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Theorem List for Intuitionistic Logic Explorer - 2401-2500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremralimi 2401 Inference quantifying both antecedent and consequent, with strong hypothesis. (Contributed by NM, 4-Mar-1997.)
(𝜑𝜓)       (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓)

Theoremral2imi 2402 Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝐴 𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))

Theoremralimdaa 2403 Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-Sep-2003.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))

Theoremralimdva 2404* Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))

Theoremralimdv 2405* Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 8-Oct-2003.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))

Theoremralimdv2 2406* Inference quantifying both antecedent and consequent. (Contributed by NM, 1-Feb-2005.)
(𝜑 → ((𝑥𝐴𝜓) → (𝑥𝐵𝜒)))       (𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐵 𝜒))

Theoremralrimi 2407 Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 10-Oct-1999.)
𝑥𝜑    &   (𝜑 → (𝑥𝐴𝜓))       (𝜑 → ∀𝑥𝐴 𝜓)

Theoremralrimiv 2408* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.)
(𝜑 → (𝑥𝐴𝜓))       (𝜑 → ∀𝑥𝐴 𝜓)

Theoremralrimiva 2409* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Jan-2006.)
((𝜑𝑥𝐴) → 𝜓)       (𝜑 → ∀𝑥𝐴 𝜓)

Theoremralrimivw 2410* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)
(𝜑𝜓)       (𝜑 → ∀𝑥𝐴 𝜓)

Theoremr19.21t 2411 Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers (closed theorem version). (Contributed by NM, 1-Mar-2008.)
(Ⅎ𝑥𝜑 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓)))

Theoremr19.21 2412 Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers. (Contributed by Scott Fenton, 30-Mar-2011.)
𝑥𝜑       (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓))

Theoremr19.21v 2413* Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
(∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓))

Theoremralrimd 2414 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 16-Feb-2004.)
𝑥𝜑    &   𝑥𝜓    &   (𝜑 → (𝜓 → (𝑥𝐴𝜒)))       (𝜑 → (𝜓 → ∀𝑥𝐴 𝜒))

Theoremralrimdv 2415* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 27-May-1998.)
(𝜑 → (𝜓 → (𝑥𝐴𝜒)))       (𝜑 → (𝜓 → ∀𝑥𝐴 𝜒))

Theoremralrimdva 2416* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Feb-2008.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝐴 𝜒))

Theoremralrimivv 2417* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 24-Jul-2004.)
(𝜑 → ((𝑥𝐴𝑦𝐵) → 𝜓))       (𝜑 → ∀𝑥𝐴𝑦𝐵 𝜓)

Theoremralrimivva 2418* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by Jeff Madsen, 19-Jun-2011.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝜓)       (𝜑 → ∀𝑥𝐴𝑦𝐵 𝜓)

Theoremralrimivvva 2419* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with triple quantification.) (Contributed by Mario Carneiro, 9-Jul-2014.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵𝑧𝐶)) → 𝜓)       (𝜑 → ∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜓)

Theoremralrimdvv 2420* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 1-Jun-2005.)
(𝜑 → (𝜓 → ((𝑥𝐴𝑦𝐵) → 𝜒)))       (𝜑 → (𝜓 → ∀𝑥𝐴𝑦𝐵 𝜒))

Theoremralrimdvva 2421* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 2-Feb-2008.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝐴𝑦𝐵 𝜒))

Theoremrgen2 2422* Generalization rule for restricted quantification. (Contributed by NM, 30-May-1999.)
((𝑥𝐴𝑦𝐵) → 𝜑)       𝑥𝐴𝑦𝐵 𝜑

Theoremrgen3 2423* Generalization rule for restricted quantification. (Contributed by NM, 12-Jan-2008.)
((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑)       𝑥𝐴𝑦𝐵𝑧𝐶 𝜑

Theoremr19.21bi 2424 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.)
(𝜑 → ∀𝑥𝐴 𝜓)       ((𝜑𝑥𝐴) → 𝜓)

Theoremrspec2 2425 Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.)
𝑥𝐴𝑦𝐵 𝜑       ((𝑥𝐴𝑦𝐵) → 𝜑)

Theoremrspec3 2426 Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.)
𝑥𝐴𝑦𝐵𝑧𝐶 𝜑       ((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑)

Theoremr19.21be 2427 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 21-Nov-1994.)
(𝜑 → ∀𝑥𝐴 𝜓)       𝑥𝐴 (𝜑𝜓)

Theoremnrex 2428 Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
(𝑥𝐴 → ¬ 𝜓)        ¬ ∃𝑥𝐴 𝜓

Theoremnrexdv 2429* Deduction adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
((𝜑𝑥𝐴) → ¬ 𝜓)       (𝜑 → ¬ ∃𝑥𝐴 𝜓)

Theoremrexim 2430 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 2431 Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.)
(𝑥𝐴 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓)

Theoremreximi2 2432 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 2433 Inference quantifying both antecedent and consequent. (Contributed by NM, 18-Oct-1996.)
(𝜑𝜓)       (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓)

Theoremreximdai 2434 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 2435* 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 2436* 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 2437* 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 2438* 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))

Theoremr19.12 2439* 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 2440 Closed theorem form of r19.23 2441. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
(Ⅎ𝑥𝜓 → (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))

Theoremr19.23 2441 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 2442* Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)
(∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))

Theoremrexlimi 2443 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 2444* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.)
(𝑥𝐴 → (𝜑𝜓))       (∃𝑥𝐴 𝜑𝜓)

Theoremrexlimiva 2445* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Dec-2006.)
((𝑥𝐴𝜑) → 𝜓)       (∃𝑥𝐴 𝜑𝜓)

Theoremrexlimivw 2446* Weaker version of rexlimiv 2444. (Contributed by FL, 19-Sep-2011.)
(𝜑𝜓)       (∃𝑥𝐴 𝜑𝜓)

Theoremrexlimd 2447 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 2448 Version of rexlimd 2447 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 2449* 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 2450* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 20-Jan-2007.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓𝜒))

Theoremrexlimdvaa 2451* 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 2452* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 2449. (Contributed by NM, 7-Jun-2015.)
((𝜑𝑥𝐴𝜓) → 𝜒)       (𝜑 → (∃𝑥𝐴 𝜓𝜒))

Theoremrexlimdvw 2453* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Jun-2014.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓𝜒))

Theoremrexlimddv 2454* Restricted existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
(𝜑 → ∃𝑥𝐴 𝜓)    &   ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜒)       (𝜑𝜒)

Theoremrexlimivv 2455* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 17-Feb-2004.)
((𝑥𝐴𝑦𝐵) → (𝜑𝜓))       (∃𝑥𝐴𝑦𝐵 𝜑𝜓)

Theoremrexlimdvv 2456* 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 2457* 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 2458 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))

Theoremr19.26-2 2459 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 2460 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 2461 Theorem 19.26 of [Margaris] p. 90 with mixed quantifiers. (Contributed by NM, 22-Feb-2004.)
(∀𝑥((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓)) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜓))

Theoremralbi 2462 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 2463 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 2464 Split a biconditional and distribute quantifier. (Contributed by NM, 3-Jun-2012.)
(∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 (𝜑𝜓) ∧ ∀𝑥𝐴 (𝜓𝜑)))

Theoremr19.27av 2465* 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 2466* 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 2467 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 2468 Variation of Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)
((∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) → ∃𝑥𝐴 (𝜑𝜓))

Theoremr19.29af2 2469 A commonly used pattern based on r19.29 2467 (Contributed by Thierry Arnoux, 17-Dec-2017.)
𝑥𝜑    &   𝑥𝜒    &   (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴 𝜓)       (𝜑𝜒)

Theoremr19.29af 2470* A commonly used pattern based on r19.29 2467 (Contributed by Thierry Arnoux, 29-Nov-2017.)
𝑥𝜑    &   (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴 𝜓)       (𝜑𝜒)

Theoremr19.29a 2471* A commonly used pattern based on r19.29 2467 (Contributed by Thierry Arnoux, 22-Nov-2017.)
(((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴 𝜓)       (𝜑𝜒)

Theoremr19.29d2r 2472 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 2473* A commonly used pattern based on r19.29 2467, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.)
((((𝜑𝑥𝐴) ∧ 𝑦𝐵) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)       (𝜑𝜒)

Theoremr19.32r 2474 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 2475* 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 2476. (Contributed by Jim Kingdon, 19-Aug-2018.)
((𝜑 ∨ ∀𝑥𝐴 𝜓) → ∀𝑥𝐴 (𝜑𝜓))

Theoremr19.32vdc 2476* 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 2477 Restricted quantifier version of 19.35-1 1531. (Contributed by Jim Kingdon, 4-Jun-2018.)
(∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))

Theoremr19.36av 2478* 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 2479 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 2480* 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 2481 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 2482 Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 1-Nov-2010.)
𝑥𝜓       (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))

Theoremr19.41v 2483* Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 17-Dec-2003.)
(∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))

Theoremr19.42v 2484* Restricted version of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
(∃𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝐴 𝜓))

Theoremr19.43 2485 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 2486* 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 2487* 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 2488* Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑦𝐴    &   𝑥𝐵       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥𝐴 𝜑)

Theoremrexcomf 2489* Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑦𝐴    &   𝑥𝐵       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 𝜑)

Theoremralcom 2490* Commutation of restricted quantifiers. (Contributed by NM, 13-Oct-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
(∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥𝐴 𝜑)

Theoremrexcom 2491* Commutation of restricted quantifiers. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
(∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 𝜑)

Theoremrexcom13 2492* Swap 1st and 3rd restricted existential quantifiers. (Contributed by NM, 8-Apr-2015.)
(∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∃𝑧𝐶𝑦𝐵𝑥𝐴 𝜑)

Theoremrexrot4 2493* Rotate existential restricted quantifiers twice. (Contributed by NM, 8-Apr-2015.)
(∃𝑥𝐴𝑦𝐵𝑧𝐶𝑤𝐷 𝜑 ↔ ∃𝑧𝐶𝑤𝐷𝑥𝐴𝑦𝐵 𝜑)

Theoremralcom3 2494 A commutative law for restricted quantifiers that swaps the domain of the restriction. (Contributed by NM, 22-Feb-2004.)
(∀𝑥𝐴 (𝑥𝐵𝜑) ↔ ∀𝑥𝐵 (𝑥𝐴𝜑))

Theoremreean 2495* 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 2496* Rearrange existential quantifiers. (Contributed by NM, 9-May-1999.)
(∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓))

Theorem3reeanv 2497* Rearrange three existential quantifiers. (Contributed by Jeff Madsen, 11-Jun-2010.)
(∃𝑥𝐴𝑦𝐵𝑧𝐶 (𝜑𝜓𝜒) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓 ∧ ∃𝑧𝐶 𝜒))

Theoremnfreu1 2498 𝑥 is not free in ∃!𝑥𝐴𝜑. (Contributed by NM, 19-Mar-1997.)
𝑥∃!𝑥𝐴 𝜑

Theoremnfrmo1 2499 𝑥 is not free in ∃*𝑥𝐴𝜑. (Contributed by NM, 16-Jun-2017.)
𝑥∃*𝑥𝐴 𝜑

Theoremnfreudxy 2500* 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)

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