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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
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