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

Theoremrexbidva 2301* 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 2302 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 2303 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 2304* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 20-Nov-1994.)
(φ → (ψχ))       (φ → (x A ψx A χ))

Theoremrexbidv 2305* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 20-Nov-1994.)
(φ → (ψχ))       (φ → (x A ψx A χ))

Theoremralbidv2 2306* 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 2307* 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 2308 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 2309 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 2310 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 2311 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 2312 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 2313 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 2314 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 2315 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 2316 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 2317 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 2318* 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 2319* Double restricted universal quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)
yA       (x A y B φxy((x A y B) → φ))

Theoremr2exf 2320* Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)
yA       (x A y B φxy((x A y B) φ))

Theoremr2al 2321* Double restricted universal quantification. (Contributed by NM, 19-Nov-1995.)
(x A y B φxy((x A y B) → φ))

Theoremr2ex 2322* Double restricted existential quantification. (Contributed by NM, 11-Nov-1995.)
(x A y B φxy((x A y B) φ))

Theorem2ralbida 2323* 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 2324* 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 2325* 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 2326* 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 2327* 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 2328* 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 2329 A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)
(x A (φ → ¬ ψ) ↔ ¬ x A (φ ψ))

Theoremrisset 2330* Two ways to say "A belongs to B." (Contributed by NM, 22-Nov-1994.)
(A Bx B x = A)

Theoremhbral 2331 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 2332 x is not free in x Aφ. (Contributed by NM, 18-Oct-1996.)
(x A φxx A φ)

Theoremnfra1 2333 x is not free in x Aφ. (Contributed by NM, 18-Oct-1996.) (Revised by Mario Carneiro, 7-Oct-2016.)
xx A φ

Theoremnfraldxy 2334* Not-free for restricted universal quantification where x and y are distinct. See nfraldya 2336 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 2335* Not-free for restricted existential quantification where x and y are distinct. See nfrexdya 2337 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 2336* Not-free for restricted universal quantification where y and A are distinct. See nfraldxy 2334 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 2337* Not-free for restricted existential quantification where y and A are distinct. See nfrexdxy 2335 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 2338* Not-free for restricted universal quantification where x and y are distinct. See nfralya 2340 for a version with y and A distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
xA    &   xφ       xy A φ

Theoremnfrexxy 2339* Not-free for restricted existential quantification where x and y are distinct. See nfrexya 2341 for a version with y and A distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
xA    &   xφ       xy A φ

Theoremnfralya 2340* Not-free for restricted universal quantification where y and A are distinct. See nfralxy 2338 for a version with x and y distinct instead. (Contributed by Jim Kingdon, 3-Jun-2018.)
xA    &   xφ       xy A φ

Theoremnfrexya 2341* Not-free for restricted existential quantification where y and A are distinct. See nfrexxy 2339 for a version with x and y distinct instead. (Contributed by Jim Kingdon, 3-Jun-2018.)
xA    &   xφ       xy A φ

Theoremnfra2xy 2342* Not-free given two restricted quantifiers. (Contributed by Jim Kingdon, 20-Aug-2018.)
yx A y B φ

Theoremnfre1 2343 x is not free in x Aφ. (Contributed by NM, 19-Mar-1997.) (Revised by Mario Carneiro, 7-Oct-2016.)
xx A φ

Theoremr3al 2344* 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 2345 Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)
(xφx A φ)

Theoremrexex 2346 Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)
(x A φxφ)

Theoremrsp 2347 Restricted specialization. (Contributed by NM, 17-Oct-1996.)
(x A φ → (x Aφ))

Theoremrspe 2348 Restricted specialization. (Contributed by NM, 12-Oct-1999.)
((x A φ) → x A φ)

Theoremrsp2 2349 Restricted specialization. (Contributed by NM, 11-Feb-1997.)
(x A y B φ → ((x A y B) → φ))

Theoremrsp2e 2350 Restricted specialization. (Contributed by FL, 4-Jun-2012.)
((x A y B φ) → x A y B φ)

Theoremrspec 2351 Specialization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
x A φ       (x Aφ)

Theoremrgen 2352 Generalization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
(x Aφ)       x A φ

Theoremrgen2a 2353* Generalization rule for restricted quantification. Note that x and y needn't be distinct (and illustrates the use of dvelimor 1876). (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 2354 Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)
φ       x A φ

Theoremrgen2w 2355 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 2356 Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.)
(x A φψ)    &   (x Aφ)       ψ

Theoremmprgbir 2357 Modus ponens on biconditional combined with restricted generalization. (Contributed by NM, 21-Mar-2004.)
(φx A ψ)    &   (x Aψ)       φ

Theoremralim 2358 Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997.)
(x A (φψ) → (x A φx A ψ))

Theoremralimi2 2359 Inference quantifying both antecedent and consequent. (Contributed by NM, 22-Feb-2004.)
((x Aφ) → (x Bψ))       (x A φx B ψ)

Theoremralimia 2360 Inference quantifying both antecedent and consequent. (Contributed by NM, 19-Jul-1996.)
(x A → (φψ))       (x A φx A ψ)

Theoremralimiaa 2361 Inference quantifying both antecedent and consequent. (Contributed by NM, 4-Aug-2007.)
((x A φ) → ψ)       (x A φx A ψ)

Theoremralimi 2362 Inference quantifying both antecedent and consequent, with strong hypothesis. (Contributed by NM, 4-Mar-1997.)
(φψ)       (x A φx A ψ)

Theoremral2imi 2363 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 2364 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 2365* 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 2366* 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 2367* Inference quantifying both antecedent and consequent. (Contributed by NM, 1-Feb-2005.)
(φ → ((x Aψ) → (x Bχ)))       (φ → (x A ψx B χ))

Theoremralrimi 2368 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 2369* 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 2370* 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 2371* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)
(φψ)       (φx A ψ)

Theoremr19.21t 2372 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 2373 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 2374* 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 2375 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 2376* 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 2377* 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 2378* 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 2379* 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 2380* 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 2381* 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 2382* 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 2383* Generalization rule for restricted quantification. (Contributed by NM, 30-May-1999.)
((x A y B) → φ)       x A y B φ

Theoremrgen3 2384* 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

Theoremr19.21bi 2385 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.)
(φx A ψ)       ((φ x A) → ψ)

Theoremrspec2 2386 Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.)
x A y B φ       ((x A y B) → φ)

Theoremrspec3 2387 Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.)
x A y B z 𝐶 φ       ((x A y B z 𝐶) → φ)

Theoremr19.21be 2388 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 21-Nov-1994.)
(φx A ψ)       x A (φψ)

Theoremnrex 2389 Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
(x A → ¬ ψ)        ¬ x A ψ

Theoremnrexdv 2390* Deduction adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
((φ x A) → ¬ ψ)       (φ → ¬ x A ψ)

Theoremrexim 2391 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 (φψ) → (x A φx A ψ))

Theoremreximia 2392 Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.)
(x A → (φψ))       (x A φx A ψ)

Theoremreximi2 2393 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 φx B ψ)

Theoremreximi 2394 Inference quantifying both antecedent and consequent. (Contributed by NM, 18-Oct-1996.)
(φψ)       (x A φx A ψ)

Theoremreximdai 2395 Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 31-Aug-1999.)
xφ    &   (φ → (x A → (ψχ)))       (φ → (x A ψx A χ))

Theoremreximdv2 2396* 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 ψx B χ))

Theoremreximdvai 2397* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 14-Nov-2002.)
(φ → (x A → (ψχ)))       (φ → (x A ψx A χ))

Theoremreximdv 2398* Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version with strong hypothesis.) (Contributed by NM, 24-Jun-1998.)
(φ → (ψχ))       (φ → (x A ψx A χ))

Theoremreximdva 2399* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.)
((φ x A) → (ψχ))       (φ → (x A ψx A χ))

Theoremr19.12 2400* Theorem 19.12 of [Margaris] p. 89 with restricted quantifiers. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
(x A y B φy B x A φ)

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