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Theorem List for Intuitionistic Logic Explorer - 2401-2500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnfne 2401 Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥 𝐴𝐵
 
Theoremnfned 2402 Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴𝐵)
 
2.1.4.2  Negated membership
 
Syntaxwnel 2403 Extend wff notation to include negated membership.
wff 𝐴𝐵
 
Definitiondf-nel 2404 Define negated membership. (Contributed by NM, 7-Aug-1994.)
(𝐴𝐵 ↔ ¬ 𝐴𝐵)
 
Theoremneli 2405 Inference associated with df-nel 2404. (Contributed by BJ, 7-Jul-2018.)
𝐴𝐵        ¬ 𝐴𝐵
 
Theoremnelir 2406 Inference associated with df-nel 2404. (Contributed by BJ, 7-Jul-2018.)
¬ 𝐴𝐵       𝐴𝐵
 
Theoremneleq1 2407 Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremneleq2 2408 Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremneleq12d 2409 Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶𝐵𝐷))
 
Theoremnfnel 2410 Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥 𝐴𝐵
 
Theoremnfneld 2411 Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴𝐵)
 
Theoremelnelne1 2412 Two classes are different if they don't contain the same element. (Contributed by AV, 28-Jan-2020.)
((𝐴𝐵𝐴𝐶) → 𝐵𝐶)
 
Theoremelnelne2 2413 Two classes are different if they don't belong to the same class. (Contributed by AV, 28-Jan-2020.)
((𝐴𝐶𝐵𝐶) → 𝐴𝐵)
 
Theoremnelcon3d 2414 Contrapositive law deduction for negated membership. (Contributed by AV, 28-Jan-2020.)
(𝜑 → (𝐴𝐵𝐶𝐷))       (𝜑 → (𝐶𝐷𝐴𝐵))
 
Theoremelnelall 2415 A contradiction concerning membership implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
(𝐴𝐵 → (𝐴𝐵𝜑))
 
2.1.5  Restricted quantification
 
Syntaxwral 2416 Extend wff notation to include restricted universal quantification.
wff 𝑥𝐴 𝜑
 
Syntaxwrex 2417 Extend wff notation to include restricted existential quantification.
wff 𝑥𝐴 𝜑
 
Syntaxwreu 2418 Extend wff notation to include restricted existential uniqueness.
wff ∃!𝑥𝐴 𝜑
 
Syntaxwrmo 2419 Extend wff notation to include restricted "at most one."
wff ∃*𝑥𝐴 𝜑
 
Syntaxcrab 2420 Extend class notation to include the restricted class abstraction (class builder).
class {𝑥𝐴𝜑}
 
Definitiondf-ral 2421 Define restricted universal quantification. Special case of Definition 4.15(3) of [TakeutiZaring] p. 22. (Contributed by NM, 19-Aug-1993.)
(∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
 
Definitiondf-rex 2422 Define restricted existential quantification. Special case of Definition 4.15(4) of [TakeutiZaring] p. 22. (Contributed by NM, 30-Aug-1993.)
(∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
 
Definitiondf-reu 2423 Define restricted existential uniqueness. (Contributed by NM, 22-Nov-1994.)
(∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
 
Definitiondf-rmo 2424 Define restricted "at most one". (Contributed by NM, 16-Jun-2017.)
(∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
 
Definitiondf-rab 2425 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.)
{𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
 
Theoremralnex 2426 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)
(∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴 𝜑)
 
Theoremrexnalim 2427 Relationship between restricted universal and existential quantifiers. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 17-Aug-2018.)
(∃𝑥𝐴 ¬ 𝜑 → ¬ ∀𝑥𝐴 𝜑)
 
Theoremdfrex2dc 2428 Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 29-Jun-2022.)
(DECID𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜑))
 
Theoremralexim 2429 Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 17-Aug-2018.)
(∀𝑥𝐴 𝜑 → ¬ ∃𝑥𝐴 ¬ 𝜑)
 
Theoremrexalim 2430 Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 17-Aug-2018.)
(∃𝑥𝐴 𝜑 → ¬ ∀𝑥𝐴 ¬ 𝜑)
 
Theoremralbida 2431 Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Oct-2003.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))
 
Theoremrexbida 2432 Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 6-Oct-2003.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
 
Theoremralbidva 2433* Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 4-Mar-1997.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))
 
Theoremrexbidva 2434* Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 9-Mar-1997.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
 
Theoremralbid 2435 Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 27-Jun-1998.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))
 
Theoremrexbid 2436 Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 27-Jun-1998.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
 
Theoremralbidv 2437* Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 20-Nov-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))
 
Theoremrexbidv 2438* Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 20-Nov-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
 
Theoremralbidv2 2439* Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Apr-1997.)
(𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
 
Theoremrexbidv2 2440* Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 22-May-1999.)
(𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))
 
Theoremralbii 2441 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)       (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜓)
 
Theoremrexbii 2442 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)       (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 𝜓)
 
Theorem2ralbii 2443 Inference adding two restricted universal quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
(𝜑𝜓)       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝐴𝑦𝐵 𝜓)
 
Theorem2rexbii 2444 Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 11-Nov-1995.)
(𝜑𝜓)       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓)
 
Theoremralbii2 2445 Inference adding different restricted universal quantifiers to each side of an equivalence. (Contributed by NM, 15-Aug-2005.)
((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓)
 
Theoremrexbii2 2446 Inference adding different restricted existential quantifiers to each side of an equivalence. (Contributed by NM, 4-Feb-2004.)
((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜓)
 
Theoremraleqbii 2447 Equality deduction for restricted universal quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)
𝐴 = 𝐵    &   (𝜓𝜒)       (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒)
 
Theoremrexeqbii 2448 Equality deduction for restricted existential quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)
𝐴 = 𝐵    &   (𝜓𝜒)       (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒)
 
Theoremralbiia 2449 Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 26-Nov-2000.)
(𝑥𝐴 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜓)
 
Theoremrexbiia 2450 Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 26-Oct-1999.)
(𝑥𝐴 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 𝜓)
 
Theorem2rexbiia 2451* Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
((𝑥𝐴𝑦𝐵) → (𝜑𝜓))       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓)
 
Theoremr2alf 2452* Double restricted universal quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑦𝐴       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
 
Theoremr2exf 2453* Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑦𝐴       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
 
Theoremr2al 2454* Double restricted universal quantification. (Contributed by NM, 19-Nov-1995.)
(∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
 
Theoremr2ex 2455* Double restricted existential quantification. (Contributed by NM, 11-Nov-1995.)
(∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
 
Theorem2ralbida 2456* Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 24-Feb-2004.)
𝑥𝜑    &   𝑦𝜑    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 ↔ ∀𝑥𝐴𝑦𝐵 𝜒))
 
Theorem2ralbidva 2457* Formula-building rule for restricted universal quantifiers (deduction form). (Contributed by NM, 4-Mar-1997.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 ↔ ∀𝑥𝐴𝑦𝐵 𝜒))
 
Theorem2rexbidva 2458* Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 15-Dec-2004.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))
 
Theorem2ralbidv 2459* Formula-building rule for restricted universal quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.) (Revised by Szymon Jaroszewicz, 16-Mar-2007.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 ↔ ∀𝑥𝐴𝑦𝐵 𝜒))
 
Theorem2rexbidv 2460* Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))
 
Theoremrexralbidv 2461* Formula-building rule for restricted quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))
 
Theoremralinexa 2462 A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)
(∀𝑥𝐴 (𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥𝐴 (𝜑𝜓))
 
Theoremrisset 2463* Two ways to say "𝐴 belongs to 𝐵." (Contributed by NM, 22-Nov-1994.)
(𝐴𝐵 ↔ ∃𝑥𝐵 𝑥 = 𝐴)
 
Theoremhbral 2464 Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by David Abernethy, 13-Dec-2009.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)    &   (𝜑 → ∀𝑥𝜑)       (∀𝑦𝐴 𝜑 → ∀𝑥𝑦𝐴 𝜑)
 
Theoremhbra1 2465 𝑥 is not free in 𝑥𝐴𝜑. (Contributed by NM, 18-Oct-1996.)
(∀𝑥𝐴 𝜑 → ∀𝑥𝑥𝐴 𝜑)
 
Theoremnfra1 2466 𝑥 is not free in 𝑥𝐴𝜑. (Contributed by NM, 18-Oct-1996.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝑥𝐴 𝜑
 
Theoremnfraldxy 2467* Not-free for restricted universal quantification where 𝑥 and 𝑦 are distinct. See nfraldya 2469 for a version with 𝑦 and 𝐴 distinct instead. (Contributed by Jim Kingdon, 29-May-2018.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
 
Theoremnfrexdxy 2468* Not-free for restricted existential quantification where 𝑥 and 𝑦 are distinct. See nfrexdya 2470 for a version with 𝑦 and 𝐴 distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
 
Theoremnfraldya 2469* Not-free for restricted universal quantification where 𝑦 and 𝐴 are distinct. See nfraldxy 2467 for a version with 𝑥 and 𝑦 distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
 
Theoremnfrexdya 2470* Not-free for restricted existential quantification where 𝑦 and 𝐴 are distinct. See nfrexdxy 2468 for a version with 𝑥 and 𝑦 distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
 
Theoremnfralxy 2471* Not-free for restricted universal quantification where 𝑥 and 𝑦 are distinct. See nfralya 2473 for a version with 𝑦 and 𝐴 distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
𝑥𝐴    &   𝑥𝜑       𝑥𝑦𝐴 𝜑
 
Theoremnfrexxy 2472* Not-free for restricted existential quantification where 𝑥 and 𝑦 are distinct. See nfrexya 2474 for a version with 𝑦 and 𝐴 distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
𝑥𝐴    &   𝑥𝜑       𝑥𝑦𝐴 𝜑
 
Theoremnfralya 2473* Not-free for restricted universal quantification where 𝑦 and 𝐴 are distinct. See nfralxy 2471 for a version with 𝑥 and 𝑦 distinct instead. (Contributed by Jim Kingdon, 3-Jun-2018.)
𝑥𝐴    &   𝑥𝜑       𝑥𝑦𝐴 𝜑
 
Theoremnfrexya 2474* Not-free for restricted existential quantification where 𝑦 and 𝐴 are distinct. See nfrexxy 2472 for a version with 𝑥 and 𝑦 distinct instead. (Contributed by Jim Kingdon, 3-Jun-2018.)
𝑥𝐴    &   𝑥𝜑       𝑥𝑦𝐴 𝜑
 
Theoremnfra2xy 2475* Not-free given two restricted quantifiers. (Contributed by Jim Kingdon, 20-Aug-2018.)
𝑦𝑥𝐴𝑦𝐵 𝜑
 
Theoremnfre1 2476 𝑥 is not free in 𝑥𝐴𝜑. (Contributed by NM, 19-Mar-1997.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝑥𝐴 𝜑
 
Theoremr3al 2477* Triple restricted universal quantification. (Contributed by NM, 19-Nov-1995.)
(∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑))
 
Theoremalral 2478 Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)
(∀𝑥𝜑 → ∀𝑥𝐴 𝜑)
 
Theoremrexex 2479 Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)
(∃𝑥𝐴 𝜑 → ∃𝑥𝜑)
 
Theoremrsp 2480 Restricted specialization. (Contributed by NM, 17-Oct-1996.)
(∀𝑥𝐴 𝜑 → (𝑥𝐴𝜑))
 
Theoremrspe 2481 Restricted specialization. (Contributed by NM, 12-Oct-1999.)
((𝑥𝐴𝜑) → ∃𝑥𝐴 𝜑)
 
Theoremrsp2 2482 Restricted specialization. (Contributed by NM, 11-Feb-1997.)
(∀𝑥𝐴𝑦𝐵 𝜑 → ((𝑥𝐴𝑦𝐵) → 𝜑))
 
Theoremrsp2e 2483 Restricted specialization. (Contributed by FL, 4-Jun-2012.)
((𝑥𝐴𝑦𝐵𝜑) → ∃𝑥𝐴𝑦𝐵 𝜑)
 
Theoremrspec 2484 Specialization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
𝑥𝐴 𝜑       (𝑥𝐴𝜑)
 
Theoremrgen 2485 Generalization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
(𝑥𝐴𝜑)       𝑥𝐴 𝜑
 
Theoremrgen2a 2486* Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 are not required to be disjoint. This proof illustrates the use of dvelim 1992. Usage of rgen2 2518 instead is highly encouraged. (Contributed by NM, 23-Nov-1994.) (Proof rewritten by Jim Kingdon, 1-Jun-2018.) (New usage is discouraged.)
((𝑥𝐴𝑦𝐴) → 𝜑)       𝑥𝐴𝑦𝐴 𝜑
 
Theoremrgenw 2487 Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)
𝜑       𝑥𝐴 𝜑
 
Theoremrgen2w 2488 Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by NM, 18-Jun-2014.)
𝜑       𝑥𝐴𝑦𝐵 𝜑
 
Theoremmprg 2489 Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.)
(∀𝑥𝐴 𝜑𝜓)    &   (𝑥𝐴𝜑)       𝜓
 
Theoremmprgbir 2490 Modus ponens on biconditional combined with restricted generalization. (Contributed by NM, 21-Mar-2004.)
(𝜑 ↔ ∀𝑥𝐴 𝜓)    &   (𝑥𝐴𝜓)       𝜑
 
Theoremralim 2491 Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997.)
(∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓))
 
Theoremralimi2 2492 Inference quantifying both antecedent and consequent. (Contributed by NM, 22-Feb-2004.)
((𝑥𝐴𝜑) → (𝑥𝐵𝜓))       (∀𝑥𝐴 𝜑 → ∀𝑥𝐵 𝜓)
 
Theoremralimia 2493 Inference quantifying both antecedent and consequent. (Contributed by NM, 19-Jul-1996.)
(𝑥𝐴 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓)
 
Theoremralimiaa 2494 Inference quantifying both antecedent and consequent. (Contributed by NM, 4-Aug-2007.)
((𝑥𝐴𝜑) → 𝜓)       (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓)
 
Theoremralimi 2495 Inference quantifying both antecedent and consequent, with strong hypothesis. (Contributed by NM, 4-Mar-1997.)
(𝜑𝜓)       (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓)
 
Theorem2ralimi 2496 Inference quantifying both antecedent and consequent two times, with strong hypothesis. (Contributed by AV, 3-Dec-2021.)
(𝜑𝜓)       (∀𝑥𝐴𝑦𝐵 𝜑 → ∀𝑥𝐴𝑦𝐵 𝜓)
 
Theoremral2imi 2497 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 2498 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 2499* 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 2500* 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))
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