 Home Intuitionistic Logic ExplorerTheorem List (p. 24 of 95) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 2301-2400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxwrex 2301 Extend wff notation to include restricted existential quantification.
wff x A φ

Syntaxwreu 2302 Extend wff notation to include restricted existential uniqueness.
wff ∃!x A φ

Syntaxwrmo 2303 Extend wff notation to include restricted "at most one."
wff ∃*x A φ

Syntaxcrab 2304 Extend class notation to include the restricted class abstraction (class builder).
class {x Aφ}

Definitiondf-ral 2305 Define restricted universal quantification. Special case of Definition 4.15(3) of [TakeutiZaring] p. 22. (Contributed by NM, 19-Aug-1993.)
(x A φx(x Aφ))

Definitiondf-rex 2306 Define restricted existential quantification. Special case of Definition 4.15(4) of [TakeutiZaring] p. 22. (Contributed by NM, 30-Aug-1993.)
(x A φx(x A φ))

Definitiondf-reu 2307 Define restricted existential uniqueness. (Contributed by NM, 22-Nov-1994.)
(∃!x A φ∃!x(x A φ))

Definitiondf-rmo 2308 Define restricted "at most one". (Contributed by NM, 16-Jun-2017.)
(∃*x A φ∃*x(x A φ))

Definitiondf-rab 2309 Define a restricted class abstraction (class builder), which is the class of all x in A such that φ is true. Definition of [TakeutiZaring] p. 20. (Contributed by NM, 22-Nov-1994.)
{x Aφ} = {x ∣ (x A φ)}

Theoremralnex 2310 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)
(x A ¬ φ ↔ ¬ x A φ)

Theoremrexnalim 2311 Relationship between restricted universal and existential quantifiers. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 17-Aug-2018.)
(x A ¬ φ → ¬ x A φ)

Theoremralexim 2312 Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 17-Aug-2018.)
(x A φ → ¬ x A ¬ φ)

Theoremrexalim 2313 Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 17-Aug-2018.)
(x A φ → ¬ x A ¬ φ)

Theoremralbida 2314 Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 6-Oct-2003.)
xφ    &   ((φ x A) → (ψχ))       (φ → (x A ψx A χ))

Theoremrexbida 2315 Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 6-Oct-2003.)
xφ    &   ((φ x A) → (ψχ))       (φ → (x A ψx A χ))

Theoremralbidva 2316* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 4-Mar-1997.)
((φ x A) → (ψχ))       (φ → (x A ψx A χ))

Theoremrexbidva 2317* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 9-Mar-1997.)
((φ x A) → (ψχ))       (φ → (x A ψx A χ))

Theoremralbid 2318 Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 27-Jun-1998.)
xφ    &   (φ → (ψχ))       (φ → (x A ψx A χ))

Theoremrexbid 2319 Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 27-Jun-1998.)
xφ    &   (φ → (ψχ))       (φ → (x A ψx A χ))

Theoremralbidv 2320* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 20-Nov-1994.)
(φ → (ψχ))       (φ → (x A ψx A χ))

Theoremrexbidv 2321* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 20-Nov-1994.)
(φ → (ψχ))       (φ → (x A ψx A χ))

Theoremralbidv2 2322* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 6-Apr-1997.)
(φ → ((x Aψ) ↔ (x Bχ)))       (φ → (x A ψx B χ))

Theoremrexbidv2 2323* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 22-May-1999.)
(φ → ((x A ψ) ↔ (x B χ)))       (φ → (x A ψx B χ))

Theoremralbii 2324 Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 23-Nov-1994.) (Revised by Mario Carneiro, 17-Oct-2016.)
(φψ)       (x A φx A ψ)

Theoremrexbii 2325 Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 23-Nov-1994.) (Revised by Mario Carneiro, 17-Oct-2016.)
(φψ)       (x A φx A ψ)

Theorem2ralbii 2326 Inference adding two restricted universal quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
(φψ)       (x A y B φx A y B ψ)

Theorem2rexbii 2327 Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 11-Nov-1995.)
(φψ)       (x A y B φx A y B ψ)

Theoremralbii2 2328 Inference adding different restricted universal quantifiers to each side of an equivalence. (Contributed by NM, 15-Aug-2005.)
((x Aφ) ↔ (x Bψ))       (x A φx B ψ)

Theoremrexbii2 2329 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 φx B ψ)

Theoremraleqbii 2330 Equality deduction for restricted universal quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)
A = B    &   (ψχ)       (x A ψx B χ)

Theoremrexeqbii 2331 Equality deduction for restricted existential quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)
A = B    &   (ψχ)       (x A ψx B χ)

Theoremralbiia 2332 Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 26-Nov-2000.)
(x A → (φψ))       (x A φx A ψ)

Theoremrexbiia 2333 Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 26-Oct-1999.)
(x A → (φψ))       (x A φx A ψ)

Theorem2rexbiia 2334* Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
((x A y B) → (φψ))       (x A y B φx A y B ψ)

Theoremr2alf 2335* Double restricted universal quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)
yA       (x A y B φxy((x A y B) → φ))

Theoremr2exf 2336* Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)
yA       (x A y B φxy((x A y B) φ))

Theoremr2al 2337* Double restricted universal quantification. (Contributed by NM, 19-Nov-1995.)
(x A y B φxy((x A y B) → φ))

Theoremr2ex 2338* Double restricted existential quantification. (Contributed by NM, 11-Nov-1995.)
(x A y B φxy((x A y B) φ))

Theorem2ralbida 2339* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 24-Feb-2004.)
xφ    &   yφ    &   ((φ (x A y B)) → (ψχ))       (φ → (x A y B ψx A y B χ))

Theorem2ralbidva 2340* Formula-building rule for restricted universal quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.)
((φ (x A y B)) → (ψχ))       (φ → (x A y B ψx A y B χ))

Theorem2rexbidva 2341* Formula-building rule for restricted existential quantifiers (deduction rule). (Contributed by NM, 15-Dec-2004.)
((φ (x A y B)) → (ψχ))       (φ → (x A y B ψx A y B χ))

Theorem2ralbidv 2342* Formula-building rule for restricted universal quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.) (Revised by Szymon Jaroszewicz, 16-Mar-2007.)
(φ → (ψχ))       (φ → (x A y B ψx A y B χ))

Theorem2rexbidv 2343* Formula-building rule for restricted existential quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.)
(φ → (ψχ))       (φ → (x A y B ψx A y B χ))

Theoremrexralbidv 2344* Formula-building rule for restricted quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.)
(φ → (ψχ))       (φ → (x A y B ψx A y B χ))

Theoremralinexa 2345 A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)
(x A (φ → ¬ ψ) ↔ ¬ x A (φ ψ))

Theoremrisset 2346* Two ways to say "A belongs to B." (Contributed by NM, 22-Nov-1994.)
(A Bx B x = A)

Theoremhbral 2347 Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by David Abernethy, 13-Dec-2009.)
(y Ax y A)    &   (φxφ)       (y A φxy A φ)

Theoremhbra1 2348 x is not free in x Aφ. (Contributed by NM, 18-Oct-1996.)
(x A φxx A φ)

Theoremnfra1 2349 x is not free in x Aφ. (Contributed by NM, 18-Oct-1996.) (Revised by Mario Carneiro, 7-Oct-2016.)
xx A φ

Theoremnfraldxy 2350* Not-free for restricted universal quantification where x and y are distinct. See nfraldya 2352 for a version with y and A distinct instead. (Contributed by Jim Kingdon, 29-May-2018.)
yφ    &   (φxA)    &   (φ → Ⅎxψ)       (φ → Ⅎxy A ψ)

Theoremnfrexdxy 2351* Not-free for restricted existential quantification where x and y are distinct. See nfrexdya 2353 for a version with y and A distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
yφ    &   (φxA)    &   (φ → Ⅎxψ)       (φ → Ⅎxy A ψ)

Theoremnfraldya 2352* Not-free for restricted universal quantification where y and A are distinct. See nfraldxy 2350 for a version with x and y distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
yφ    &   (φxA)    &   (φ → Ⅎxψ)       (φ → Ⅎxy A ψ)

Theoremnfrexdya 2353* Not-free for restricted existential quantification where y and A are distinct. See nfrexdxy 2351 for a version with x and y distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
yφ    &   (φxA)    &   (φ → Ⅎxψ)       (φ → Ⅎxy A ψ)

Theoremnfralxy 2354* Not-free for restricted universal quantification where x and y are distinct. See nfralya 2356 for a version with y and A distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
xA    &   xφ       xy A φ

Theoremnfrexxy 2355* Not-free for restricted existential quantification where x and y are distinct. See nfrexya 2357 for a version with y and A distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
xA    &   xφ       xy A φ

Theoremnfralya 2356* Not-free for restricted universal quantification where y and A are distinct. See nfralxy 2354 for a version with x and y distinct instead. (Contributed by Jim Kingdon, 3-Jun-2018.)
xA    &   xφ       xy A φ

Theoremnfrexya 2357* Not-free for restricted existential quantification where y and A are distinct. See nfrexxy 2355 for a version with x and y distinct instead. (Contributed by Jim Kingdon, 3-Jun-2018.)
xA    &   xφ       xy A φ

Theoremnfra2xy 2358* Not-free given two restricted quantifiers. (Contributed by Jim Kingdon, 20-Aug-2018.)
yx A y B φ

Theoremnfre1 2359 x is not free in x Aφ. (Contributed by NM, 19-Mar-1997.) (Revised by Mario Carneiro, 7-Oct-2016.)
xx A φ

Theoremr3al 2360* Triple restricted universal quantification. (Contributed by NM, 19-Nov-1995.)
(x A y B z 𝐶 φxyz((x A y B z 𝐶) → φ))

Theoremalral 2361 Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)
(xφx A φ)

Theoremrexex 2362 Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)
(x A φxφ)

Theoremrsp 2363 Restricted specialization. (Contributed by NM, 17-Oct-1996.)
(x A φ → (x Aφ))

Theoremrspe 2364 Restricted specialization. (Contributed by NM, 12-Oct-1999.)
((x A φ) → x A φ)

Theoremrsp2 2365 Restricted specialization. (Contributed by NM, 11-Feb-1997.)
(x A y B φ → ((x A y B) → φ))

Theoremrsp2e 2366 Restricted specialization. (Contributed by FL, 4-Jun-2012.)
((x A y B φ) → x A y B φ)

Theoremrspec 2367 Specialization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
x A φ       (x Aφ)

Theoremrgen 2368 Generalization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
(x Aφ)       x A φ

Theoremrgen2a 2369* Generalization rule for restricted quantification. Note that x and y needn't be distinct (and illustrates the use of dvelimor 1891). (Contributed by NM, 23-Nov-1994.) (Proof rewritten by Jim Kingdon, 1-Jun-2018.)
((x A y A) → φ)       x A y A φ

Theoremrgenw 2370 Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)
φ       x A φ

Theoremrgen2w 2371 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

Theoremmprg 2372 Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.)
(x A φψ)    &   (x Aφ)       ψ

Theoremmprgbir 2373 Modus ponens on biconditional combined with restricted generalization. (Contributed by NM, 21-Mar-2004.)
(φx A ψ)    &   (x Aψ)       φ

Theoremralim 2374 Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997.)
(x A (φψ) → (x A φx A ψ))

Theoremralimi2 2375 Inference quantifying both antecedent and consequent. (Contributed by NM, 22-Feb-2004.)
((x Aφ) → (x Bψ))       (x A φx B ψ)

Theoremralimia 2376 Inference quantifying both antecedent and consequent. (Contributed by NM, 19-Jul-1996.)
(x A → (φψ))       (x A φx A ψ)

Theoremralimiaa 2377 Inference quantifying both antecedent and consequent. (Contributed by NM, 4-Aug-2007.)
((x A φ) → ψ)       (x A φx A ψ)

Theoremralimi 2378 Inference quantifying both antecedent and consequent, with strong hypothesis. (Contributed by NM, 4-Mar-1997.)
(φψ)       (x A φx A ψ)

Theoremral2imi 2379 Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.)
(φ → (ψχ))       (x A φ → (x A ψx A χ))

Theoremralimdaa 2380 Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-Sep-2003.)
xφ    &   ((φ x A) → (ψχ))       (φ → (x A ψx A χ))

Theoremralimdva 2381* Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.)
((φ x A) → (ψχ))       (φ → (x A ψx A χ))

Theoremralimdv 2382* Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 8-Oct-2003.)
(φ → (ψχ))       (φ → (x A ψx A χ))

Theoremralimdv2 2383* Inference quantifying both antecedent and consequent. (Contributed by NM, 1-Feb-2005.)
(φ → ((x Aψ) → (x Bχ)))       (φ → (x A ψx B χ))

Theoremralrimi 2384 Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 10-Oct-1999.)
xφ    &   (φ → (x Aψ))       (φx A ψ)

Theoremralrimiv 2385* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.)
(φ → (x Aψ))       (φx A ψ)

Theoremralrimiva 2386* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Jan-2006.)
((φ x A) → ψ)       (φx A ψ)

Theoremralrimivw 2387* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)
(φψ)       (φx A ψ)

Theoremr19.21t 2388 Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers (closed theorem version). (Contributed by NM, 1-Mar-2008.)
(Ⅎxφ → (x A (φψ) ↔ (φx A ψ)))

Theoremr19.21 2389 Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers. (Contributed by Scott Fenton, 30-Mar-2011.)
xφ       (x A (φψ) ↔ (φx A ψ))

Theoremr19.21v 2390* Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
(x A (φψ) ↔ (φx A ψ))

Theoremralrimd 2391 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 16-Feb-2004.)
xφ    &   xψ    &   (φ → (ψ → (x Aχ)))       (φ → (ψx A χ))

Theoremralrimdv 2392* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 27-May-1998.)
(φ → (ψ → (x Aχ)))       (φ → (ψx A χ))

Theoremralrimdva 2393* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Feb-2008.)
((φ x A) → (ψχ))       (φ → (ψx A χ))

Theoremralrimivv 2394* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 24-Jul-2004.)
(φ → ((x A y B) → ψ))       (φx A y B ψ)

Theoremralrimivva 2395* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by Jeff Madsen, 19-Jun-2011.)
((φ (x A y B)) → ψ)       (φx A y B ψ)

Theoremralrimivvva 2396* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with triple quantification.) (Contributed by Mario Carneiro, 9-Jul-2014.)
((φ (x A y B z 𝐶)) → ψ)       (φx A y B z 𝐶 ψ)

Theoremralrimdvv 2397* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 1-Jun-2005.)
(φ → (ψ → ((x A y B) → χ)))       (φ → (ψx A y B χ))

Theoremralrimdvva 2398* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 2-Feb-2008.)
((φ (x A y B)) → (ψχ))       (φ → (ψx A y B χ))

Theoremrgen2 2399* Generalization rule for restricted quantification. (Contributed by NM, 30-May-1999.)
((x A y B) → φ)       x A y B φ

Theoremrgen3 2400* Generalization rule for restricted quantification. (Contributed by NM, 12-Jan-2008.)
((x A y B z 𝐶) → φ)       x A y B z 𝐶 φ

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9457
 Copyright terms: Public domain < Previous  Next >