 
Theoremr19.44av 2565* One direction of a restricted quantifier version of Theorem 19.44 of [Margaris] p. 90. The other direction doesn't hold when 𝐴 is empty. (Contributed by NM, 2-Apr-2004.)
(∃𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑𝜓))
 
Theoremr19.45av 2566* Restricted version of one direction of Theorem 19.45 of [Margaris] p. 90. (The other direction doesn't hold when 𝐴 is empty.) (Contributed by NM, 2-Apr-2004.)
(∃𝑥𝐴 (𝜑𝜓) → (𝜑 ∨ ∃𝑥𝐴 𝜓))
 
Theoremralcomf 2567* Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑦𝐴    &   𝑥𝐵       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥𝐴 𝜑)
 
Theoremrexcomf 2568* Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑦𝐴    &   𝑥𝐵       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 𝜑)
 
Theoremralcom 2569* Commutation of restricted quantifiers. (Contributed by NM, 13-Oct-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
(∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥𝐴 𝜑)
 
Theoremrexcom 2570* Commutation of restricted quantifiers. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
(∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 𝜑)
 
Theoremrexcom13 2571* Swap 1st and 3rd restricted existential quantifiers. (Contributed by NM, 8-Apr-2015.)
(∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∃𝑧𝐶𝑦𝐵𝑥𝐴 𝜑)
 
Theoremrexrot4 2572* Rotate existential restricted quantifiers twice. (Contributed by NM, 8-Apr-2015.)
(∃𝑥𝐴𝑦𝐵𝑧𝐶𝑤𝐷 𝜑 ↔ ∃𝑧𝐶𝑤𝐷𝑥𝐴𝑦𝐵 𝜑)
 
Theoremralcom3 2573 A commutative law for restricted quantifiers that swaps the domain of the restriction. (Contributed by NM, 22-Feb-2004.)
(∀𝑥𝐴 (𝑥𝐵𝜑) ↔ ∀𝑥𝐵 (𝑥𝐴𝜑))
 
Theoremreean 2574* Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Andrew Salmon, 30-May-2011.)
𝑦𝜑    &   𝑥𝜓       (∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓))
 
Theoremreeanv 2575* Rearrange existential quantifiers. (Contributed by NM, 9-May-1999.)
(∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓))
 
Theorem3reeanv 2576* Rearrange three existential quantifiers. (Contributed by Jeff Madsen, 11-Jun-2010.)
(∃𝑥𝐴𝑦𝐵𝑧𝐶 (𝜑𝜓𝜒) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓 ∧ ∃𝑧𝐶 𝜒))
 
Theoremnfreu1 2577 𝑥 is not free in ∃!𝑥𝐴𝜑. (Contributed by NM, 19-Mar-1997.)
𝑥∃!𝑥𝐴 𝜑
 
Theoremnfrmo1 2578 𝑥 is not free in ∃*𝑥𝐴𝜑. (Contributed by NM, 16-Jun-2017.)
𝑥∃*𝑥𝐴 𝜑
 
Theoremnfreudxy 2579* Not-free deduction for restricted uniqueness. This is a version where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃!𝑦𝐴 𝜓)
 
Theoremnfreuxy 2580* Not-free for restricted uniqueness. This is a version where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.)
𝑥𝐴    &   𝑥𝜑       𝑥∃!𝑦𝐴 𝜑
 
Theoremrabid 2581 An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of [Quine] p. 16. (Contributed by NM, 9-Oct-2003.)
(𝑥 ∈ {𝑥𝐴𝜑} ↔ (𝑥𝐴𝜑))
 
Theoremrabid2 2582* An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
(𝐴 = {𝑥𝐴𝜑} ↔ ∀𝑥𝐴 𝜑)
 
Theoremrabbi 2583 Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2646. (Contributed by NM, 25-Nov-2013.)
(∀𝑥𝐴 (𝜓𝜒) ↔ {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
 
Theoremrabswap 2584 Swap with a membership relation in a restricted class abstraction. (Contributed by NM, 4-Jul-2005.)
{𝑥𝐴𝑥𝐵} = {𝑥𝐵𝑥𝐴}
 
Theoremnfrab1 2585 The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
𝑥{𝑥𝐴𝜑}
 
Theoremnfrabxy 2586* A variable not free in a wff remains so in a restricted class abstraction. (Contributed by Jim Kingdon, 19-Jul-2018.)
𝑥𝜑    &   𝑥𝐴       𝑥{𝑦𝐴𝜑}
 
Theoremreubida 2587 Formula-building rule for restricted existential quantifier (deduction form). (Contributed by Mario Carneiro, 19-Nov-2016.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))
 
Theoremreubidva 2588* Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 13-Nov-2004.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))
 
Theoremreubidv 2589* Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 17-Oct-1996.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))
 
Theoremreubiia 2590 Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 14-Nov-2004.)
(𝑥𝐴 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐴 𝜓)
 
Theoremreubii 2591 Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 22-Oct-1999.)
(𝜑𝜓)       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐴 𝜓)
 
Theoremrmobida 2592 Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))
 
Theoremrmobidva 2593* Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))
 
Theoremrmobidv 2594* Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))
 
Theoremrmobiia 2595 Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 16-Jun-2017.)
(𝑥𝐴 → (𝜑𝜓))       (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐴 𝜓)
 
Theoremrmobii 2596 Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 16-Jun-2017.)
(𝜑𝜓)       (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐴 𝜓)
 
Theoremraleqf 2597 Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑))
 
Theoremrexeqf 2598 Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))
 
Theoremreueq1f 2599 Equality theorem for restricted unique existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜑))
 
Theoremrmoeq1f 2600 Equality theorem for restricted at-most-one quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))
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