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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | el2v 3501 | If a proposition is implied by 𝑥 ∈ V and 𝑦 ∈ V (which is true, see vex 3497), then it is true. (Contributed by Peter Mazsa, 13-Oct-2018.) |
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → 𝜑) ⇒ ⊢ 𝜑 | ||
Theorem | eqv 3502* | The universe contains every set. (Contributed by NM, 11-Sep-2006.) Remove dependency on ax-10 2145, ax-11 2161, ax-13 2390. (Revised by BJ, 10-Aug-2022.) |
⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴) | ||
Theorem | eqvf 3503 | The universe contains every set. (Contributed by BJ, 15-Jul-2021.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴) | ||
Theorem | abv 3504 | The class of sets verifying a property is the universal class if and only if that property is a tautology. The reverse implication (bj-abv 34226) requires fewer axioms. (Contributed by BJ, 19-Mar-2021.) |
⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑) | ||
Theorem | elisset 3505* | An element of a class exists. (Contributed by NM, 1-May-1995.) Reduce dependencies on axioms. (Revised by BJ, 29-Apr-2019.) |
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | ||
Theorem | isset 3506* |
Two ways to say "𝐴 is a set": A class 𝐴 is a
member of the
universal class V (see df-v 3496)
if and only if the class 𝐴
exists (i.e. there exists some set 𝑥 equal to class 𝐴).
Theorem 6.9 of [Quine] p. 43.
Notational convention: We will use the
notational device "𝐴 ∈ V " to mean "𝐴 is a
set" very
frequently, for example in uniex 7467. Note that a class 𝐴 which
is
not a set is called a proper class. In some theorems, such as
uniexg 7466, in order to shorten certain proofs we use
the more general
antecedent 𝐴 ∈ 𝑉 instead of 𝐴 ∈ V to mean
"𝐴 is a set."
Note that a constant is implicitly considered distinct from all variables. This is why V is not included in the distinct variable list, even though df-clel 2893 requires that the expression substituted for 𝐵 not contain 𝑥. (Also, the Metamath spec does not allow constants in the distinct variable list.) (Contributed by NM, 26-May-1993.) |
⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | ||
Theorem | issetf 3507 | A version of isset 3506 that does not require 𝑥 and 𝐴 to be distinct. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | ||
Theorem | isseti 3508* | A way to say "𝐴 is a set" (inference form). (Contributed by NM, 24-Jun-1993.) Remove dependencies on axioms. (Revised by BJ, 13-Jul-2019.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∃𝑥 𝑥 = 𝐴 | ||
Theorem | issetiOLD 3509* | Obsolete version of isseti 3508 as of 28-Aug-2023. A way to say "𝐴 is a set" (inference form). (Contributed by NM, 24-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∃𝑥 𝑥 = 𝐴 | ||
Theorem | issetri 3510* | A way to say "𝐴 is a set" (inference form). (Contributed by NM, 21-Jun-1993.) |
⊢ ∃𝑥 𝑥 = 𝐴 ⇒ ⊢ 𝐴 ∈ V | ||
Theorem | eqvisset 3511 | A class equal to a variable is a set. Note the absence of disjoint variable condition, contrary to isset 3506 and issetri 3510. (Contributed by BJ, 27-Apr-2019.) |
⊢ (𝑥 = 𝐴 → 𝐴 ∈ V) | ||
Theorem | elex 3512 | If a class is a member of another class, then it is a set. Theorem 6.12 of [Quine] p. 44. (Contributed by NM, 26-May-1993.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | ||
Theorem | elexi 3513 | If a class is a member of another class, then it is a set. Inference associated with elex 3512. (Contributed by NM, 11-Jun-1994.) |
⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ 𝐴 ∈ V | ||
Theorem | elexd 3514 | If a class is a member of another class, then it is a set. Deduction associated with elex 3512. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐴 ∈ V) | ||
Theorem | elissetOLD 3515* | Obsolete version of elisset 3505 as of 28-Aug-2023. An element of a class exists. (Contributed by NM, 1-May-1995.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | ||
Theorem | elex2 3516* | If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.) |
⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵) | ||
Theorem | elex22 3517* | If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.) |
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) | ||
Theorem | prcnel 3518 | A proper class doesn't belong to any class. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Proof shortened by AV, 14-Nov-2020.) |
⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ 𝑉) | ||
Theorem | ralv 3519 | A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.) |
⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑) | ||
Theorem | rexv 3520 | An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.) |
⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑) | ||
Theorem | reuv 3521 | A unique existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.) |
⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑) | ||
Theorem | rmov 3522 | An at-most-one quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥𝜑) | ||
Theorem | rabab 3523 | 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} = {𝑥 ∣ 𝜑} | ||
Theorem | rexcom4b 3524* | Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑) | ||
Theorem | ralcom4OLD 3525* | Obsolete version of ralcom4 3235 as of 26-Aug-2023. Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rexcom4OLD 3526* | Obsolete version of rexcom4 3249 as of 26-Aug-2023. Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑) | ||
Theorem | ceqsalt 3527* | Closed theorem version of ceqsalg 3529. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.) |
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) | ||
Theorem | ceqsralt 3528* | Restricted quantifier version of ceqsalt 3527. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.) |
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) | ||
Theorem | ceqsalg 3529* | A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. For an alternate proof, see ceqsalgALT 3530. (Contributed by NM, 29-Oct-2003.) (Proof shortened by BJ, 29-Sep-2019.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) | ||
Theorem | ceqsalgALT 3530* | Alternate proof of ceqsalg 3529, not using ceqsalt 3527. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by BJ, 29-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) | ||
Theorem | ceqsal 3531* | A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.) |
⊢ Ⅎ𝑥𝜓 & ⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓) | ||
Theorem | ceqsalv 3532* | 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 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓) | ||
Theorem | ceqsralv 3533* | Restricted quantifier version of ceqsalv 3532. (Contributed by NM, 21-Jun-2013.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) | ||
Theorem | gencl 3534* | Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.) |
⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝐵)) & ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓)) & ⊢ (𝜒 → 𝜑) ⇒ ⊢ (𝜃 → 𝜓) | ||
Theorem | 2gencl 3535* | Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.) |
⊢ (𝐶 ∈ 𝑆 ↔ ∃𝑥 ∈ 𝑅 𝐴 = 𝐶) & ⊢ (𝐷 ∈ 𝑆 ↔ ∃𝑦 ∈ 𝑅 𝐵 = 𝐷) & ⊢ (𝐴 = 𝐶 → (𝜑 ↔ 𝜓)) & ⊢ (𝐵 = 𝐷 → (𝜓 ↔ 𝜒)) & ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) → 𝜑) ⇒ ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆) → 𝜒) | ||
Theorem | 3gencl 3536* | Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.) |
⊢ (𝐷 ∈ 𝑆 ↔ ∃𝑥 ∈ 𝑅 𝐴 = 𝐷) & ⊢ (𝐹 ∈ 𝑆 ↔ ∃𝑦 ∈ 𝑅 𝐵 = 𝐹) & ⊢ (𝐺 ∈ 𝑆 ↔ ∃𝑧 ∈ 𝑅 𝐶 = 𝐺) & ⊢ (𝐴 = 𝐷 → (𝜑 ↔ 𝜓)) & ⊢ (𝐵 = 𝐹 → (𝜓 ↔ 𝜒)) & ⊢ (𝐶 = 𝐺 → (𝜒 ↔ 𝜃)) & ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅) → 𝜑) ⇒ ⊢ ((𝐷 ∈ 𝑆 ∧ 𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆) → 𝜃) | ||
Theorem | cgsexg 3537* | Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.) |
⊢ (𝑥 = 𝐴 → 𝜒) & ⊢ (𝜒 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝜒 ∧ 𝜑) ↔ 𝜓)) | ||
Theorem | cgsex2g 3538* | Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝜒) & ⊢ (𝜒 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦(𝜒 ∧ 𝜑) ↔ 𝜓)) | ||
Theorem | cgsex4g 3539* | An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.) Avoid ax-10 2145. (Revised by Gino Giotto, 20-Aug-2023.) |
⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → 𝜒) & ⊢ (𝜒 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑) ↔ 𝜓)) | ||
Theorem | ceqsex 3540* | Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝜓 & ⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓) | ||
Theorem | ceqsexv 3541* | Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) |
⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓) | ||
Theorem | ceqsexv2d 3542* | Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016.) |
⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ 𝜓 ⇒ ⊢ ∃𝑥𝜑 | ||
Theorem | ceqsex2 3543* | Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.) |
⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑦𝜒 & ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ 𝜒) | ||
Theorem | ceqsex2v 3544* | Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.) Avoid ax-10 2145 and ax-11 2161. (Revised by Gino Giotto, 20-Aug-2023.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ 𝜒) | ||
Theorem | ceqsex3v 3545* | Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) ⇒ ⊢ (∃𝑥∃𝑦∃𝑧((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜑) ↔ 𝜃) | ||
Theorem | ceqsex4v 3546* | Elimination of four existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ (𝑤 = 𝐷 → (𝜃 ↔ 𝜏)) ⇒ ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷) ∧ 𝜑) ↔ 𝜏) | ||
Theorem | ceqsex6v 3547* | 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 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ (𝑤 = 𝐷 → (𝜃 ↔ 𝜏)) & ⊢ (𝑣 = 𝐸 → (𝜏 ↔ 𝜂)) & ⊢ (𝑢 = 𝐹 → (𝜂 ↔ 𝜁)) ⇒ ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ (𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜑) ↔ 𝜁) | ||
Theorem | ceqsex8v 3548* | 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 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ (𝑤 = 𝐷 → (𝜃 ↔ 𝜏)) & ⊢ (𝑣 = 𝐸 → (𝜏 ↔ 𝜂)) & ⊢ (𝑢 = 𝐹 → (𝜂 ↔ 𝜁)) & ⊢ (𝑡 = 𝐺 → (𝜁 ↔ 𝜎)) & ⊢ (𝑠 = 𝐻 → (𝜎 ↔ 𝜌)) ⇒ ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤∃𝑣∃𝑢∃𝑡∃𝑠(((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻)) ∧ 𝜑) ↔ 𝜌) | ||
Theorem | gencbvex 3549* | 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 → (𝜑 ↔ 𝜓)) & ⊢ (𝐴 = 𝑦 → (𝜒 ↔ 𝜃)) & ⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝑦)) ⇒ ⊢ (∃𝑥(𝜒 ∧ 𝜑) ↔ ∃𝑦(𝜃 ∧ 𝜓)) | ||
Theorem | gencbvex2 3550* | Restatement of gencbvex 3549 with weaker hypotheses. (Contributed by Jeff Hankins, 6-Dec-2006.) |
⊢ 𝐴 ∈ V & ⊢ (𝐴 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝐴 = 𝑦 → (𝜒 ↔ 𝜃)) & ⊢ (𝜃 → ∃𝑥(𝜒 ∧ 𝐴 = 𝑦)) ⇒ ⊢ (∃𝑥(𝜒 ∧ 𝜑) ↔ ∃𝑦(𝜃 ∧ 𝜓)) | ||
Theorem | gencbval 3551* | Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) |
⊢ 𝐴 ∈ V & ⊢ (𝐴 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝐴 = 𝑦 → (𝜒 ↔ 𝜃)) & ⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝑦)) ⇒ ⊢ (∀𝑥(𝜒 → 𝜑) ↔ ∀𝑦(𝜃 → 𝜓)) | ||
Theorem | sbhypf 3552* | Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3911. (Contributed by Raph Levien, 10-Apr-2004.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓)) | ||
Theorem | vtoclgft 3553 | Closed theorem form of vtoclgf 3565. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by JJ, 11-Aug-2021.) Avoid ax-13 2390. (Revised by Gino Giotto, 6-Oct-2023.) |
⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ 𝑉) → 𝜓) | ||
Theorem | vtoclgftOLD 3554 | Obsolete version of vtoclgft 3553. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by JJ, 11-Aug-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ 𝑉) → 𝜓) | ||
Theorem | vtocldf 3555 | Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → 𝜓) & ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝜒) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | vtocld 3556* | Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | vtocl2d 3557* | Implicit substitution of two classes for two setvar variables. (Contributed by Thierry Arnoux, 25-Aug-2020.) (Revised by BTernaryTau, 19-Oct-2023.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | vtoclf 3558* | Implicit substitution of a class for a setvar variable. This is a generalization of chvar 2413. (Contributed by NM, 30-Aug-1993.) |
⊢ Ⅎ𝑥𝜓 & ⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ 𝜑 ⇒ ⊢ 𝜓 | ||
Theorem | vtocl 3559* | Implicit substitution of a class for a setvar variable. See also vtoclALT 3560. (Contributed by NM, 30-Aug-1993.) Remove dependency on ax-10 2145. (Revised by BJ, 29-Nov-2020.) (Proof shortened by SN, 20-Apr-2024.) |
⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ 𝜑 ⇒ ⊢ 𝜓 | ||
Theorem | vtoclALT 3560* | Alternate proof of vtocl 3559. Shorter but requires more axioms. (Contributed by NM, 30-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ 𝜑 ⇒ ⊢ 𝜓 | ||
Theorem | vtocl2 3561* | 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 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) & ⊢ 𝜑 ⇒ ⊢ 𝜓 | ||
Theorem | vtocl2OLD 3562* | Obsolete proof of vtocl2 3561 as of 23-Aug-2023. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) & ⊢ 𝜑 ⇒ ⊢ 𝜓 | ||
Theorem | vtocl3 3563* | Implicit substitution of classes for setvar variables. (Contributed by NM, 3-Jun-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) Avoid ax-10 2145 and ax-11 2161. (Revised by Gino Giotto, 20-Aug-2023.) (Proof shortened by Wolf Lammen, 23-Aug-2023.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) & ⊢ 𝜑 ⇒ ⊢ 𝜓 | ||
Theorem | vtoclb 3564* | Implicit substitution of a class for a setvar variable. (Contributed by NM, 23-Dec-1993.) |
⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃)) & ⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (𝜒 ↔ 𝜃) | ||
Theorem | vtoclgf 3565 | Implicit substitution of a class for a setvar variable, with bound-variable hypotheses in place of disjoint variable restrictions. (Contributed by NM, 21-Sep-2003.) (Proof shortened by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ 𝜑 ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝜓) | ||
Theorem | vtoclg1f 3566* | Version of vtoclgf 3565 with one non-freeness hypothesis replaced with a disjoint variable condition, thus avoiding dependency on ax-11 2161 and ax-13 2390. (Contributed by BJ, 1-May-2019.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ 𝜑 ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝜓) | ||
Theorem | vtoclg 3567* | Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995.) Avoid ax-12 2177. (Revised by SN, 20-Apr-2024.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ 𝜑 ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝜓) | ||
Theorem | vtoclgOLD 3568* | Obsolete version of vtoclg 3567 as of 20-Apr-2024. (Contributed by NM, 17-Apr-1995.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ 𝜑 ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝜓) | ||
Theorem | vtoclbg 3569* | 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 3570 | 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 3571 | 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 3572* | Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 25-Apr-1995.) Remove dependency on ax-10 2145, ax-11 2161, and ax-13 2390. (Revised by Steven Nguyen, 29-Nov-2022.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ 𝜑 ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜒) | ||
Theorem | vtoclgaf 3573* | 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 3574* | Implicit substitution of a class for a setvar variable. (Contributed by NM, 20-Aug-1995.) Avoid ax-10 2145 and ax-11 2161. (Revised by Gino Giotto, 20-Aug-2023.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 ∈ 𝐵 → 𝜑) ⇒ ⊢ (𝐴 ∈ 𝐵 → 𝜓) | ||
Theorem | vtocl2ga 3575* | Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.) Avoid ax-10 2145 and ax-11 2161. (Revised by Gino Giotto, 20-Aug-2023.) (Proof shortened by Wolf Lammen, 23-Aug-2023.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜑) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒) | ||
Theorem | vtocl2gaf 3576* | 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 | vtocl3gaf 3577* | 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 3578* | 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 | vtocl4g 3579* | Implicit substitution of 4 classes for 4 setvar variables. (Contributed by AV, 22-Jan-2019.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜌)) & ⊢ (𝑤 = 𝐷 → (𝜌 ↔ 𝜃)) & ⊢ 𝜑 ⇒ ⊢ (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇)) → 𝜃) | ||
Theorem | vtocl4ga 3580* | Implicit substitution of 4 classes for 4 setvar variables. (Contributed by AV, 22-Jan-2019.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜌)) & ⊢ (𝑤 = 𝐷 → (𝜌 ↔ 𝜃)) & ⊢ (((𝑥 ∈ 𝑄 ∧ 𝑦 ∈ 𝑅) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇)) → 𝜑) ⇒ ⊢ (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇)) → 𝜃) | ||
Theorem | vtocleg 3581* | Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝑥 = 𝐴 → 𝜑) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝜑) | ||
Theorem | vtoclegft 3582* | Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 3583.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.) |
⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → 𝜑) | ||
Theorem | vtoclef 3583* | Implicit substitution of a class for a setvar variable. (Contributed by NM, 18-Aug-1993.) |
⊢ Ⅎ𝑥𝜑 & ⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → 𝜑) ⇒ ⊢ 𝜑 | ||
Theorem | vtocle 3584* | Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.) |
⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → 𝜑) ⇒ ⊢ 𝜑 | ||
Theorem | vtoclri 3585* | 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 3586 | A closed version of spcimgf 3588. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓))) | ||
Theorem | spcgft 3587 | A closed version of spcgf 3590. (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 | spcimgf 3588 | 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 3589 | Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑)) | ||
Theorem | spcgf 3590 | 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 3591 | Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑)) | ||
Theorem | spcimdv 3592* | Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.) Avoid ax-10 2145 and ax-11 2161. (Revised by Gino Giotto, 20-Aug-2023.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒)) | ||
Theorem | spcdv 3593* | Rule of specialization, using implicit substitution. Analogous to rspcdv 3615. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒)) | ||
Theorem | spcimedv 3594* | 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 3595* | Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.) Avoid ax-10 2145, ax-11 2161. (Revised by Wolf Lammen, 25-Aug-2023.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)) | ||
Theorem | spcgvOLD 3596* | Obsolete version of spcgv 3595 as of 25-Aug-2023. (Contributed by NM, 22-Jun-1994.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)) | ||
Theorem | spcegv 3597* | Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.) Avoid ax-10 2145, ax-11 2161. (Revised by Wolf Lammen, 25-Aug-2023.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑)) | ||
Theorem | spcegvOLD 3598* | Obsolete version of spcegv 3597 as of 25-Aug-2023. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑)) | ||
Theorem | spcedv 3599* | Existential specialization, using implicit substitution, deduction version. (Contributed by RP, 12-Aug-2020.) |
⊢ (𝜑 → 𝑋 ∈ V) & ⊢ (𝜑 → 𝜒) & ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
Theorem | spc2egv 3600* | Existential specialization with two quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝜓 → ∃𝑥∃𝑦𝜑)) |
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