Theorem List for Intuitionistic Logic Explorer - 2401-2500 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | ralrimivva 2401* |
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) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓) |
|
Theorem | ralrimivvva 2402* |
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) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓) |
|
Theorem | ralrimdvv 2403* |
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 → (𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)) |
|
Theorem | ralrimdvva 2404* |
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)) → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)) |
|
Theorem | rgen2 2405* |
Generalization rule for restricted quantification. (Contributed by NM,
30-May-1999.)
|
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑) ⇒ ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 |
|
Theorem | rgen3 2406* |
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 |
|
Theorem | r19.21bi 2407 |
Inference from Theorem 19.21 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 20-Nov-1994.)
|
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓) |
|
Theorem | rspec2 2408 |
Specialization rule for restricted quantification. (Contributed by NM,
20-Nov-1994.)
|
⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ⇒ ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑) |
|
Theorem | rspec3 2409 |
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) |
|
Theorem | r19.21be 2410 |
Inference from Theorem 19.21 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 21-Nov-1994.)
|
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) ⇒ ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) |
|
Theorem | nrex 2411 |
Inference adding restricted existential quantifier to negated wff.
(Contributed by NM, 16-Oct-2003.)
|
⊢ (𝑥 ∈ 𝐴 → ¬ 𝜓) ⇒ ⊢ ¬ ∃𝑥 ∈ 𝐴 𝜓 |
|
Theorem | nrexdv 2412* |
Deduction adding restricted existential quantifier to negated wff.
(Contributed by NM, 16-Oct-2003.)
|
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ 𝜓) ⇒ ⊢ (𝜑 → ¬ ∃𝑥 ∈ 𝐴 𝜓) |
|
Theorem | rexim 2413 |
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 (𝜑 → 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) |
|
Theorem | reximia 2414 |
Inference quantifying both antecedent and consequent. (Contributed by
NM, 10-Feb-1997.)
|
⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) |
|
Theorem | reximi2 2415 |
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 𝜑 → ∃𝑥 ∈ 𝐵 𝜓) |
|
Theorem | reximi 2416 |
Inference quantifying both antecedent and consequent. (Contributed by
NM, 18-Oct-1996.)
|
⊢ (𝜑 → 𝜓) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) |
|
Theorem | reximdai 2417 |
Deduction from Theorem 19.22 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 31-Aug-1999.)
|
⊢ Ⅎ𝑥𝜑
& ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒)) |
|
Theorem | reximdv2 2418* |
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 𝜓 → ∃𝑥 ∈ 𝐵 𝜒)) |
|
Theorem | reximdvai 2419* |
Deduction quantifying both antecedent and consequent, based on Theorem
19.22 of [Margaris] p. 90.
(Contributed by NM, 14-Nov-2002.)
|
⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒)) |
|
Theorem | reximdv 2420* |
Deduction from Theorem 19.22 of [Margaris] p.
90. (Restricted
quantifier version with strong hypothesis.) (Contributed by NM,
24-Jun-1998.)
|
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒)) |
|
Theorem | reximdva 2421* |
Deduction quantifying both antecedent and consequent, based on Theorem
19.22 of [Margaris] p. 90.
(Contributed by NM, 22-May-1999.)
|
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒)) |
|
Theorem | r19.12 2422* |
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 𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
|
Theorem | r19.23t 2423 |
Closed theorem form of r19.23 2424. (Contributed by NM, 4-Mar-2013.)
(Revised by Mario Carneiro, 8-Oct-2016.)
|
⊢ (Ⅎ𝑥𝜓 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓))) |
|
Theorem | r19.23 2424 |
Theorem 19.23 of [Margaris] p. 90 with
restricted quantifiers.
(Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro,
8-Oct-2016.)
|
⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)) |
|
Theorem | r19.23v 2425* |
Theorem 19.23 of [Margaris] p. 90 with
restricted quantifiers.
(Contributed by NM, 31-Aug-1999.)
|
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)) |
|
Theorem | rexlimi 2426 |
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.)
|
⊢ Ⅎ𝑥𝜓
& ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓) |
|
Theorem | rexlimiv 2427* |
Inference from Theorem 19.23 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 20-Nov-1994.)
|
⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓) |
|
Theorem | rexlimiva 2428* |
Inference from Theorem 19.23 of [Margaris] p.
90 (restricted quantifier
version). (Contributed by NM, 18-Dec-2006.)
|
⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓) |
|
Theorem | rexlimivw 2429* |
Weaker version of rexlimiv 2427. (Contributed by FL, 19-Sep-2011.)
|
⊢ (𝜑 → 𝜓) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓) |
|
Theorem | rexlimd 2430 |
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.)
|
⊢ Ⅎ𝑥𝜑
& ⊢ Ⅎ𝑥𝜒
& ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
|
Theorem | rexlimd2 2431 |
Version of rexlimd 2430 with deduction version of second hypothesis.
(Contributed by NM, 21-Jul-2013.) (Revised by Mario Carneiro,
8-Oct-2016.)
|
⊢ Ⅎ𝑥𝜑
& ⊢ (𝜑 → Ⅎ𝑥𝜒)
& ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
|
Theorem | rexlimdv 2432* |
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 → (𝜓 → 𝜒))) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
|
Theorem | rexlimdva 2433* |
Inference from Theorem 19.23 of [Margaris] p.
90 (restricted quantifier
version). (Contributed by NM, 20-Jan-2007.)
|
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
|
Theorem | rexlimdvaa 2434* |
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 𝜓 → 𝜒)) |
|
Theorem | rexlimdv3a 2435* |
Inference from Theorem 19.23 of [Margaris] p.
90 (restricted quantifier
version). Frequently-used variant of rexlimdv 2432. (Contributed by NM,
7-Jun-2015.)
|
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓) → 𝜒) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
|
Theorem | rexlimdvw 2436* |
Inference from Theorem 19.23 of [Margaris] p.
90 (restricted quantifier
version). (Contributed by NM, 18-Jun-2014.)
|
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
|
Theorem | rexlimddv 2437* |
Restricted existential elimination rule of natural deduction.
(Contributed by Mario Carneiro, 15-Jun-2016.)
|
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝜒) ⇒ ⊢ (𝜑 → 𝜒) |
|
Theorem | rexlimivv 2438* |
Inference from Theorem 19.23 of [Margaris] p.
90 (restricted quantifier
version). (Contributed by NM, 17-Feb-2004.)
|
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜑 → 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → 𝜓) |
|
Theorem | rexlimdvv 2439* |
Inference from Theorem 19.23 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 22-Jul-2004.)
|
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒))) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → 𝜒)) |
|
Theorem | rexlimdvva 2440* |
Inference from Theorem 19.23 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 18-Jun-2014.)
|
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → 𝜒)) |
|
Theorem | r19.26 2441 |
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)) |
|
Theorem | r19.26-2 2442 |
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)) |
|
Theorem | r19.26-3 2443 |
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)) |
|
Theorem | r19.26m 2444 |
Theorem 19.26 of [Margaris] p. 90 with mixed
quantifiers. (Contributed by
NM, 22-Feb-2004.)
|
⊢ (∀𝑥((𝑥 ∈ 𝐴 → 𝜑) ∧ (𝑥 ∈ 𝐵 → 𝜓)) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜓)) |
|
Theorem | ralbi 2445 |
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 (𝜑 ↔ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)) |
|
Theorem | rexbi 2446 |
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 (𝜑 ↔ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓)) |
|
Theorem | ralbiim 2447 |
Split a biconditional and distribute quantifier. (Contributed by NM,
3-Jun-2012.)
|
⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑))) |
|
Theorem | r19.27av 2448* |
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)) |
|
Theorem | r19.28av 2449* |
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)) |
|
Theorem | r19.29 2450 |
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)) |
|
Theorem | r19.29r 2451 |
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)) |
|
Theorem | r19.29af2 2452 |
A commonly used pattern based on r19.29 2450 (Contributed by Thierry
Arnoux, 17-Dec-2017.)
|
⊢ Ⅎ𝑥𝜑
& ⊢ Ⅎ𝑥𝜒
& ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)
& ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) ⇒ ⊢ (𝜑 → 𝜒) |
|
Theorem | r19.29af 2453* |
A commonly used pattern based on r19.29 2450 (Contributed by Thierry
Arnoux, 29-Nov-2017.)
|
⊢ Ⅎ𝑥𝜑
& ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)
& ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) ⇒ ⊢ (𝜑 → 𝜒) |
|
Theorem | r19.29a 2454* |
A commonly used pattern based on r19.29 2450 (Contributed by Thierry
Arnoux, 22-Nov-2017.)
|
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)
& ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) ⇒ ⊢ (𝜑 → 𝜒) |
|
Theorem | r19.29d2r 2455 |
Theorem 19.29 of [Margaris] p. 90 with two
restricted quantifiers,
deduction version (Contributed by Thierry Arnoux, 30-Jan-2017.)
|
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)
& ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝜒)) |
|
Theorem | r19.29vva 2456* |
A commonly used pattern based on r19.29 2450, version with two restricted
quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.)
|
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → 𝜒)
& ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) ⇒ ⊢ (𝜑 → 𝜒) |
|
Theorem | r19.32r 2457 |
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.)
|
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ ((𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓)) |
|
Theorem | r19.32vr 2458* |
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 2459. (Contributed by Jim Kingdon, 19-Aug-2018.)
|
⊢ ((𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓)) |
|
Theorem | r19.32vdc 2459* |
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))) |
|
Theorem | r19.35-1 2460 |
Restricted quantifier version of 19.35-1 1515. (Contributed by Jim Kingdon,
4-Jun-2018.)
|
⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) |
|
Theorem | r19.36av 2461* |
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 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → 𝜓)) |
|
Theorem | r19.37 2462 |
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.)
|
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) |
|
Theorem | r19.37av 2463* |
Restricted version of one direction of Theorem 19.37 of [Margaris]
p. 90. (Contributed by NM, 2-Apr-2004.)
|
⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) |
|
Theorem | r19.40 2464 |
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)) |
|
Theorem | r19.41 2465 |
Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90.
(Contributed by NM, 1-Nov-2010.)
|
⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)) |
|
Theorem | r19.41v 2466* |
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)) |
|
Theorem | r19.42v 2467* |
Restricted version of Theorem 19.42 of [Margaris] p. 90. (Contributed
by NM, 27-May-1998.)
|
⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓)) |
|
Theorem | r19.43 2468 |
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)) |
|
Theorem | r19.44av 2469* |
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)) |
|
Theorem | r19.45av 2470* |
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)) |
|
Theorem | ralcomf 2471* |
Commutation of restricted quantifiers. (Contributed by Mario Carneiro,
14-Oct-2016.)
|
⊢ Ⅎ𝑦𝐴
& ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑) |
|
Theorem | rexcomf 2472* |
Commutation of restricted quantifiers. (Contributed by Mario Carneiro,
14-Oct-2016.)
|
⊢ Ⅎ𝑦𝐴
& ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
|
Theorem | ralcom 2473* |
Commutation of restricted quantifiers. (Contributed by NM,
13-Oct-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
|
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑) |
|
Theorem | rexcom 2474* |
Commutation of restricted quantifiers. (Contributed by NM,
19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
|
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
|
Theorem | rexcom13 2475* |
Swap 1st and 3rd restricted existential quantifiers. (Contributed by
NM, 8-Apr-2015.)
|
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
|
Theorem | rexrot4 2476* |
Rotate existential restricted quantifiers twice. (Contributed by NM,
8-Apr-2015.)
|
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) |
|
Theorem | ralcom3 2477 |
A commutative law for restricted quantifiers that swaps the domain of
the restriction. (Contributed by NM, 22-Feb-2004.)
|
⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 → 𝜑)) |
|
Theorem | reean 2478* |
Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.)
(Proof shortened by Andrew Salmon, 30-May-2011.)
|
⊢ Ⅎ𝑦𝜑
& ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓)) |
|
Theorem | reeanv 2479* |
Rearrange existential quantifiers. (Contributed by NM, 9-May-1999.)
|
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓)) |
|
Theorem | 3reeanv 2480* |
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)) |
|
Theorem | nfreu1 2481 |
𝑥
is not free in ∃!𝑥 ∈ 𝐴𝜑. (Contributed by NM,
19-Mar-1997.)
|
⊢ Ⅎ𝑥∃!𝑥 ∈ 𝐴 𝜑 |
|
Theorem | nfrmo1 2482 |
𝑥
is not free in ∃*𝑥 ∈ 𝐴𝜑. (Contributed by NM,
16-Jun-2017.)
|
⊢ Ⅎ𝑥∃*𝑥 ∈ 𝐴 𝜑 |
|
Theorem | nfreudxy 2483* |
Not-free deduction for restricted uniqueness. This is a version where
𝑥 and 𝑦 are distinct.
(Contributed by Jim Kingdon,
6-Jun-2018.)
|
⊢ Ⅎ𝑦𝜑
& ⊢ (𝜑 → Ⅎ𝑥𝐴)
& ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜓) |
|
Theorem | nfreuxy 2484* |
Not-free for restricted uniqueness. This is a version where 𝑥 and
𝑦 are distinct. (Contributed by Jim
Kingdon, 6-Jun-2018.)
|
⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑 |
|
Theorem | rabid 2485 |
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 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
|
Theorem | rabid2 2486* |
An "identity" law for restricted class abstraction. (Contributed by
NM,
9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
|
⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑) |
|
Theorem | rabbi 2487 |
Equivalent wff's correspond to equal restricted class abstractions.
Closed theorem form of rabbidva 2548. (Contributed by NM,
25-Nov-2013.)
|
⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) |
|
Theorem | rabswap 2488 |
Swap with a membership relation in a restricted class abstraction.
(Contributed by NM, 4-Jul-2005.)
|
⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝐴} |
|
Theorem | nfrab1 2489 |
The abstraction variable in a restricted class abstraction isn't free.
(Contributed by NM, 19-Mar-1997.)
|
⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑} |
|
Theorem | nfrabxy 2490* |
A variable not free in a wff remains so in a restricted class
abstraction. (Contributed by Jim Kingdon, 19-Jul-2018.)
|
⊢ Ⅎ𝑥𝜑
& ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑} |
|
Theorem | reubida 2491 |
Formula-building rule for restricted existential quantifier (deduction
rule). (Contributed by Mario Carneiro, 19-Nov-2016.)
|
⊢ Ⅎ𝑥𝜑
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒)) |
|
Theorem | reubidva 2492* |
Formula-building rule for restricted existential quantifier (deduction
rule). (Contributed by NM, 13-Nov-2004.)
|
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒)) |
|
Theorem | reubidv 2493* |
Formula-building rule for restricted existential quantifier (deduction
rule). (Contributed by NM, 17-Oct-1996.)
|
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒)) |
|
Theorem | reubiia 2494 |
Formula-building rule for restricted existential quantifier (inference
rule). (Contributed by NM, 14-Nov-2004.)
|
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐴 𝜓) |
|
Theorem | reubii 2495 |
Formula-building rule for restricted existential quantifier (inference
rule). (Contributed by NM, 22-Oct-1999.)
|
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐴 𝜓) |
|
Theorem | rmobida 2496 |
Formula-building rule for restricted existential quantifier (deduction
rule). (Contributed by NM, 16-Jun-2017.)
|
⊢ Ⅎ𝑥𝜑
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒)) |
|
Theorem | rmobidva 2497* |
Formula-building rule for restricted existential quantifier (deduction
rule). (Contributed by NM, 16-Jun-2017.)
|
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒)) |
|
Theorem | rmobidv 2498* |
Formula-building rule for restricted existential quantifier (deduction
rule). (Contributed by NM, 16-Jun-2017.)
|
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒)) |
|
Theorem | rmobiia 2499 |
Formula-building rule for restricted existential quantifier (inference
rule). (Contributed by NM, 16-Jun-2017.)
|
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐴 𝜓) |
|
Theorem | rmobii 2500 |
Formula-building rule for restricted existential quantifier (inference
rule). (Contributed by NM, 16-Jun-2017.)
|
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐴 𝜓) |