Theorem List for Intuitionistic Logic Explorer - 2701-2800 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | vtoclg 2701* |
Implicit substitution of a class expression for a setvar variable.
(Contributed by NM, 17-Apr-1995.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ 𝜑 ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝜓) |
|
Theorem | vtoclbg 2702* |
Implicit substitution of a class for a setvar variable. (Contributed by
NM, 29-Apr-1994.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃)) & ⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝜒 ↔ 𝜃)) |
|
Theorem | vtocl2gf 2703 |
Implicit substitution of a class for a setvar variable. (Contributed by
NM, 25-Apr-1995.)
|
⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑦𝐴
& ⊢ Ⅎ𝑦𝐵
& ⊢ Ⅎ𝑥𝜓
& ⊢ Ⅎ𝑦𝜒
& ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ 𝜑 ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜒) |
|
Theorem | vtocl3gf 2704 |
Implicit substitution of a class for a setvar variable. (Contributed by
NM, 10-Aug-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
|
⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑦𝐴
& ⊢ Ⅎ𝑧𝐴
& ⊢ Ⅎ𝑦𝐵
& ⊢ Ⅎ𝑧𝐵
& ⊢ Ⅎ𝑧𝐶
& ⊢ Ⅎ𝑥𝜓
& ⊢ Ⅎ𝑦𝜒
& ⊢ Ⅎ𝑧𝜃
& ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ 𝜑 ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝜃) |
|
Theorem | vtocl2g 2705* |
Implicit substitution of 2 classes for 2 setvar variables. (Contributed
by NM, 25-Apr-1995.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ 𝜑 ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜒) |
|
Theorem | vtoclgaf 2706* |
Implicit substitution of a class for a setvar variable. (Contributed by
NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.)
|
⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝜓
& ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 ∈ 𝐵 → 𝜑) ⇒ ⊢ (𝐴 ∈ 𝐵 → 𝜓) |
|
Theorem | vtoclga 2707* |
Implicit substitution of a class for a setvar variable. (Contributed by
NM, 20-Aug-1995.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 ∈ 𝐵 → 𝜑) ⇒ ⊢ (𝐴 ∈ 𝐵 → 𝜓) |
|
Theorem | vtocl2gaf 2708* |
Implicit substitution of 2 classes for 2 setvar variables. (Contributed
by NM, 10-Aug-2013.)
|
⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑦𝐴
& ⊢ Ⅎ𝑦𝐵
& ⊢ Ⅎ𝑥𝜓
& ⊢ Ⅎ𝑦𝜒
& ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜑) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒) |
|
Theorem | vtocl2ga 2709* |
Implicit substitution of 2 classes for 2 setvar variables. (Contributed
by NM, 20-Aug-1995.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜑) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒) |
|
Theorem | vtocl3gaf 2710* |
Implicit substitution of 3 classes for 3 setvar variables. (Contributed
by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 11-Oct-2016.)
|
⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑦𝐴
& ⊢ Ⅎ𝑧𝐴
& ⊢ Ⅎ𝑦𝐵
& ⊢ Ⅎ𝑧𝐵
& ⊢ Ⅎ𝑧𝐶
& ⊢ Ⅎ𝑥𝜓
& ⊢ Ⅎ𝑦𝜒
& ⊢ Ⅎ𝑧𝜃
& ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇) → 𝜑) ⇒ ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → 𝜃) |
|
Theorem | vtocl3ga 2711* |
Implicit substitution of 3 classes for 3 setvar variables. (Contributed
by NM, 20-Aug-1995.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜑) ⇒ ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 𝜃) |
|
Theorem | vtocleg 2712* |
Implicit substitution of a class for a setvar variable. (Contributed by
NM, 10-Jan-2004.)
|
⊢ (𝑥 = 𝐴 → 𝜑) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝜑) |
|
Theorem | vtoclegft 2713* |
Implicit substitution of a class for a setvar variable. (Closed theorem
version of vtoclef 2714.) (Contributed by NM, 7-Nov-2005.) (Revised
by
Mario Carneiro, 11-Oct-2016.)
|
⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → 𝜑) |
|
Theorem | vtoclef 2714* |
Implicit substitution of a class for a setvar variable. (Contributed by
NM, 18-Aug-1993.)
|
⊢ Ⅎ𝑥𝜑
& ⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → 𝜑) ⇒ ⊢ 𝜑 |
|
Theorem | vtocle 2715* |
Implicit substitution of a class for a setvar variable. (Contributed by
NM, 9-Sep-1993.)
|
⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → 𝜑) ⇒ ⊢ 𝜑 |
|
Theorem | vtoclri 2716* |
Implicit substitution of a class for a setvar variable. (Contributed by
NM, 21-Nov-1994.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ ∀𝑥 ∈ 𝐵 𝜑 ⇒ ⊢ (𝐴 ∈ 𝐵 → 𝜓) |
|
Theorem | spcimgft 2717 |
A closed version of spcimgf 2721. (Contributed by Mario Carneiro,
4-Jan-2017.)
|
⊢ Ⅎ𝑥𝜓
& ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓))) |
|
Theorem | spcgft 2718 |
A closed version of spcgf 2723. (Contributed by Andrew Salmon,
6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.)
|
⊢ Ⅎ𝑥𝜓
& ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓))) |
|
Theorem | spcimegft 2719 |
A closed version of spcimegf 2722. (Contributed by Mario Carneiro,
4-Jan-2017.)
|
⊢ Ⅎ𝑥𝜓
& ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜑)) → (𝐴 ∈ 𝐵 → (𝜓 → ∃𝑥𝜑))) |
|
Theorem | spcegft 2720 |
A closed version of spcegf 2724. (Contributed by Jim Kingdon,
22-Jun-2018.)
|
⊢ Ⅎ𝑥𝜓
& ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝐵 → (𝜓 → ∃𝑥𝜑))) |
|
Theorem | spcimgf 2721 |
Rule of specialization, using implicit substitution. Compare Theorem
7.3 of [Quine] p. 44. (Contributed by
Mario Carneiro, 4-Jan-2017.)
|
⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝜓
& ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)) |
|
Theorem | spcimegf 2722 |
Existential specialization, using implicit substitution. (Contributed
by Mario Carneiro, 4-Jan-2017.)
|
⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝜓
& ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑)) |
|
Theorem | spcgf 2723 |
Rule of specialization, using implicit substitution. Compare Theorem
7.3 of [Quine] p. 44. (Contributed by NM,
2-Feb-1997.) (Revised by
Andrew Salmon, 12-Aug-2011.)
|
⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝜓
& ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)) |
|
Theorem | spcegf 2724 |
Existential specialization, using implicit substitution. (Contributed
by NM, 2-Feb-1997.)
|
⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝜓
& ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑)) |
|
Theorem | spcimdv 2725* |
Restricted specialization, using implicit substitution. (Contributed
by Mario Carneiro, 4-Jan-2017.)
|
⊢ (𝜑 → 𝐴 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒)) |
|
Theorem | spcdv 2726* |
Rule of specialization, using implicit substitution. Analogous to
rspcdv 2747. (Contributed by David Moews, 1-May-2017.)
|
⊢ (𝜑 → 𝐴 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒)) |
|
Theorem | spcimedv 2727* |
Restricted existential specialization, using implicit substitution.
(Contributed by Mario Carneiro, 4-Jan-2017.)
|
⊢ (𝜑 → 𝐴 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓)) ⇒ ⊢ (𝜑 → (𝜒 → ∃𝑥𝜓)) |
|
Theorem | spcgv 2728* |
Rule of specialization, using implicit substitution. Compare Theorem
7.3 of [Quine] p. 44. (Contributed by NM,
22-Jun-1994.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)) |
|
Theorem | spcegv 2729* |
Existential specialization, using implicit substitution. (Contributed
by NM, 14-Aug-1994.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑)) |
|
Theorem | spc2egv 2730* |
Existential specialization with 2 quantifiers, using implicit
substitution. (Contributed by NM, 3-Aug-1995.)
|
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝜓 → ∃𝑥∃𝑦𝜑)) |
|
Theorem | spc2gv 2731* |
Specialization with 2 quantifiers, using implicit substitution.
(Contributed by NM, 27-Apr-2004.)
|
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥∀𝑦𝜑 → 𝜓)) |
|
Theorem | spc3egv 2732* |
Existential specialization with 3 quantifiers, using implicit
substitution. (Contributed by NM, 12-May-2008.)
|
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝜓 → ∃𝑥∃𝑦∃𝑧𝜑)) |
|
Theorem | spc3gv 2733* |
Specialization with 3 quantifiers, using implicit substitution.
(Contributed by NM, 12-May-2008.)
|
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∀𝑥∀𝑦∀𝑧𝜑 → 𝜓)) |
|
Theorem | spcv 2734* |
Rule of specialization, using implicit substitution. (Contributed by
NM, 22-Jun-1994.)
|
⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) |
|
Theorem | spcev 2735* |
Existential specialization, using implicit substitution. (Contributed
by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
|
⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝜓 → ∃𝑥𝜑) |
|
Theorem | spc2ev 2736* |
Existential specialization, using implicit substitution. (Contributed
by NM, 3-Aug-1995.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝜓 → ∃𝑥∃𝑦𝜑) |
|
Theorem | rspct 2737* |
A closed version of rspc 2738. (Contributed by Andrew Salmon,
6-Jun-2011.)
|
⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓))) |
|
Theorem | rspc 2738* |
Restricted specialization, using implicit substitution. (Contributed by
NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)
|
⊢ Ⅎ𝑥𝜓
& ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓)) |
|
Theorem | rspce 2739* |
Restricted existential specialization, using implicit substitution.
(Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro,
11-Oct-2016.)
|
⊢ Ⅎ𝑥𝜓
& ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑) |
|
Theorem | rspcv 2740* |
Restricted specialization, using implicit substitution. (Contributed by
NM, 26-May-1998.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓)) |
|
Theorem | rspccv 2741* |
Restricted specialization, using implicit substitution. (Contributed by
NM, 2-Feb-2006.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓)) |
|
Theorem | rspcva 2742* |
Restricted specialization, using implicit substitution. (Contributed by
NM, 13-Sep-2005.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝜑) → 𝜓) |
|
Theorem | rspccva 2743* |
Restricted specialization, using implicit substitution. (Contributed by
NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((∀𝑥 ∈ 𝐵 𝜑 ∧ 𝐴 ∈ 𝐵) → 𝜓) |
|
Theorem | rspcev 2744* |
Restricted existential specialization, using implicit substitution.
(Contributed by NM, 26-May-1998.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑) |
|
Theorem | rspcimdv 2745* |
Restricted specialization, using implicit substitution. (Contributed
by Mario Carneiro, 4-Jan-2017.)
|
⊢ (𝜑 → 𝐴 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒)) |
|
Theorem | rspcimedv 2746* |
Restricted existential specialization, using implicit substitution.
(Contributed by Mario Carneiro, 4-Jan-2017.)
|
⊢ (𝜑 → 𝐴 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓)) ⇒ ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓)) |
|
Theorem | rspcdv 2747* |
Restricted specialization, using implicit substitution. (Contributed by
NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
|
⊢ (𝜑 → 𝐴 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒)) |
|
Theorem | rspcedv 2748* |
Restricted existential specialization, using implicit substitution.
(Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro,
4-Jan-2017.)
|
⊢ (𝜑 → 𝐴 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓)) |
|
Theorem | rspcdva 2749* |
Restricted specialization, using implicit substitution. (Contributed by
Thierry Arnoux, 21-Jun-2020.)
|
⊢ (𝑥 = 𝐶 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
& ⊢ (𝜑 → 𝐶 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝜒) |
|
Theorem | rspcedvd 2750* |
Restricted existential specialization, using implicit substitution.
Variant of rspcedv 2748. (Contributed by AV, 27-Nov-2019.)
|
⊢ (𝜑 → 𝐴 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → 𝜒) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓) |
|
Theorem | rspceaimv 2751* |
Restricted existential specialization of a universally quantified
implication. (Contributed by BJ, 24-Aug-2022.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐶 (𝜓 → 𝜒)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 (𝜑 → 𝜒)) |
|
Theorem | rspcedeq1vd 2752* |
Restricted existential specialization, using implicit substitution.
Variant of rspcedvd 2750 for equations, in which the left hand side
depends on the quantified variable. (Contributed by AV,
24-Dec-2019.)
|
⊢ (𝜑 → 𝐴 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐶 = 𝐷) |
|
Theorem | rspcedeq2vd 2753* |
Restricted existential specialization, using implicit substitution.
Variant of rspcedvd 2750 for equations, in which the right hand side
depends on the quantified variable. (Contributed by AV,
24-Dec-2019.)
|
⊢ (𝜑 → 𝐴 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐶 = 𝐷) |
|
Theorem | rspc2 2754* |
2-variable restricted specialization, using implicit substitution.
(Contributed by NM, 9-Nov-2012.)
|
⊢ Ⅎ𝑥𝜒
& ⊢ Ⅎ𝑦𝜓
& ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑 → 𝜓)) |
|
Theorem | rspc2gv 2755* |
Restricted specialization with two quantifiers, using implicit
substitution. (Contributed by BJ, 2-Dec-2021.)
|
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 𝜑 → 𝜓)) |
|
Theorem | rspc2v 2756* |
2-variable restricted specialization, using implicit substitution.
(Contributed by NM, 13-Sep-1999.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑 → 𝜓)) |
|
Theorem | rspc2va 2757* |
2-variable restricted specialization, using implicit substitution.
(Contributed by NM, 18-Jun-2014.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓)) ⇒ ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑) → 𝜓) |
|
Theorem | rspc2ev 2758* |
2-variable restricted existential specialization, using implicit
substitution. (Contributed by NM, 16-Oct-1999.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝜓) → ∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 𝜑) |
|
Theorem | rspc3v 2759* |
3-variable restricted specialization, using implicit substitution.
(Contributed by NM, 10-May-2005.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃)) & ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 𝜑 → 𝜓)) |
|
Theorem | rspc3ev 2760* |
3-variable restricted existentional specialization, using implicit
substitution. (Contributed by NM, 25-Jul-2012.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃)) & ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜓)) ⇒ ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ 𝜓) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 ∃𝑧 ∈ 𝑇 𝜑) |
|
Theorem | rspceeqv 2761* |
Restricted existential specialization in an equality, using implicit
substitution. (Contributed by BJ, 2-Sep-2022.)
|
⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷) ⇒ ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐸 = 𝐷) → ∃𝑥 ∈ 𝐵 𝐸 = 𝐶) |
|
Theorem | eqvinc 2762* |
A variable introduction law for class equality. (Contributed by NM,
14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)) |
|
Theorem | eqvincg 2763* |
A variable introduction law for class equality, deduction version.
(Contributed by Thierry Arnoux, 2-Mar-2017.)
|
⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵))) |
|
Theorem | eqvincf 2764 |
A variable introduction law for class equality, using bound-variable
hypotheses instead of distinct variable conditions. (Contributed by NM,
14-Sep-2003.)
|
⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐵
& ⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)) |
|
Theorem | alexeq 2765* |
Two ways to express substitution of 𝐴 for 𝑥 in 𝜑.
(Contributed by NM, 2-Mar-1995.)
|
⊢ 𝐴 ∈ V ⇒ ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
|
Theorem | ceqex 2766* |
Equality implies equivalence with substitution. (Contributed by NM,
2-Mar-1995.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))) |
|
Theorem | ceqsexg 2767* |
A representation of explicit substitution of a class for a variable,
inferred from an implicit substitution hypothesis. (Contributed by NM,
11-Oct-2004.)
|
⊢ Ⅎ𝑥𝜓
& ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)) |
|
Theorem | ceqsexgv 2768* |
Elimination of an existential quantifier, using implicit substitution.
(Contributed by NM, 29-Dec-1996.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)) |
|
Theorem | ceqsrexv 2769* |
Elimination of a restricted existential quantifier, using implicit
substitution. (Contributed by NM, 30-Apr-2004.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝐵 → (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)) |
|
Theorem | ceqsrexbv 2770* |
Elimination of a restricted existential quantifier, using implicit
substitution. (Contributed by Mario Carneiro, 14-Mar-2014.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) |
|
Theorem | ceqsrex2v 2771* |
Elimination of a restricted existential quantifier, using implicit
substitution. (Contributed by NM, 29-Oct-2005.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑) ↔ 𝜒)) |
|
Theorem | clel2 2772* |
An alternate definition of class membership when the class is a set.
(Contributed by NM, 18-Aug-1993.)
|
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) |
|
Theorem | clel3g 2773* |
An alternate definition of class membership when the class is a set.
(Contributed by NM, 13-Aug-2005.)
|
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥))) |
|
Theorem | clel3 2774* |
An alternate definition of class membership when the class is a set.
(Contributed by NM, 18-Aug-1993.)
|
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥)) |
|
Theorem | clel4 2775* |
An alternate definition of class membership when the class is a set.
(Contributed by NM, 18-Aug-1993.)
|
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥)) |
|
Theorem | pm13.183 2776* |
Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only 𝐴 is
required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.)
|
⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 ↔ ∀𝑧(𝑧 = 𝐴 ↔ 𝑧 = 𝐵))) |
|
Theorem | rr19.3v 2777* |
Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89.
(Contributed by NM, 25-Oct-2012.)
|
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑) |
|
Theorem | rr19.28v 2778* |
Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90.
(Contributed by NM, 29-Oct-2012.)
|
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)) |
|
Theorem | elabgt 2779* |
Membership in a class abstraction, using implicit substitution. (Closed
theorem version of elabg 2783.) (Contributed by NM, 7-Nov-2005.) (Proof
shortened by Andrew Salmon, 8-Jun-2011.)
|
⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
|
Theorem | elabgf 2780 |
Membership in a class abstraction, using implicit substitution. Compare
Theorem 6.13 of [Quine] p. 44. This
version has bound-variable
hypotheses in place of distinct variable restrictions. (Contributed by
NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.)
|
⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝜓
& ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
|
Theorem | elabf 2781* |
Membership in a class abstraction, using implicit substitution.
(Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro,
12-Oct-2016.)
|
⊢ Ⅎ𝑥𝜓
& ⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
|
Theorem | elab 2782* |
Membership in a class abstraction, using implicit substitution. Compare
Theorem 6.13 of [Quine] p. 44.
(Contributed by NM, 1-Aug-1994.)
|
⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
|
Theorem | elabg 2783* |
Membership in a class abstraction, using implicit substitution. Compare
Theorem 6.13 of [Quine] p. 44.
(Contributed by NM, 14-Apr-1995.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
|
Theorem | elab2g 2784* |
Membership in a class abstraction, using implicit substitution.
(Contributed by NM, 13-Sep-1995.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ 𝐵 = {𝑥 ∣ 𝜑} ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ 𝜓)) |
|
Theorem | elab2 2785* |
Membership in a class abstraction, using implicit substitution.
(Contributed by NM, 13-Sep-1995.)
|
⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ 𝐵 = {𝑥 ∣ 𝜑} ⇒ ⊢ (𝐴 ∈ 𝐵 ↔ 𝜓) |
|
Theorem | elab4g 2786* |
Membership in a class abstraction, using implicit substitution.
(Contributed by NM, 17-Oct-2012.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ 𝐵 = {𝑥 ∣ 𝜑} ⇒ ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ V ∧ 𝜓)) |
|
Theorem | elab3gf 2787 |
Membership in a class abstraction, with a weaker antecedent than
elabgf 2780. (Contributed by NM, 6-Sep-2011.)
|
⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝜓
& ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
|
Theorem | elab3g 2788* |
Membership in a class abstraction, with a weaker antecedent than
elabg 2783. (Contributed by NM, 29-Aug-2006.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
|
Theorem | elab3 2789* |
Membership in a class abstraction using implicit substitution.
(Contributed by NM, 10-Nov-2000.)
|
⊢ (𝜓 → 𝐴 ∈ V) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
|
Theorem | elrabi 2790* |
Implication for the membership in a restricted class abstraction.
(Contributed by Alexander van der Vekens, 31-Dec-2017.)
|
⊢ (𝐴 ∈ {𝑥 ∈ 𝑉 ∣ 𝜑} → 𝐴 ∈ 𝑉) |
|
Theorem | elrabf 2791 |
Membership in a restricted class abstraction, using implicit
substitution. This version has bound-variable hypotheses in place of
distinct variable restrictions. (Contributed by NM, 21-Sep-2003.)
|
⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐵
& ⊢ Ⅎ𝑥𝜓
& ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) |
|
Theorem | elrab3t 2792* |
Membership in a restricted class abstraction, using implicit
substitution. (Closed theorem version of elrab3 2794.) (Contributed by
Thierry Arnoux, 31-Aug-2017.)
|
⊢ ((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝜓)) |
|
Theorem | elrab 2793* |
Membership in a restricted class abstraction, using implicit
substitution. (Contributed by NM, 21-May-1999.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) |
|
Theorem | elrab3 2794* |
Membership in a restricted class abstraction, using implicit
substitution. (Contributed by NM, 5-Oct-2006.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝜓)) |
|
Theorem | elrabd 2795* |
Membership in a restricted class abstraction, using implicit
substitution. Deduction version of elrab 2793. (Contributed by Glauco
Siliprandi, 23-Oct-2021.)
|
⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → 𝐴 ∈ 𝐵)
& ⊢ (𝜑 → 𝜒) ⇒ ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓}) |
|
Theorem | elrab2 2796* |
Membership in a class abstraction, using implicit substitution.
(Contributed by NM, 2-Nov-2006.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ 𝜑} ⇒ ⊢ (𝐴 ∈ 𝐶 ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) |
|
Theorem | ralab 2797* |
Universal quantification over a class abstraction. (Contributed by Jeff
Madsen, 10-Jun-2010.)
|
⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∀𝑥(𝜓 → 𝜒)) |
|
Theorem | ralrab 2798* |
Universal quantification over a restricted class abstraction.
(Contributed by Jeff Madsen, 10-Jun-2010.)
|
⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜒 ↔ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) |
|
Theorem | rexab 2799* |
Existential quantification over a class abstraction. (Contributed by
Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro,
3-Sep-2015.)
|
⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∃𝑥(𝜓 ∧ 𝜒)) |
|
Theorem | rexrab 2800* |
Existential quantification over a class abstraction. (Contributed by
Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.)
|
⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜒 ↔ ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒)) |