![]() |
Intuitionistic Logic Explorer Theorem List (p. 30 of 149) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > ILE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rexrab 2901* | 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)) | ||
Theorem | ralab2 2902* | Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∀𝑦(𝜑 → 𝜒)) | ||
Theorem | ralrab2 2903* | Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∀𝑦 ∈ 𝐴 (𝜑 → 𝜒)) | ||
Theorem | rexab2 2904* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∃𝑦(𝜑 ∧ 𝜒)) | ||
Theorem | rexrab2 2905* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝜒)) | ||
Theorem | abidnf 2906* | Identity used to create closed-form versions of bound-variable hypothesis builders for class expressions. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Mario Carneiro, 12-Oct-2016.) |
⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) | ||
Theorem | dedhb 2907* | A deduction theorem for converting the inference ⊢ Ⅎ𝑥𝐴 => ⊢ 𝜑 into a closed theorem. Use nfa1 1541 and nfab 2324 to eliminate the hypothesis of the substitution instance 𝜓 of the inference. For converting the inference form into a deduction form, abidnf 2906 is useful. (Contributed by NM, 8-Dec-2006.) |
⊢ (𝐴 = {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} → (𝜑 ↔ 𝜓)) & ⊢ 𝜓 ⇒ ⊢ (Ⅎ𝑥𝐴 → 𝜑) | ||
Theorem | eqeu 2908* | A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓 ∧ ∀𝑥(𝜑 → 𝑥 = 𝐴)) → ∃!𝑥𝜑) | ||
Theorem | eueq 2909* | Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.) |
⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴) | ||
Theorem | eueq1 2910* | Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∃!𝑥 𝑥 = 𝐴 | ||
Theorem | eueq2dc 2911* | Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (DECID 𝜑 → ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵))) | ||
Theorem | eueq3dc 2912* | Equality has existential uniqueness (split into 3 cases). (Contributed by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro, 28-Sep-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ ¬ (𝜑 ∧ 𝜓) ⇒ ⊢ (DECID 𝜑 → (DECID 𝜓 → ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)))) | ||
Theorem | moeq 2913* | There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.) |
⊢ ∃*𝑥 𝑥 = 𝐴 | ||
Theorem | moeq3dc 2914* | "At most one" property of equality (split into 3 cases). (Contributed by Jim Kingdon, 7-Jul-2018.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ ¬ (𝜑 ∧ 𝜓) ⇒ ⊢ (DECID 𝜑 → (DECID 𝜓 → ∃*𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)))) | ||
Theorem | mosubt 2915* | "At most one" remains true after substitution. (Contributed by Jim Kingdon, 18-Jan-2019.) |
⊢ (∀𝑦∃*𝑥𝜑 → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑)) | ||
Theorem | mosub 2916* | "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.) |
⊢ ∃*𝑥𝜑 ⇒ ⊢ ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑) | ||
Theorem | mo2icl 2917* | Theorem for inferring "at most one". (Contributed by NM, 17-Oct-1996.) |
⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑) | ||
Theorem | mob2 2918* | Consequence of "at most one". (Contributed by NM, 2-Jan-2015.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑 ∧ 𝜑) → (𝑥 = 𝐴 ↔ 𝜓)) | ||
Theorem | moi2 2919* | Consequence of "at most one". (Contributed by NM, 29-Jun-2008.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑) ∧ (𝜑 ∧ 𝜓)) → 𝑥 = 𝐴) | ||
Theorem | mob 2920* | Equality implied by "at most one". (Contributed by NM, 18-Feb-2006.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) ⇒ ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∃*𝑥𝜑 ∧ 𝜓) → (𝐴 = 𝐵 ↔ 𝜒)) | ||
Theorem | moi 2921* | Equality implied by "at most one". (Contributed by NM, 18-Feb-2006.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) ⇒ ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∃*𝑥𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝐴 = 𝐵) | ||
Theorem | morex 2922* | Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ 𝐵 ∈ V & ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥𝜑) → (𝜓 → 𝐵 ∈ 𝐴)) | ||
Theorem | euxfr2dc 2923* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 14-Nov-2004.) |
⊢ 𝐴 ∈ V & ⊢ ∃*𝑦 𝑥 = 𝐴 ⇒ ⊢ (DECID ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑)) | ||
Theorem | euxfrdc 2924* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 14-Nov-2004.) |
⊢ 𝐴 ∈ V & ⊢ ∃!𝑦 𝑥 = 𝐴 & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (DECID ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)) | ||
Theorem | euind 2925* | Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.) |
⊢ 𝐵 ∈ V & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ ((∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) ∧ ∃𝑥𝜑) → ∃!𝑧∀𝑥(𝜑 → 𝑧 = 𝐴)) | ||
Theorem | reu2 2926* | A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.) |
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))) | ||
Theorem | reu6 2927* | A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.) |
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)) | ||
Theorem | reu3 2928* | A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.) |
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦))) | ||
Theorem | reu6i 2929* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ ((𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜑) | ||
Theorem | eqreu 2930* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐵 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rmo4 2931* | Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) | ||
Theorem | reu4 2932* | Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))) | ||
Theorem | reu7 2933* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) | ||
Theorem | reu8 2934* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) | ||
Theorem | rmo3f 2935* | Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) | ||
Theorem | rmo4f 2936* | Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) | ||
Theorem | reueq 2937* | Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.) |
⊢ (𝐵 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 𝑥 = 𝐵) | ||
Theorem | rmoan 2938 | Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.) |
⊢ (∃*𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑)) | ||
Theorem | rmoim 2939 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | rmoimia 2940 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rmoimi2 2941 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜓)) ⇒ ⊢ (∃*𝑥 ∈ 𝐵 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑) | ||
Theorem | 2reuswapdc 2942* | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.) |
⊢ (DECID ∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) → (∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 → (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑))) | ||
Theorem | reuind 2943* | Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ ((∀𝑥∀𝑦(((𝐴 ∈ 𝐶 ∧ 𝜑) ∧ (𝐵 ∈ 𝐶 ∧ 𝜓)) → 𝐴 = 𝐵) ∧ ∃𝑥(𝐴 ∈ 𝐶 ∧ 𝜑)) → ∃!𝑧 ∈ 𝐶 ∀𝑥((𝐴 ∈ 𝐶 ∧ 𝜑) → 𝑧 = 𝐴)) | ||
Theorem | 2rmorex 2944* | Double restricted quantification with "at most one," analogous to 2moex 2112. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝜑) | ||
Theorem | nelrdva 2945* | Deduce negative membership from an implication. (Contributed by Thierry Arnoux, 27-Nov-2017.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ 𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴) | ||
This is a very useless definition, which "abbreviates" (𝑥 = 𝑦 → 𝜑) as CondEq(𝑥 = 𝑦 → 𝜑). What this display hides, though, is that the first expression, even though it has a shorter constant string, is actually much more complicated in its parse tree: it is parsed as (wi (wceq (cv vx) (cv vy)) wph), while the CondEq version is parsed as (wcdeq vx vy wph). It also allows us to give a name to the specific ternary operation (𝑥 = 𝑦 → 𝜑). This is all used as part of a metatheorem: we want to say that ⊢ (𝑥 = 𝑦 → (𝜑(𝑥) ↔ 𝜑(𝑦))) and ⊢ (𝑥 = 𝑦 → 𝐴(𝑥) = 𝐴(𝑦)) are provable, for any expressions 𝜑(𝑥) or 𝐴(𝑥) in the language. The proof is by induction, so the base case is each of the primitives, which is why you will see a theorem for each of the set.mm primitive operations. The metatheorem comes with a disjoint variables condition: every variable in 𝜑(𝑥) is assumed disjoint from 𝑥 except 𝑥 itself. For such a proof by induction, we must consider each of the possible forms of 𝜑(𝑥). If it is a variable other than 𝑥, then we have CondEq(𝑥 = 𝑦 → 𝐴 = 𝐴) or CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜑)), which is provable by cdeqth 2950 and reflexivity. Since we are only working with class and wff expressions, it can't be 𝑥 itself in set.mm, but if it was we'd have to also prove CondEq(𝑥 = 𝑦 → 𝑥 = 𝑦) (where set equality is being used on the right). Otherwise, it is a primitive operation applied to smaller expressions. In these cases, for each setvar variable parameter to the operation, we must consider if it is equal to 𝑥 or not, which yields 2^n proof obligations. Luckily, all primitive operations in set.mm have either zero or one set variable, so we only need to prove one statement for the non-set constructors (like implication) and two for the constructors taking a set (the forall and the class builder). In each of the primitive proofs, we are allowed to assume that 𝑦 is disjoint from 𝜑(𝑥) and vice versa, because this is maintained through the induction. This is how we satisfy the disjoint variable conditions of cdeqab1 2955 and cdeqab 2953. | ||
Syntax | wcdeq 2946 | Extend wff notation to include conditional equality. This is a technical device used in the proof that Ⅎ is the not-free predicate, and that definitions are conservative as a result. |
wff CondEq(𝑥 = 𝑦 → 𝜑) | ||
Definition | df-cdeq 2947 | Define conditional equality. All the notation to the left of the ↔ is fake; the parentheses and arrows are all part of the notation, which could equally well be written CondEq𝑥𝑦𝜑. On the right side is the actual implication arrow. The reason for this definition is to "flatten" the structure on the right side (whose tree structure is something like (wi (wceq (cv vx) (cv vy)) wph) ) into just (wcdeq vx vy wph). (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (CondEq(𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑)) | ||
Theorem | cdeqi 2948 | Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (𝑥 = 𝑦 → 𝜑) ⇒ ⊢ CondEq(𝑥 = 𝑦 → 𝜑) | ||
Theorem | cdeqri 2949 | Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → 𝜑) ⇒ ⊢ (𝑥 = 𝑦 → 𝜑) | ||
Theorem | cdeqth 2950 | Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ 𝜑 ⇒ ⊢ CondEq(𝑥 = 𝑦 → 𝜑) | ||
Theorem | cdeqnot 2951 | Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓)) | ||
Theorem | cdeqal 2952* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓)) | ||
Theorem | cdeqab 2953* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → {𝑧 ∣ 𝜑} = {𝑧 ∣ 𝜓}) | ||
Theorem | cdeqal1 2954* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) | ||
Theorem | cdeqab1 2955* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}) | ||
Theorem | cdeqim 2956 | Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ CondEq(𝑥 = 𝑦 → (𝜒 ↔ 𝜃)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))) | ||
Theorem | cdeqcv 2957 | Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → 𝑥 = 𝑦) | ||
Theorem | cdeqeq 2958 | Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ CondEq(𝑥 = 𝑦 → 𝐶 = 𝐷) ⇒ ⊢ CondEq(𝑥 = 𝑦 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) | ||
Theorem | cdeqel 2959 | Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ CondEq(𝑥 = 𝑦 → 𝐶 = 𝐷) ⇒ ⊢ CondEq(𝑥 = 𝑦 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) | ||
Theorem | nfcdeq 2960* | If we have a conditional equality proof, where 𝜑 is 𝜑(𝑥) and 𝜓 is 𝜑(𝑦), and 𝜑(𝑥) in fact does not have 𝑥 free in it according to Ⅎ, then 𝜑(𝑥) ↔ 𝜑(𝑦) unconditionally. This proves that Ⅎ𝑥𝜑 is actually a not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝜑 ↔ 𝜓) | ||
Theorem | nfccdeq 2961* | Variation of nfcdeq 2960 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ 𝐴 = 𝐵 | ||
Theorem | ru 2962 |
Russell's Paradox. Proposition 4.14 of [TakeutiZaring] p. 14.
In the late 1800s, Frege's Axiom of (unrestricted) Comprehension, expressed in our notation as 𝐴 ∈ V, asserted that any collection of sets 𝐴 is a set i.e. belongs to the universe V of all sets. In particular, by substituting {𝑥 ∣ 𝑥 ∉ 𝑥} (the "Russell class") for 𝐴, it asserted {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V, meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V. This contradiction was discovered by Russell in 1901 (published in 1903), invalidating the Comprehension Axiom and leading to the collapse of Frege's system. In 1908, Zermelo rectified this fatal flaw by replacing Comprehension with a weaker Subset (or Separation) Axiom asserting that 𝐴 is a set only when it is smaller than some other set 𝐵. The intuitionistic set theory IZF includes such a separation axiom, Axiom 6 of [Crosilla] p. "Axioms of CZF and IZF", which we include as ax-sep 4122. (Contributed by NM, 7-Aug-1994.) |
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V | ||
Syntax | wsbc 2963 | Extend wff notation to include the proper substitution of a class for a set. Read this notation as "the proper substitution of class 𝐴 for setvar variable 𝑥 in wff 𝜑". |
wff [𝐴 / 𝑥]𝜑 | ||
Definition | df-sbc 2964 |
Define the proper substitution of a class for a set.
When 𝐴 is a proper class, our definition evaluates to false. This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 2989 for our definition, which always evaluates to true for proper classes. Our definition also does not produce the same results as discussed in the proof of Theorem 6.6 of [Quine] p. 42 (although Theorem 6.6 itself does hold, as shown by dfsbcq 2965 below). Unfortunately, Quine's definition requires a recursive syntactical breakdown of 𝜑, and it does not seem possible to express it with a single closed formula. If we did not want to commit to any specific proper class behavior, we could use this definition only to prove Theorem dfsbcq 2965, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2964 in the form of sbc8g 2971. However, the behavior of Quine's definition at proper classes is similarly arbitrary, and for practical reasons (to avoid having to prove sethood of 𝐴 in every use of this definition) we allow direct reference to df-sbc 2964 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class. The related definition df-csb defines proper substitution into a class variable (as opposed to a wff variable). (Contributed by NM, 14-Apr-1995.) (Revised by NM, 25-Dec-2016.) |
⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) | ||
Theorem | dfsbcq 2965 |
This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds
under both our definition and Quine's, provides us with a weak definition
of the proper substitution of a class for a set. Since our df-sbc 2964 does
not result in the same behavior as Quine's for proper classes, if we
wished to avoid conflict with Quine's definition we could start with this
theorem and dfsbcq2 2966 instead of df-sbc 2964. (dfsbcq2 2966 is needed because
unlike Quine we do not overload the df-sb 1763 syntax.) As a consequence of
these theorems, we can derive sbc8g 2971, which is a weaker version of
df-sbc 2964 that leaves substitution undefined when 𝐴 is a
proper class.
However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 2971, so we will allow direct use of df-sbc 2964. Proper substiution with a proper class is rarely needed, and when it is, we can simply use the expansion of Quine's definition. (Contributed by NM, 14-Apr-1995.) |
⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑥]𝜑)) | ||
Theorem | dfsbcq2 2966 | This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, relates logic substitution df-sb 1763 and substitution for class variables df-sbc 2964. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2965. (Contributed by NM, 31-Dec-2016.) |
⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | ||
Theorem | sbsbc 2967 | Show that df-sb 1763 and df-sbc 2964 are equivalent when the class term 𝐴 in df-sbc 2964 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1763 for proofs involving df-sbc 2964. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.) |
⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | ||
Theorem | sbceq1d 2968 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜓)) | ||
Theorem | sbceq1dd 2969 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → [𝐴 / 𝑥]𝜓) ⇒ ⊢ (𝜑 → [𝐵 / 𝑥]𝜓) | ||
Theorem | sbceqbid 2970* | Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜒)) | ||
Theorem | sbc8g 2971 | This is the closest we can get to df-sbc 2964 if we start from dfsbcq 2965 (see its comments) and dfsbcq2 2966. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})) | ||
Theorem | sbcex 2972 | By our definition of proper substitution, it can only be true if the substituted expression is a set. (Contributed by Mario Carneiro, 13-Oct-2016.) |
⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) | ||
Theorem | sbceq1a 2973 | Equality theorem for class substitution. Class version of sbequ12 1771. (Contributed by NM, 26-Sep-2003.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑)) | ||
Theorem | sbceq2a 2974 | Equality theorem for class substitution. Class version of sbequ12r 1772. (Contributed by NM, 4-Jan-2017.) |
⊢ (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) | ||
Theorem | spsbc 2975 | Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See also stdpc4 1775 and rspsbc 3046. (Contributed by NM, 16-Jan-2004.) |
⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)) | ||
Theorem | spsbcd 2976 | Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See also stdpc4 1775 and rspsbc 3046. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑥𝜓) ⇒ ⊢ (𝜑 → [𝐴 / 𝑥]𝜓) | ||
Theorem | sbcth 2977 | A substitution into a theorem remains true (when 𝐴 is a set). (Contributed by NM, 5-Nov-2005.) |
⊢ 𝜑 ⇒ ⊢ (𝐴 ∈ 𝑉 → [𝐴 / 𝑥]𝜑) | ||
Theorem | sbcthdv 2978* | Deduction version of sbcth 2977. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → [𝐴 / 𝑥]𝜓) | ||
Theorem | sbcid 2979 | An identity theorem for substitution. See sbid 1774. (Contributed by Mario Carneiro, 18-Feb-2017.) |
⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) | ||
Theorem | nfsbc1d 2980 | Deduction version of nfsbc1 2981. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝜓) | ||
Theorem | nfsbc1 2981 | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑 | ||
Theorem | nfsbc1v 2982* | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |
⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑 | ||
Theorem | nfsbcd 2983 | Deduction version of nfsbc 2984. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓) | ||
Theorem | nfsbc 2984 | Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑 | ||
Theorem | sbcco 2985* | A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |
⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑) | ||
Theorem | sbcco2 2986* | A composition law for class substitution. Importantly, 𝑥 may occur free in the class expression substituted for 𝐴. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑) | ||
Theorem | sbc5 2987* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) |
⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) | ||
Theorem | sbc6g 2988* | An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))) | ||
Theorem | sbc6 2989* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)) | ||
Theorem | sbc7 2990* | An equivalence for class substitution in the spirit of df-clab 2164. Note that 𝑥 and 𝐴 don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) |
⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜑)) | ||
Theorem | cbvsbcw 2991* | Version of cbvsbc 2992 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓) | ||
Theorem | cbvsbc 2992 | Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓) | ||
Theorem | cbvsbcv 2993* | Change the bound variable of a class substitution using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓) | ||
Theorem | sbciegft 2994* | Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 2995.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) | ||
Theorem | sbciegf 2995* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) | ||
Theorem | sbcieg 2996* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) | ||
Theorem | sbcie2g 2997* | Conversion of implicit substitution to explicit class substitution. This version of sbcie 2998 avoids a disjointness condition on 𝑥 and 𝐴 by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜒)) | ||
Theorem | sbcie 2998* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.) |
⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) | ||
Theorem | sbciedf 2999* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) & ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜒) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)) | ||
Theorem | sbcied 3000* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |