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Type | Label | Description |
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Statement | ||
Theorem | ra5 3001 | Restricted quantifier version of Axiom 5 of [Mendelson] p. 69. This is an axiom of a predicate calculus for a restricted domain. Compare the unrestricted stdpc5 1564. (Contributed by NM, 16-Jan-2004.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) | ||
Theorem | rmo2ilem 3002* | Condition implying restricted at-most-one quantifier. (Contributed by Jim Kingdon, 14-Jul-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) → ∃*𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rmo2i 3003* | Condition implying restricted at-most-one quantifier. (Contributed by NM, 17-Jun-2017.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) → ∃*𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rmo3 3004* | Restricted at-most-one quantifier using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) | ||
Theorem | rmob 3005* | Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.) |
⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜒)) ⇒ ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) → (𝐵 = 𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝜒))) | ||
Theorem | rmoi 3006* | Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) |
⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜒)) ⇒ ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓) ∧ (𝐶 ∈ 𝐴 ∧ 𝜒)) → 𝐵 = 𝐶) | ||
Syntax | csb 3007 | Extend class notation to include the proper substitution of a class for a set into another class. |
class ⦋𝐴 / 𝑥⦌𝐵 | ||
Definition | df-csb 3008* | Define the proper substitution of a class for a set into another class. The underlined brackets distinguish it from the substitution into a wff, wsbc 2913, to prevent ambiguity. Theorem sbcel1g 3026 shows an example of how ambiguity could arise if we didn't use distinguished brackets. Theorem sbccsbg 3036 recreates substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.) |
⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} | ||
Theorem | csb2 3009* | Alternate expression for the proper substitution into a class, without referencing substitution into a wff. Note that 𝑥 can be free in 𝐵 but cannot occur in 𝐴. (Contributed by NM, 2-Dec-2013.) |
⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑥(𝑥 = 𝐴 ∧ 𝑦 ∈ 𝐵)} | ||
Theorem | csbeq1 3010 | Analog of dfsbcq 2915 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.) |
⊢ (𝐴 = 𝐵 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶) | ||
Theorem | cbvcsbw 3011* | Version of cbvcsb 3012 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑦𝐶 & ⊢ Ⅎ𝑥𝐷 & ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) ⇒ ⊢ ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑦⦌𝐷 | ||
Theorem | cbvcsb 3012 | Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on 𝐴. (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro, 11-Dec-2016.) |
⊢ Ⅎ𝑦𝐶 & ⊢ Ⅎ𝑥𝐷 & ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) ⇒ ⊢ ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑦⦌𝐷 | ||
Theorem | cbvcsbv 3013* | Change the bound variable of a proper substitution into a class using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) |
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑦⦌𝐶 | ||
Theorem | csbeq1d 3014 | Equality deduction for proper substitution into a class. (Contributed by NM, 3-Dec-2005.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶) | ||
Theorem | csbid 3015 | Analog of sbid 1748 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.) |
⊢ ⦋𝑥 / 𝑥⦌𝐴 = 𝐴 | ||
Theorem | csbeq1a 3016 | Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.) |
⊢ (𝑥 = 𝐴 → 𝐵 = ⦋𝐴 / 𝑥⦌𝐵) | ||
Theorem | csbco 3017* |
Composition law for chained substitutions into a class.
Use the weaker csbcow 3018 when possible. (Contributed by NM, 10-Nov-2005.) (New usage is discouraged.) |
⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵 | ||
Theorem | csbcow 3018* | Composition law for chained substitutions into a class. Version of csbco 3017 with a disjoint variable condition. Although currently the proof is a direct reference to csbco 3017, we expect that the additional distinct variable condition will eventually enable us to remove usage of ax-bndl 1487 here. (Contributed by NM, 10-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵 | ||
Theorem | csbtt 3019 | Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). (Contributed by Mario Carneiro, 14-Oct-2016.) |
⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐵) → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵) | ||
Theorem | csbconstgf 3020 | Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). (Contributed by NM, 10-Nov-2005.) |
⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵) | ||
Theorem | csbconstg 3021* | Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). csbconstgf 3020 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵) | ||
Theorem | sbcel12g 3022 | Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶)) | ||
Theorem | sbceqg 3023 | Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)) | ||
Theorem | sbcnel12g 3024 | Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ∉ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∉ ⦋𝐴 / 𝑥⦌𝐶)) | ||
Theorem | sbcne12g 3025 | Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶)) | ||
Theorem | sbcel1g 3026* | Move proper substitution in and out of a membership relation. Note that the scope of [𝐴 / 𝑥] is the wff 𝐵 ∈ 𝐶, whereas the scope of ⦋𝐴 / 𝑥⦌ is the class 𝐵. (Contributed by NM, 10-Nov-2005.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝐶)) | ||
Theorem | sbceq1g 3027* | Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) | ||
Theorem | sbcel2g 3028* | Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ 𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶)) | ||
Theorem | sbceq2g 3029* | Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ 𝐵 = ⦋𝐴 / 𝑥⦌𝐶)) | ||
Theorem | csbcomg 3030* | Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶) | ||
Theorem | csbeq2 3031 | Substituting into equivalent classes gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.) |
⊢ (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) | ||
Theorem | csbeq2d 3032 | Formula-building deduction for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) | ||
Theorem | csbeq2dv 3033* | Formula-building deduction for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.) |
⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) | ||
Theorem | csbeq2i 3034 | Formula-building inference for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.) |
⊢ 𝐵 = 𝐶 ⇒ ⊢ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶 | ||
Theorem | csbvarg 3035 | The proper substitution of a class for setvar variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴) | ||
Theorem | sbccsbg 3036* | Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑})) | ||
Theorem | sbccsb2g 3037 | Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑})) | ||
Theorem | nfcsb1d 3038 | Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵) | ||
Theorem | nfcsb1 3039 | Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵 | ||
Theorem | nfcsb1v 3040* | Bound-variable hypothesis builder for substitution into a class. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 12-Oct-2016.) |
⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵 | ||
Theorem | nfcsbd 3041 | Deduction version of nfcsb 3042. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵) | ||
Theorem | nfcsb 3042 | Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵 | ||
Theorem | csbhypf 3043* | Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2738 for class substitution version. (Contributed by NM, 19-Dec-2008.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶) | ||
Theorem | csbiebt 3044* | Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3048.) (Contributed by NM, 11-Nov-2005.) |
⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) | ||
Theorem | csbiedf 3045* | Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝐶) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) | ||
Theorem | csbieb 3046* | Bidirectional conversion between an implicit class substitution hypothesis 𝑥 = 𝐴 → 𝐵 = 𝐶 and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008.) |
⊢ 𝐴 ∈ V & ⊢ Ⅎ𝑥𝐶 ⇒ ⊢ (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) | ||
Theorem | csbiebg 3047* | Bidirectional conversion between an implicit class substitution hypothesis 𝑥 = 𝐴 → 𝐵 = 𝐶 and its explicit substitution equivalent. (Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro, 11-Dec-2016.) |
⊢ Ⅎ𝑥𝐶 ⇒ ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) | ||
Theorem | csbiegf 3048* | Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
⊢ (𝐴 ∈ 𝑉 → Ⅎ𝑥𝐶) & ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) | ||
Theorem | csbief 3049* | Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
⊢ 𝐴 ∈ V & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶 | ||
Theorem | csbie 3050* | Conversion of implicit substitution to explicit substitution into a class. (Contributed by AV, 2-Dec-2019.) |
⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶 | ||
Theorem | csbied 3051* | Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) | ||
Theorem | csbied2 3052* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = 𝐷) | ||
Theorem | csbie2t 3053* | Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3054). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = 𝐷) | ||
Theorem | csbie2 3054* | Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) ⇒ ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = 𝐷 | ||
Theorem | csbie2g 3055* | Conversion of implicit substitution to explicit class substitution. This version of sbcie 2947 avoids a disjointness condition on 𝑥 and 𝐴 by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.) |
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) & ⊢ (𝑦 = 𝐴 → 𝐶 = 𝐷) ⇒ ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐷) | ||
Theorem | sbcnestgf 3056 | Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.) |
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑)) | ||
Theorem | csbnestgf 3057 | Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.) |
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝐶) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶) | ||
Theorem | sbcnestg 3058* | Nest the composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑)) | ||
Theorem | csbnestg 3059* | Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶) | ||
Theorem | csbnest1g 3060 | Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑥⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑥⦌𝐶) | ||
Theorem | csbidmg 3061* | Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵) | ||
Theorem | sbcco3g 3062* | Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.) |
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜑)) | ||
Theorem | csbco3g 3063* | Composition of two class substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.) |
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = ⦋𝐶 / 𝑦⦌𝐷) | ||
Theorem | rspcsbela 3064* | Special case related to rspsbc 2995. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.) |
⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷) → ⦋𝐴 / 𝑥⦌𝐶 ∈ 𝐷) | ||
Theorem | sbnfc2 3065* | Two ways of expressing "𝑥 is (effectively) not free in 𝐴." (Contributed by Mario Carneiro, 14-Oct-2016.) |
⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦∀𝑧⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴) | ||
Theorem | csbabg 3066* | Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} = {𝑦 ∣ [𝐴 / 𝑥]𝜑}) | ||
Theorem | cbvralcsf 3067 | A more general version of cbvralf 2651 that doesn't require 𝐴 and 𝐵 to be distinct from 𝑥 or 𝑦. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓) | ||
Theorem | cbvrexcsf 3068 | A more general version of cbvrexf 2652 that has no distinct variable restrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.) (Proof shortened by Mario Carneiro, 7-Dec-2014.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓) | ||
Theorem | cbvreucsf 3069 | A more general version of cbvreuv 2659 that has no distinct variable rextrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐵 𝜓) | ||
Theorem | cbvrabcsf 3070 | A more general version of cbvrab 2687 with no distinct variable restrictions. (Contributed by Andrew Salmon, 13-Jul-2011.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓} | ||
Theorem | cbvralv2 3071* | Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒) | ||
Theorem | cbvrexv2 3072* | Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒) | ||
Syntax | cdif 3073 | Extend class notation to include class difference (read: "𝐴 minus 𝐵"). |
class (𝐴 ∖ 𝐵) | ||
Syntax | cun 3074 | Extend class notation to include union of two classes (read: "𝐴 union 𝐵"). |
class (𝐴 ∪ 𝐵) | ||
Syntax | cin 3075 | Extend class notation to include the intersection of two classes (read: "𝐴 intersect 𝐵"). |
class (𝐴 ∩ 𝐵) | ||
Syntax | wss 3076 | Extend wff notation to include the subclass relation. This is read "𝐴 is a subclass of 𝐵 " or "𝐵 includes 𝐴." When 𝐴 exists as a set, it is also read "𝐴 is a subset of 𝐵." |
wff 𝐴 ⊆ 𝐵 | ||
Theorem | difjust 3077* | Soundness justification theorem for df-dif 3078. (Contributed by Rodolfo Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)} | ||
Definition | df-dif 3078* | Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. Contrast this operation with union (𝐴 ∪ 𝐵) (df-un 3080) and intersection (𝐴 ∩ 𝐵) (df-in 3082). Several notations are used in the literature; we chose the ∖ convention used in Definition 5.3 of [Eisenberg] p. 67 instead of the more common minus sign to reserve the latter for later use in, e.g., arithmetic. We will use the terminology "𝐴 excludes 𝐵 " to mean 𝐴 ∖ 𝐵. We will use "𝐵 is removed from 𝐴 " to mean 𝐴 ∖ {𝐵} i.e. the removal of an element or equivalently the exclusion of a singleton. (Contributed by NM, 29-Apr-1994.) |
⊢ (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)} | ||
Theorem | unjust 3079* | Soundness justification theorem for df-un 3080. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)} | ||
Definition | df-un 3080* | Define the union of two classes. Definition 5.6 of [TakeutiZaring] p. 16. Contrast this operation with difference (𝐴 ∖ 𝐵) (df-dif 3078) and intersection (𝐴 ∩ 𝐵) (df-in 3082). (Contributed by NM, 23-Aug-1993.) |
⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)} | ||
Theorem | injust 3081* | Soundness justification theorem for df-in 3082. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} | ||
Definition | df-in 3082* | Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. Contrast this operation with union (𝐴 ∪ 𝐵) (df-un 3080) and difference (𝐴 ∖ 𝐵) (df-dif 3078). (Contributed by NM, 29-Apr-1994.) |
⊢ (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} | ||
Theorem | dfin5 3083* | Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.) |
⊢ (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} | ||
Theorem | dfdif2 3084* | Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.) |
⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} | ||
Theorem | eldif 3085 | Expansion of membership in a class difference. (Contributed by NM, 29-Apr-1994.) |
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) | ||
Theorem | eldifd 3086 | If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3085. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶)) | ||
Theorem | eldifad 3087 | If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3085. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶)) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐵) | ||
Theorem | eldifbd 3088 | If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3085. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶)) ⇒ ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶) | ||
Definition | df-ss 3089 | Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18. Note that 𝐴 ⊆ 𝐴 (proved in ssid 3122). For a more traditional definition, but requiring a dummy variable, see dfss2 3091. Other possible definitions are given by dfss3 3092, ssequn1 3251, ssequn2 3254, and sseqin2 3300. (Contributed by NM, 27-Apr-1994.) |
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴) | ||
Theorem | dfss 3090 | Variant of subclass definition df-ss 3089. (Contributed by NM, 3-Sep-2004.) |
⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐴 ∩ 𝐵)) | ||
Theorem | dfss2 3091* | Alternate definition of the subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Jan-2002.) |
⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | ||
Theorem | dfss3 3092* | Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.) |
⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) | ||
Theorem | dfss2f 3093 | Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 3-Jul-1994.) (Revised by Andrew Salmon, 27-Aug-2011.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | ||
Theorem | dfss3f 3094 | Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 20-Mar-2004.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) | ||
Theorem | nfss 3095 | If 𝑥 is not free in 𝐴 and 𝐵, it is not free in 𝐴 ⊆ 𝐵. (Contributed by NM, 27-Dec-1996.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐴 ⊆ 𝐵 | ||
Theorem | ssel 3096 | Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)) | ||
Theorem | ssel2 3097 | Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐵) | ||
Theorem | sseli 3098 | Membership inference from subclass relationship. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵) | ||
Theorem | sselii 3099 | Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.) |
⊢ 𝐴 ⊆ 𝐵 & ⊢ 𝐶 ∈ 𝐴 ⇒ ⊢ 𝐶 ∈ 𝐵 | ||
Theorem | sseldi 3100 | Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.) |
⊢ 𝐴 ⊆ 𝐵 & ⊢ (𝜑 → 𝐶 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝐶 ∈ 𝐵) |
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