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Theorem List for Intuitionistic Logic Explorer - 2701-2800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelex22 2701* If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.)
((𝐴𝐵𝐴𝐶) → ∃𝑥(𝑥𝐵𝑥𝐶))
 
Theoremelex2 2702* If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.)
(𝐴𝐵 → ∃𝑥 𝑥𝐵)
 
Theoremralv 2703 A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
(∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑)
 
Theoremrexv 2704 An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
(∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
 
Theoremreuv 2705 A unique existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)
(∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑)
 
Theoremrmov 2706 An at-most-one quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥𝜑)
 
Theoremrabab 2707 A class abstraction restricted to the universe is unrestricted. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
{𝑥 ∈ V ∣ 𝜑} = {𝑥𝜑}
 
Theoremralcom4 2708* Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
(∀𝑥𝐴𝑦𝜑 ↔ ∀𝑦𝑥𝐴 𝜑)
 
Theoremrexcom4 2709* Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
(∃𝑥𝐴𝑦𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)
 
Theoremrexcom4a 2710* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
(∃𝑥𝑦𝐴 (𝜑𝜓) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥𝜓))
 
Theoremrexcom4b 2711* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
𝐵 ∈ V       (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 𝜑)
 
Theoremceqsalt 2712* Closed theorem version of ceqsalg 2714. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
 
Theoremceqsralt 2713* Restricted quantifier version of ceqsalt 2712. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
 
Theoremceqsalg 2714* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
 
Theoremceqsal 2715* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)
𝑥𝜓    &   𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
 
Theoremceqsalv 2716* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
 
Theoremceqsralv 2717* Restricted quantifier version of ceqsalv 2716. (Contributed by NM, 21-Jun-2013.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝐵 → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
 
Theoremgencl 2718* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
(𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝐵))    &   (𝐴 = 𝐵 → (𝜑𝜓))    &   (𝜒𝜑)       (𝜃𝜓)
 
Theorem2gencl 2719* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
(𝐶𝑆 ↔ ∃𝑥𝑅 𝐴 = 𝐶)    &   (𝐷𝑆 ↔ ∃𝑦𝑅 𝐵 = 𝐷)    &   (𝐴 = 𝐶 → (𝜑𝜓))    &   (𝐵 = 𝐷 → (𝜓𝜒))    &   ((𝑥𝑅𝑦𝑅) → 𝜑)       ((𝐶𝑆𝐷𝑆) → 𝜒)
 
Theorem3gencl 2720* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
(𝐷𝑆 ↔ ∃𝑥𝑅 𝐴 = 𝐷)    &   (𝐹𝑆 ↔ ∃𝑦𝑅 𝐵 = 𝐹)    &   (𝐺𝑆 ↔ ∃𝑧𝑅 𝐶 = 𝐺)    &   (𝐴 = 𝐷 → (𝜑𝜓))    &   (𝐵 = 𝐹 → (𝜓𝜒))    &   (𝐶 = 𝐺 → (𝜒𝜃))    &   ((𝑥𝑅𝑦𝑅𝑧𝑅) → 𝜑)       ((𝐷𝑆𝐹𝑆𝐺𝑆) → 𝜃)
 
Theoremcgsexg 2721* Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)
(𝑥 = 𝐴𝜒)    &   (𝜒 → (𝜑𝜓))       (𝐴𝑉 → (∃𝑥(𝜒𝜑) ↔ 𝜓))
 
Theoremcgsex2g 2722* Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.)
((𝑥 = 𝐴𝑦 = 𝐵) → 𝜒)    &   (𝜒 → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → (∃𝑥𝑦(𝜒𝜑) ↔ 𝜓))
 
Theoremcgsex4g 2723* An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.)
(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) → 𝜒)    &   (𝜒 → (𝜑𝜓))       (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → (∃𝑥𝑦𝑧𝑤(𝜒𝜑) ↔ 𝜓))
 
Theoremceqsex 2724* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.)
𝑥𝜓    &   𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
 
Theoremceqsexv 2725* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
 
Theoremceqsex2 2726* Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)
𝑥𝜓    &   𝑦𝜒    &   𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))       (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵𝜑) ↔ 𝜒)
 
Theoremceqsex2v 2727* Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))       (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵𝜑) ↔ 𝜒)
 
Theoremceqsex3v 2728* Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑧 = 𝐶 → (𝜒𝜃))       (∃𝑥𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ 𝜃)
 
Theoremceqsex4v 2729* Elimination of four existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑧 = 𝐶 → (𝜒𝜃))    &   (𝑤 = 𝐷 → (𝜃𝜏))       (∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷) ∧ 𝜑) ↔ 𝜏)
 
Theoremceqsex6v 2730* Elimination of six existential quantifiers, using implicit substitution. (Contributed by NM, 21-Sep-2011.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V    &   𝐸 ∈ V    &   𝐹 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑧 = 𝐶 → (𝜒𝜃))    &   (𝑤 = 𝐷 → (𝜃𝜏))    &   (𝑣 = 𝐸 → (𝜏𝜂))    &   (𝑢 = 𝐹 → (𝜂𝜁))       (∃𝑥𝑦𝑧𝑤𝑣𝑢((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ (𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜑) ↔ 𝜁)
 
Theoremceqsex8v 2731* Elimination of eight existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V    &   𝐸 ∈ V    &   𝐹 ∈ V    &   𝐺 ∈ V    &   𝐻 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑧 = 𝐶 → (𝜒𝜃))    &   (𝑤 = 𝐷 → (𝜃𝜏))    &   (𝑣 = 𝐸 → (𝜏𝜂))    &   (𝑢 = 𝐹 → (𝜂𝜁))    &   (𝑡 = 𝐺 → (𝜁𝜎))    &   (𝑠 = 𝐻 → (𝜎𝜌))       (∃𝑥𝑦𝑧𝑤𝑣𝑢𝑡𝑠(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑) ↔ 𝜌)
 
Theoremgencbvex 2732* Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
𝐴 ∈ V    &   (𝐴 = 𝑦 → (𝜑𝜓))    &   (𝐴 = 𝑦 → (𝜒𝜃))    &   (𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝑦))       (∃𝑥(𝜒𝜑) ↔ ∃𝑦(𝜃𝜓))
 
Theoremgencbvex2 2733* Restatement of gencbvex 2732 with weaker hypotheses. (Contributed by Jeff Hankins, 6-Dec-2006.)
𝐴 ∈ V    &   (𝐴 = 𝑦 → (𝜑𝜓))    &   (𝐴 = 𝑦 → (𝜒𝜃))    &   (𝜃 → ∃𝑥(𝜒𝐴 = 𝑦))       (∃𝑥(𝜒𝜑) ↔ ∃𝑦(𝜃𝜓))
 
Theoremgencbval 2734* Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof rewritten by Jim Kingdon, 20-Jun-2018.)
𝐴 ∈ V    &   (𝐴 = 𝑦 → (𝜑𝜓))    &   (𝐴 = 𝑦 → (𝜒𝜃))    &   (𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝑦))       (∀𝑥(𝜒𝜑) ↔ ∀𝑦(𝜃𝜓))
 
Theoremsbhypf 2735* Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf . (Contributed by Raph Levien, 10-Apr-2004.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
 
Theoremvtoclgft 2736 Closed theorem form of vtoclgf 2744. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.)
(((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴𝑉) → 𝜓)
 
Theoremvtocldf 2737 Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))    &   (𝜑𝜓)    &   𝑥𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑𝜒)
 
Theoremvtocld 2738* Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))    &   (𝜑𝜓)       (𝜑𝜒)
 
Theoremvtoclf 2739* Implicit substitution of a class for a setvar variable. This is a generalization of chvar 1730. (Contributed by NM, 30-Aug-1993.)
𝑥𝜓    &   𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       𝜓
 
Theoremvtocl 2740* Implicit substitution of a class for a setvar variable. (Contributed by NM, 30-Aug-1993.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       𝜓
 
Theoremvtocl2 2741* Implicit substitution of classes for setvar variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
𝐴 ∈ V    &   𝐵 ∈ V    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))    &   𝜑       𝜓
 
Theoremvtocl3 2742* Implicit substitution of classes for setvar variables. (Contributed by NM, 3-Jun-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))    &   𝜑       𝜓
 
Theoremvtoclb 2743* Implicit substitution of a class for a setvar variable. (Contributed by NM, 23-Dec-1993.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑥 = 𝐴 → (𝜓𝜃))    &   (𝜑𝜓)       (𝜒𝜃)
 
Theoremvtoclgf 2744 Implicit substitution of a class for a setvar variable, with bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Proof shortened by Mario Carneiro, 10-Oct-2016.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       (𝐴𝑉𝜓)
 
Theoremvtoclg1f 2745* Version of vtoclgf 2744 with one non-freeness hypothesis replaced with a disjoint variable condition, thus avoiding dependency on ax-11 1484 and ax-13 1491. (Contributed by BJ, 1-May-2019.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       (𝐴𝑉𝜓)
 
Theoremvtoclg 2746* Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       (𝐴𝑉𝜓)
 
Theoremvtoclbg 2747* Implicit substitution of a class for a setvar variable. (Contributed by NM, 29-Apr-1994.)
(𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑥 = 𝐴 → (𝜓𝜃))    &   (𝜑𝜓)       (𝐴𝑉 → (𝜒𝜃))
 
Theoremvtocl2gf 2748 Implicit substitution of a class for a setvar variable. (Contributed by NM, 25-Apr-1995.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝐵    &   𝑥𝜓    &   𝑦𝜒    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   𝜑       ((𝐴𝑉𝐵𝑊) → 𝜒)
 
Theoremvtocl3gf 2749 Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
𝑥𝐴    &   𝑦𝐴    &   𝑧𝐴    &   𝑦𝐵    &   𝑧𝐵    &   𝑧𝐶    &   𝑥𝜓    &   𝑦𝜒    &   𝑧𝜃    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑧 = 𝐶 → (𝜒𝜃))    &   𝜑       ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝜃)
 
Theoremvtocl2g 2750* Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 25-Apr-1995.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   𝜑       ((𝐴𝑉𝐵𝑊) → 𝜒)
 
Theoremvtoclgaf 2751* Implicit substitution of a class for a setvar variable. (Contributed by NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥𝐵𝜑)       (𝐴𝐵𝜓)
 
Theoremvtoclga 2752* Implicit substitution of a class for a setvar variable. (Contributed by NM, 20-Aug-1995.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥𝐵𝜑)       (𝐴𝐵𝜓)
 
Theoremvtocl2gaf 2753* Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 10-Aug-2013.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝐵    &   𝑥𝜓    &   𝑦𝜒    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   ((𝑥𝐶𝑦𝐷) → 𝜑)       ((𝐴𝐶𝐵𝐷) → 𝜒)
 
Theoremvtocl2ga 2754* Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   ((𝑥𝐶𝑦𝐷) → 𝜑)       ((𝐴𝐶𝐵𝐷) → 𝜒)
 
Theoremvtocl3gaf 2755* Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 11-Oct-2016.)
𝑥𝐴    &   𝑦𝐴    &   𝑧𝐴    &   𝑦𝐵    &   𝑧𝐵    &   𝑧𝐶    &   𝑥𝜓    &   𝑦𝜒    &   𝑧𝜃    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑧 = 𝐶 → (𝜒𝜃))    &   ((𝑥𝑅𝑦𝑆𝑧𝑇) → 𝜑)       ((𝐴𝑅𝐵𝑆𝐶𝑇) → 𝜃)
 
Theoremvtocl3ga 2756* Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑧 = 𝐶 → (𝜒𝜃))    &   ((𝑥𝐷𝑦𝑅𝑧𝑆) → 𝜑)       ((𝐴𝐷𝐵𝑅𝐶𝑆) → 𝜃)
 
Theoremvtocleg 2757* Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Jan-2004.)
(𝑥 = 𝐴𝜑)       (𝐴𝑉𝜑)
 
Theoremvtoclegft 2758* Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 2759.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)
((𝐴𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → 𝜑)
 
Theoremvtoclef 2759* Implicit substitution of a class for a setvar variable. (Contributed by NM, 18-Aug-1993.)
𝑥𝜑    &   𝐴 ∈ V    &   (𝑥 = 𝐴𝜑)       𝜑
 
Theoremvtocle 2760* Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.)
𝐴 ∈ V    &   (𝑥 = 𝐴𝜑)       𝜑
 
Theoremvtoclri 2761* Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Nov-1994.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   𝑥𝐵 𝜑       (𝐴𝐵𝜓)
 
Theoremspcimgft 2762 A closed version of spcimgf 2766. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑥𝜓    &   𝑥𝐴       (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝜑𝜓)))
 
Theoremspcgft 2763 A closed version of spcgf 2768. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.)
𝑥𝜓    &   𝑥𝐴       (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝜑𝜓)))
 
Theoremspcimegft 2764 A closed version of spcimegf 2767. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑥𝜓    &   𝑥𝐴       (∀𝑥(𝑥 = 𝐴 → (𝜓𝜑)) → (𝐴𝐵 → (𝜓 → ∃𝑥𝜑)))
 
Theoremspcegft 2765 A closed version of spcegf 2769. (Contributed by Jim Kingdon, 22-Jun-2018.)
𝑥𝜓    &   𝑥𝐴       (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (𝜓 → ∃𝑥𝜑)))
 
Theoremspcimgf 2766 Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥𝜑𝜓))
 
Theoremspcimegf 2767 Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜓𝜑))       (𝐴𝑉 → (𝜓 → ∃𝑥𝜑))
 
Theoremspcgf 2768 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.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥𝜑𝜓))
 
Theoremspcegf 2769 Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (𝜓 → ∃𝑥𝜑))
 
Theoremspcimdv 2770* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓𝜒))
 
Theoremspcdv 2771* Rule of specialization, using implicit substitution. Analogous to rspcdv 2792. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓𝜒))
 
Theoremspcimedv 2772* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜒𝜓))       (𝜑 → (𝜒 → ∃𝑥𝜓))
 
Theoremspcgv 2773* Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥𝜑𝜓))
 
Theoremspcegv 2774* Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (𝜓 → ∃𝑥𝜑))
 
Theoremspc2egv 2775* Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → (𝜓 → ∃𝑥𝑦𝜑))
 
Theoremspc2gv 2776* Specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑦𝜑𝜓))
 
Theoremspc3egv 2777* Existential specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝜓 → ∃𝑥𝑦𝑧𝜑))
 
Theoremspc3gv 2778* Specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∀𝑥𝑦𝑧𝜑𝜓))
 
Theoremspcv 2779* Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥𝜑𝜓)
 
Theoremspcev 2780* Existential specialization, using implicit substitution. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝜓 → ∃𝑥𝜑)
 
Theoremspc2ev 2781* Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
𝐴 ∈ V    &   𝐵 ∈ V    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       (𝜓 → ∃𝑥𝑦𝜑)
 
Theoremrspct 2782* A closed version of rspc 2783. (Contributed by Andrew Salmon, 6-Jun-2011.)
𝑥𝜓       (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓)))
 
Theoremrspc 2783* Restricted specialization, using implicit substitution. (Contributed by NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))
 
Theoremrspce 2784* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
 
Theoremrspcv 2785* Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))
 
Theoremrspccv 2786* Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥𝐵 𝜑 → (𝐴𝐵𝜓))
 
Theoremrspcva 2787* Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵 ∧ ∀𝑥𝐵 𝜑) → 𝜓)
 
Theoremrspccva 2788* Restricted specialization, using implicit substitution. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((∀𝑥𝐵 𝜑𝐴𝐵) → 𝜓)
 
Theoremrspcev 2789* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
 
Theoremrspcimdv 2790* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓𝜒))
 
Theoremrspcimedv 2791* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜒𝜓))       (𝜑 → (𝜒 → ∃𝑥𝐵 𝜓))
 
Theoremrspcdv 2792* Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓𝜒))
 
Theoremrspcedv 2793* Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (𝜒 → ∃𝑥𝐵 𝜓))
 
Theoremrspcdva 2794* Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 21-Jun-2020.)
(𝑥 = 𝐶 → (𝜓𝜒))    &   (𝜑 → ∀𝑥𝐴 𝜓)    &   (𝜑𝐶𝐴)       (𝜑𝜒)
 
Theoremrspcedvd 2795* Restricted existential specialization, using implicit substitution. Variant of rspcedv 2793. (Contributed by AV, 27-Nov-2019.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))    &   (𝜑𝜒)       (𝜑 → ∃𝑥𝐵 𝜓)
 
Theoremrspcime 2796* Prove a restricted existential. (Contributed by Rohan Ridenour, 3-Aug-2023.)
((𝜑𝑥 = 𝐴) → 𝜓)    &   (𝜑𝐴𝐵)       (𝜑 → ∃𝑥𝐵 𝜓)
 
Theoremrspceaimv 2797* Restricted existential specialization of a universally quantified implication. (Contributed by BJ, 24-Aug-2022.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵 ∧ ∀𝑦𝐶 (𝜓𝜒)) → ∃𝑥𝐵𝑦𝐶 (𝜑𝜒))
 
Theoremrspcedeq1vd 2798* Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2795 for equations, in which the left hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → 𝐶 = 𝐷)       (𝜑 → ∃𝑥𝐵 𝐶 = 𝐷)
 
Theoremrspcedeq2vd 2799* Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2795 for equations, in which the right hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → 𝐶 = 𝐷)       (𝜑 → ∃𝑥𝐵 𝐶 = 𝐷)
 
Theoremrspc2 2800* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 9-Nov-2012.)
𝑥𝜒    &   𝑦𝜓    &   (𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑦 = 𝐵 → (𝜒𝜓))       ((𝐴𝐶𝐵𝐷) → (∀𝑥𝐶𝑦𝐷 𝜑𝜓))
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