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

Theoremr19.23t 2401 Closed theorem form of r19.23 2402. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
(Ⅎxψ → (x A (φψ) ↔ (x A φψ)))

Theoremr19.23 2402 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       (x A (φψ) ↔ (x A φψ))

Theoremr19.23v 2403* Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)
(x A (φψ) ↔ (x A φψ))

Theoremrexlimi 2404 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    &   (x A → (φψ))       (x A φψ)

Theoremrexlimiv 2405* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.)
(x A → (φψ))       (x A φψ)

Theoremrexlimiva 2406* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Dec-2006.)
((x A φ) → ψ)       (x A φψ)

Theoremrexlimivw 2407* Weaker version of rexlimiv 2405. (Contributed by FL, 19-Sep-2011.)
(φψ)       (x A φψ)

Theoremrexlimd 2408 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 → (x A → (ψχ)))       (φ → (x A ψχ))

Theoremrexlimd2 2409 Version of rexlimd 2408 with deduction version of second hypothesis. (Contributed by NM, 21-Jul-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
xφ    &   (φ → Ⅎxχ)    &   (φ → (x A → (ψχ)))       (φ → (x A ψχ))

Theoremrexlimdv 2410* 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 → (x A → (ψχ)))       (φ → (x A ψχ))

Theoremrexlimdva 2411* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 20-Jan-2007.)
((φ x A) → (ψχ))       (φ → (x A ψχ))

Theoremrexlimdvaa 2412* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Mario Carneiro, 15-Jun-2016.)
((φ (x A ψ)) → χ)       (φ → (x A ψχ))

Theoremrexlimdv3a 2413* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 2410. (Contributed by NM, 7-Jun-2015.)
((φ x A ψ) → χ)       (φ → (x A ψχ))

Theoremrexlimdvw 2414* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Jun-2014.)
(φ → (ψχ))       (φ → (x A ψχ))

Theoremrexlimddv 2415* Restricted existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
(φx A ψ)    &   ((φ (x A ψ)) → χ)       (φχ)

Theoremrexlimivv 2416* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 17-Feb-2004.)
((x A y B) → (φψ))       (x A y B φψ)

Theoremrexlimdvv 2417* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Jul-2004.)
(φ → ((x A y B) → (ψχ)))       (φ → (x A y B ψχ))

Theoremrexlimdvva 2418* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)
((φ (x A y B)) → (ψχ))       (φ → (x A y B ψχ))

Theoremr19.26 2419 Theorem 19.26 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 28-Jan-1997.) (Proof shortened by Andrew Salmon, 30-May-2011.)
(x A (φ ψ) ↔ (x A φ x A ψ))

Theoremr19.26-2 2420 Theorem 19.26 of [Margaris] p. 90 with 2 restricted quantifiers. (Contributed by NM, 10-Aug-2004.)
(x A y B (φ ψ) ↔ (x A y B φ x A y B ψ))

Theoremr19.26-3 2421 Theorem 19.26 of [Margaris] p. 90 with 3 restricted quantifiers. (Contributed by FL, 22-Nov-2010.)
(x A (φ ψ χ) ↔ (x A φ x A ψ x A χ))

Theoremr19.26m 2422 Theorem 19.26 of [Margaris] p. 90 with mixed quantifiers. (Contributed by NM, 22-Feb-2004.)
(x((x Aφ) (x Bψ)) ↔ (x A φ x B ψ))

Theoremralbi 2423 Distribute a restricted universal quantifier over a biconditional. Theorem 19.15 of [Margaris] p. 90 with restricted quantification. (Contributed by NM, 6-Oct-2003.)
(x A (φψ) → (x A φx A ψ))

Theoremrexbi 2424 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.)
(x A (φψ) → (x A φx A ψ))

Theoremralbiim 2425 Split a biconditional and distribute quantifier. (Contributed by NM, 3-Jun-2012.)
(x A (φψ) ↔ (x A (φψ) x A (ψφ)))

Theoremr19.27av 2426* 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.)
((x A φ ψ) → x A (φ ψ))

Theoremr19.28av 2427* 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 x A ψ) → x A (φ ψ))

Theoremr19.29 2428 Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.) (Proof shortened by Andrew Salmon, 30-May-2011.)
((x A φ x A ψ) → x A (φ ψ))

Theoremr19.29r 2429 Variation of Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)
((x A φ x A ψ) → x A (φ ψ))

Theoremr19.29af2 2430 A commonly used pattern based on r19.29 2428 (Contributed by Thierry Arnoux, 17-Dec-2017.)
xφ    &   xχ    &   (((φ x A) ψ) → χ)    &   (φx A ψ)       (φχ)

Theoremr19.29af 2431* A commonly used pattern based on r19.29 2428 (Contributed by Thierry Arnoux, 29-Nov-2017.)
xφ    &   (((φ x A) ψ) → χ)    &   (φx A ψ)       (φχ)

Theoremr19.29a 2432* A commonly used pattern based on r19.29 2428 (Contributed by Thierry Arnoux, 22-Nov-2017.)
(((φ x A) ψ) → χ)    &   (φx A ψ)       (φχ)

Theoremr19.29d2r 2433 Theorem 19.29 of [Margaris] p. 90 with two restricted quantifiers, deduction version (Contributed by Thierry Arnoux, 30-Jan-2017.)
(φx A y B ψ)    &   (φx A y B χ)       (φx A y B (ψ χ))

Theoremr19.29_2a 2434* A commonly used pattern based on r19.29 2428, version with two restricted quantifiers (Contributed by Thierry Arnoux, 26-Nov-2017.)
((((φ x A) y B) ψ) → χ)    &   (φx A y B ψ)       (φχ)

Theoremr19.32r 2435 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 x A ψ) → x A (φ ψ))

Theoremr19.32vr 2436* 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 2437. (Contributed by Jim Kingdon, 19-Aug-2018.)
((φ x A ψ) → x A (φ ψ))

Theoremr19.32vdc 2437* Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, where φ is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)
(DECID φ → (x A (φ ψ) ↔ (φ x A ψ)))

Theoremr19.35-1 2438 Restricted quantifier version of 19.35-1 1497. (Contributed by Jim Kingdon, 4-Jun-2018.)
(x A (φψ) → (x A φx A ψ))

Theoremr19.36av 2439* 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.)
(x A (φψ) → (x A φψ))

Theoremr19.37 2440 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       (x A (φψ) → (φx A ψ))

Theoremr19.37av 2441* Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.)
(x A (φψ) → (φx A ψ))

Theoremr19.40 2442 Restricted quantifier version of Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.)
(x A (φ ψ) → (x A φ x A ψ))

Theoremr19.41 2443 Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 1-Nov-2010.)
xψ       (x A (φ ψ) ↔ (x A φ ψ))

Theoremr19.41v 2444* Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 17-Dec-2003.)
(x A (φ ψ) ↔ (x A φ ψ))

Theoremr19.42v 2445* Restricted version of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
(x A (φ ψ) ↔ (φ x A ψ))

Theoremr19.43 2446 Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) (Proof rewritten by Jim Kingdon, 5-Jun-2018.)
(x A (φ ψ) ↔ (x A φ x A ψ))

Theoremr19.44av 2447* 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.)
(x A (φ ψ) → (x A φ ψ))

Theoremr19.45av 2448* 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.)
(x A (φ ψ) → (φ x A ψ))

Theoremralcomf 2449* Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
yA    &   xB       (x A y B φy B x A φ)

Theoremrexcomf 2450* Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
yA    &   xB       (x A y B φy B x A φ)

Theoremralcom 2451* Commutation of restricted quantifiers. (Contributed by NM, 13-Oct-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
(x A y B φy B x A φ)

Theoremrexcom 2452* Commutation of restricted quantifiers. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
(x A y B φy B x A φ)

Theoremrexcom13 2453* Swap 1st and 3rd restricted existential quantifiers. (Contributed by NM, 8-Apr-2015.)
(x A y B z 𝐶 φz 𝐶 y B x A φ)

Theoremrexrot4 2454* Rotate existential restricted quantifiers twice. (Contributed by NM, 8-Apr-2015.)
(x A y B z 𝐶 w 𝐷 φz 𝐶 w 𝐷 x A y B φ)

Theoremralcom3 2455 A commutative law for restricted quantifiers that swaps the domain of the restriction. (Contributed by NM, 22-Feb-2004.)
(x A (x Bφ) ↔ x B (x Aφ))

Theoremreean 2456* Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Andrew Salmon, 30-May-2011.)
yφ    &   xψ       (x A y B (φ ψ) ↔ (x A φ y B ψ))

Theoremreeanv 2457* Rearrange existential quantifiers. (Contributed by NM, 9-May-1999.)
(x A y B (φ ψ) ↔ (x A φ y B ψ))

Theorem3reeanv 2458* Rearrange three existential quantifiers. (Contributed by Jeff Madsen, 11-Jun-2010.)
(x A y B z 𝐶 (φ ψ χ) ↔ (x A φ y B ψ z 𝐶 χ))

Theoremnfreu1 2459 x is not free in ∃!x Aφ. (Contributed by NM, 19-Mar-1997.)
x∃!x A φ

Theoremnfrmo1 2460 x is not free in ∃*x Aφ. (Contributed by NM, 16-Jun-2017.)
x∃*x A φ

Theoremnfreudxy 2461* 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∃!y A ψ)

Theoremnfreuxy 2462* Not-free for restricted uniqueness. This is a version where x and y are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.)
xA    &   xφ       x∃!y A φ

Theoremrabid 2463 An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of [Quine] p. 16. (Contributed by NM, 9-Oct-2003.)
(x {x Aφ} ↔ (x A φ))

Theoremrabid2 2464* An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
(A = {x Aφ} ↔ x A φ)

Theoremrabbi 2465 Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2526. (Contributed by NM, 25-Nov-2013.)
(x A (ψχ) ↔ {x Aψ} = {x Aχ})

Theoremrabswap 2466 Swap with a membership relation in a restricted class abstraction. (Contributed by NM, 4-Jul-2005.)
{x Ax B} = {x Bx A}

Theoremnfrab1 2467 The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
x{x Aφ}

Theoremnfrabxy 2468* A variable not free in a wff remains so in a restricted class abstraction. (Contributed by Jim Kingdon, 19-Jul-2018.)
xφ    &   xA       x{y Aφ}

Theoremreubida 2469 Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by Mario Carneiro, 19-Nov-2016.)
xφ    &   ((φ x A) → (ψχ))       (φ → (∃!x A ψ∃!x A χ))

Theoremreubidva 2470* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 13-Nov-2004.)
((φ x A) → (ψχ))       (φ → (∃!x A ψ∃!x A χ))

Theoremreubidv 2471* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 17-Oct-1996.)
(φ → (ψχ))       (φ → (∃!x A ψ∃!x A χ))

Theoremreubiia 2472 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 14-Nov-2004.)
(x A → (φψ))       (∃!x A φ∃!x A ψ)

Theoremreubii 2473 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 22-Oct-1999.)
(φψ)       (∃!x A φ∃!x A ψ)

Theoremrmobida 2474 Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
xφ    &   ((φ x A) → (ψχ))       (φ → (∃*x A ψ∃*x A χ))

Theoremrmobidva 2475* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
((φ x A) → (ψχ))       (φ → (∃*x A ψ∃*x A χ))

Theoremrmobidv 2476* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
(φ → (ψχ))       (φ → (∃*x A ψ∃*x A χ))

Theoremrmobiia 2477 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.)
(x A → (φψ))       (∃*x A φ∃*x A ψ)

Theoremrmobii 2478 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.)
(φψ)       (∃*x A φ∃*x A ψ)

Theoremraleqf 2479 Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
xA    &   xB       (A = B → (x A φx B φ))

Theoremrexeqf 2480 Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.)
xA    &   xB       (A = B → (x A φx B φ))

Theoremreueq1f 2481 Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
xA    &   xB       (A = B → (∃!x A φ∃!x B φ))

Theoremrmoeq1f 2482 Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
xA    &   xB       (A = B → (∃*x A φ∃*x B φ))

Theoremraleq 2483* Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
(A = B → (x A φx B φ))

Theoremrexeq 2484* Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.)
(A = B → (x A φx B φ))

Theoremreueq1 2485* Equality theorem for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
(A = B → (∃!x A φ∃!x B φ))

Theoremrmoeq1 2486* Equality theorem for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(A = B → (∃*x A φ∃*x B φ))

Theoremraleqi 2487* Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.)
A = B       (x A φx B φ)

Theoremrexeqi 2488* Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.)
A = B       (x A φx B φ)

Theoremraleqdv 2489* Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.)
(φA = B)       (φ → (x A ψx B ψ))

Theoremrexeqdv 2490* Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)
(φA = B)       (φ → (x A ψx B ψ))

Theoremraleqbi1dv 2491* Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
(A = B → (φψ))       (A = B → (x A φx B ψ))

Theoremrexeqbi1dv 2492* Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.)
(A = B → (φψ))       (A = B → (x A φx B ψ))

Theoremreueqd 2493* Equality deduction for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
(A = B → (φψ))       (A = B → (∃!x A φ∃!x B ψ))

Theoremrmoeqd 2494* Equality deduction for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(A = B → (φψ))       (A = B → (∃*x A φ∃*x B ψ))

Theoremraleqbidv 2495* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
(φA = B)    &   (φ → (ψχ))       (φ → (x A ψx B χ))

Theoremrexeqbidv 2496* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
(φA = B)    &   (φ → (ψχ))       (φ → (x A ψx B χ))

Theoremraleqbidva 2497* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
(φA = B)    &   ((φ x A) → (ψχ))       (φ → (x A ψx B χ))

Theoremrexeqbidva 2498* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
(φA = B)    &   ((φ x A) → (ψχ))       (φ → (x A ψx B χ))

Theoremmormo 2499 Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.)
(∃*xφ∃*x A φ)

Theoremreu5 2500 Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.)
(∃!x A φ ↔ (x A φ ∃*x A φ))

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