Home | Metamath
Proof Explorer Theorem List (p. 340 of 450) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | Metamath Proof Explorer
(1-28698) |
Hilbert Space Explorer
(28699-30221) |
Users' Mathboxes
(30222-44913) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | bj-con2comi 33901 | Inference associated with bj-con2com 33900. Its associated inference is mt2 202. TODO: when in the main part, add to mt2 202 that it is the inference associated with bj-con2comi 33901. (Contributed by BJ, 19-Mar-2020.) |
⊢ 𝜑 ⇒ ⊢ ((𝜓 → ¬ 𝜑) → ¬ 𝜓) | ||
Theorem | bj-pm2.01i 33902 | Inference associated with the weak Clavius law pm2.01 191. (Contributed by BJ, 30-Mar-2020.) |
⊢ (𝜑 → ¬ 𝜑) ⇒ ⊢ ¬ 𝜑 | ||
Theorem | bj-nimn 33903 | If a formula is true, then it does not imply its negation. (Contributed by BJ, 19-Mar-2020.) A shorter proof is possible using id 22 and jc 163, however, the present proof uses theorems that are more basic than jc 163. (Proof modification is discouraged.) |
⊢ (𝜑 → ¬ (𝜑 → ¬ 𝜑)) | ||
Theorem | bj-nimni 33904 | Inference associated with bj-nimn 33903. (Contributed by BJ, 19-Mar-2020.) |
⊢ 𝜑 ⇒ ⊢ ¬ (𝜑 → ¬ 𝜑) | ||
Theorem | bj-peircei 33905 | Inference associated with peirce 204. (Contributed by BJ, 30-Mar-2020.) |
⊢ ((𝜑 → 𝜓) → 𝜑) ⇒ ⊢ 𝜑 | ||
Theorem | bj-looinvi 33906 | Inference associated with looinv 205. Its associated inference is bj-looinvii 33907. (Contributed by BJ, 30-Mar-2020.) |
⊢ ((𝜑 → 𝜓) → 𝜓) ⇒ ⊢ ((𝜓 → 𝜑) → 𝜑) | ||
Theorem | bj-looinvii 33907 | Inference associated with bj-looinvi 33906. (Contributed by BJ, 30-Mar-2020.) |
⊢ ((𝜑 → 𝜓) → 𝜓) & ⊢ (𝜓 → 𝜑) ⇒ ⊢ 𝜑 | ||
A few lemmas about disjunction. The fundamental theorems in this family are the dual statements pm4.71 560 and pm4.72 946. See also biort 932 and biorf 933. | ||
Theorem | bj-jaoi1 33908 | Shortens orfa2 35368 (58>53), pm1.2 900 (20>18), pm1.2 900 (20>18), pm2.4 903 (31>25), pm2.41 904 (31>25), pm2.42 939 (38>32), pm3.2ni 877 (43>39), pm4.44 993 (55>51). (Contributed by BJ, 30-Sep-2019.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜑 ∨ 𝜓) → 𝜓) | ||
Theorem | bj-jaoi2 33909 | Shortens consensus 1047 (110>106), elnn0z 11997 (336>329), pm1.2 900 (20>19), pm3.2ni 877 (43>39), pm4.44 993 (55>51). (Contributed by BJ, 30-Sep-2019.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜓 ∨ 𝜑) → 𝜓) | ||
A few other characterizations of the bicondional. The inter-definability of logical connectives offers many ways to express a given statement. Some useful theorems in this regard are df-or 844, df-an 399, pm4.64 845, imor 849, pm4.62 852 through pm4.67 401, and, for the De Morgan laws, ianor 978 through pm4.57 987. | ||
Theorem | bj-dfbi4 33910 | Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.) |
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓))) | ||
Theorem | bj-dfbi5 33911 | Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.) |
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) → (𝜑 ∧ 𝜓))) | ||
Theorem | bj-dfbi6 33912 | Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.) |
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ↔ (𝜑 ∧ 𝜓))) | ||
Theorem | bj-bijust0ALT 33913 | Alternate proof of bijust0 206; shorter but using additional intermediate results. (Contributed by NM, 11-May-1999.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Revised by BJ, 19-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ ((𝜑 → 𝜑) → ¬ (𝜑 → 𝜑)) | ||
Theorem | bj-bijust00 33914 | A self-implication does not imply the negation of a self-implication. Most general theorem of which bijust 207 is an instance (bijust0 206 and bj-bijust0ALT 33913 are therefore also instances of it). (Contributed by BJ, 7-Sep-2022.) |
⊢ ¬ ((𝜑 → 𝜑) → ¬ (𝜓 → 𝜓)) | ||
Theorem | bj-consensus 33915 | Version of consensus 1047 expressed using the conditional operator. (Remark: it may be better to express it as consensus 1047, using only binary connectives, and hinting at the fact that it is a Boolean algebra identity, like the absorption identities.) (Contributed by BJ, 30-Sep-2019.) |
⊢ ((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓 ∧ 𝜒)) ↔ if-(𝜑, 𝜓, 𝜒)) | ||
Theorem | bj-consensusALT 33916 | Alternate proof of bj-consensus 33915. (Contributed by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓 ∧ 𝜒)) ↔ if-(𝜑, 𝜓, 𝜒)) | ||
Theorem | bj-df-ifc 33917* | Candidate definition for the conditional operator for classes. This is in line with the definition of a class as the extension of a predicate in df-clab 2803. We reprove the current df-if 4471 from it in bj-dfif 33918. (Contributed by BJ, 20-Sep-2019.) (Proof modification is discouraged.) |
⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)} | ||
Theorem | bj-dfif 33918* | Alternate definition of the conditional operator for classes, which used to be the main definition. (Contributed by BJ, 26-Dec-2023.) (Proof modification is discouraged.) |
⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵))} | ||
Theorem | bj-ififc 33919 | A biconditional connecting the conditional operator for propositions and the conditional operator for classes. Note that there is no sethood hypothesis on 𝑋: it is implied by either side. (Contributed by BJ, 24-Sep-2019.) Generalize statement from setvar 𝑥 to class 𝑋. (Revised by BJ, 26-Dec-2023.) |
⊢ (𝑋 ∈ if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝑋 ∈ 𝐴, 𝑋 ∈ 𝐵)) | ||
Miscellaneous theorems of propositional calculus. | ||
Theorem | bj-imbi12 33920 | Uncurried (imported) form of imbi12 349. (Contributed by BJ, 6-May-2019.) |
⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))) | ||
Theorem | bj-biorfi 33921 | This should be labeled "biorfi" while the current biorfi 935 should be labeled "biorfri". The dual of biorf 933 is not biantr 804 but iba 530 (and ibar 531). So there should also be a "biorfr". (Note that these four statements can actually be strengthened to biconditionals.) (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.) |
⊢ ¬ 𝜑 ⇒ ⊢ (𝜓 ↔ (𝜑 ∨ 𝜓)) | ||
Theorem | bj-falor 33922 | Dual of truan 1547 (which has biconditional reversed). (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 ↔ (⊥ ∨ 𝜑)) | ||
Theorem | bj-falor2 33923 | Dual of truan 1547. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.) |
⊢ ((⊥ ∨ 𝜑) ↔ 𝜑) | ||
Theorem | bj-bibibi 33924 | A property of the biconditional. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 ↔ (𝜓 ↔ (𝜑 ↔ 𝜓))) | ||
Theorem | bj-imn3ani 33925 | Duplication of bnj1224 32077. Three-fold version of imnani 403. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by BJ, 22-Oct-2019.) (Proof modification is discouraged.) |
⊢ ¬ (𝜑 ∧ 𝜓 ∧ 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝜒) | ||
Theorem | bj-andnotim 33926 | Two ways of expressing a certain ternary connective. Note the respective positions of the three formulas on each side of the biconditional. (Contributed by BJ, 6-Oct-2018.) |
⊢ (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ ((𝜑 → 𝜓) ∨ 𝜒)) | ||
Theorem | bj-bi3ant 33927 | This used to be in the main part. (Contributed by Wolf Lammen, 14-May-2013.) (Revised by BJ, 14-Jun-2019.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (((𝜃 → 𝜏) → 𝜑) → (((𝜏 → 𝜃) → 𝜓) → ((𝜃 ↔ 𝜏) → 𝜒))) | ||
Theorem | bj-bisym 33928 | This used to be in the main part. (Contributed by Wolf Lammen, 14-May-2013.) (Revised by BJ, 14-Jun-2019.) |
⊢ (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → (((𝜓 → 𝜑) → (𝜃 → 𝜒)) → ((𝜑 ↔ 𝜓) → (𝜒 ↔ 𝜃)))) | ||
Theorem | bj-bixor 33929 | Equivalence of two ternary operations. Note the identical order and parenthesizing of the three arguments in both expressions. (Contributed by BJ, 31-Dec-2023.) |
⊢ ((𝜑 ↔ (𝜓 ⊻ 𝜒)) ↔ (𝜑 ⊻ (𝜓 ↔ 𝜒))) | ||
In this section, we prove some theorems related to modal logic. For modal logic, we refer to https://en.wikipedia.org/wiki/Kripke_semantics, https://en.wikipedia.org/wiki/Modal_logic and https://plato.stanford.edu/entries/logic-modal/. Monadic first-order logic (i.e., with quantification over only one variable) is bi-interpretable with modal logic, by mapping ∀𝑥 to "necessity" (generally denoted by a box) and ∃𝑥 to "possibility" (generally denoted by a diamond). Therefore, we use these quantifiers so as not to introduce new symbols. (To be strictly within modal logic, we should add disjoint variable conditions between 𝑥 and any other metavariables appearing in the statements.) For instance, ax-gen 1795 corresponds to the necessitation rule of modal logic, and ax-4 1809 corresponds to the distributivity axiom (K) of modal logic, also called the Kripke scheme. Modal logics satisfying these rule and axiom are called "normal modal logics", of which the most important modal logics are. The minimal normal modal logic is also denoted by (K). Here are a few normal modal logics with their axiomatizations (on top of (K)): (K) axiomatized by no supplementary axioms; (T) axiomatized by the axiom T; (K4) axiomatized by the axiom 4; (S4) axiomatized by the axioms T,4; (S5) axiomatized by the axioms T,5 or D,B,4; (GL) axiomatized by the axiom GL. The last one, called Gödel–Löb logic or provability logic, is important because it describes exactly the properties of provability in Peano arithmetic, as proved by Robert Solovay. See for instance https://plato.stanford.edu/entries/logic-provability/ 1809. A basic result in this logic is bj-gl4 33933. | ||
Theorem | bj-axdd2 33930 | This implication, proved using only ax-gen 1795 and ax-4 1809 on top of propositional calculus (hence holding, up to the standard interpretation, in any normal modal logic), shows that the axiom scheme ⊢ ∃𝑥⊤ implies the axiom scheme ⊢ (∀𝑥𝜑 → ∃𝑥𝜑). These correspond to the modal axiom (D), and in predicate calculus, they assert that the universe of discourse is nonempty. For the converse, see bj-axd2d 33931. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜓)) | ||
Theorem | bj-axd2d 33931 | This implication, proved using only ax-gen 1795 on top of propositional calculus (hence holding, up to the standard interpretation, in any modal logic), shows that the axiom scheme ⊢ (∀𝑥𝜑 → ∃𝑥𝜑) implies the axiom scheme ⊢ ∃𝑥⊤. These correspond to the modal axiom (D), and in predicate calculus, they assert that the universe of discourse is nonempty. For the converse, see bj-axdd2 33930. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((∀𝑥⊤ → ∃𝑥⊤) → ∃𝑥⊤) | ||
Theorem | bj-axtd 33932 | This implication, proved from propositional calculus only (hence holding, up to the standard interpretation, in any modal logic), shows that the axiom scheme ⊢ (∀𝑥𝜑 → 𝜑) (modal T) implies the axiom scheme ⊢ (∀𝑥𝜑 → ∃𝑥𝜑) (modal D). See also bj-axdd2 33930 and bj-axd2d 33931. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((∀𝑥 ¬ 𝜑 → ¬ 𝜑) → ((∀𝑥𝜑 → 𝜑) → (∀𝑥𝜑 → ∃𝑥𝜑))) | ||
Theorem | bj-gl4 33933 | In a normal modal logic, the modal axiom GL implies the modal axiom (4). Translated to first-order logic, Axiom GL reads ⊢ (∀𝑥(∀𝑥𝜑 → 𝜑) → ∀𝑥𝜑). Note that the antecedent of bj-gl4 33933 is an instance of the axiom GL, with 𝜑 replaced by (∀𝑥𝜑 ∧ 𝜑), which is a modality sometimes called the "strong necessity" of 𝜑. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((∀𝑥(∀𝑥(∀𝑥𝜑 ∧ 𝜑) → (∀𝑥𝜑 ∧ 𝜑)) → ∀𝑥(∀𝑥𝜑 ∧ 𝜑)) → (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)) | ||
Theorem | bj-axc4 33934 | Over minimal calculus, the modal axiom (4) (hba1 2300) and the modal axiom (K) (ax-4 1809) together imply axc4 2339. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) → ((∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥∀𝑥𝜑 → ∀𝑥𝜓)) → (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)))) | ||
In this section, we assume that, on top of propositional calculus, there is given a provability predicate Prv satisfying the three axioms ax-prv1 33936 and ax-prv2 33937 and ax-prv3 33938. Note the similarity with ax-gen 1795, ax-4 1809 and hba1 2300 respectively. These three properties of Prv are often called the Hilbert–Bernays–Löb derivability conditions, or the Hilbert–Bernays provability conditions. This corresponds to the modal logic (K4) (see previous section for modal logic). The interpretation of provability logic is the following: we are given a background first-order theory T, the wff Prv 𝜑 means "𝜑 is provable in T", and the turnstile ⊢ indicates provability in T. Beware that "provability logic" often means (K) augmented with the Gödel–Löb axiom GL, which we do not assume here (at least for the moment). See for instance https://plato.stanford.edu/entries/logic-provability/ 2300. Provability logic is worth studying because whenever T is a first-order theory containing Robinson arithmetic (a fragment of Peano arithmetic), one can prove (using Gödel numbering, and in the much weaker primitive recursive arithmetic) that there exists in T a provability predicate Prv satisfying the above three axioms. (We do not construct this predicate in this section; this is still a project.) The main theorems of this section are the "easy parts" of the proofs of Gödel's second incompleteness theorem (bj-babygodel 33941) and Löb's theorem (bj-babylob 33942). See the comments of these theorems for details. | ||
Syntax | cprvb 33935 | Syntax for the provability predicate. |
wff Prv 𝜑 | ||
Axiom | ax-prv1 33936 | First property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.) |
⊢ 𝜑 ⇒ ⊢ Prv 𝜑 | ||
Axiom | ax-prv2 33937 | Second property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.) |
⊢ (Prv (𝜑 → 𝜓) → (Prv 𝜑 → Prv 𝜓)) | ||
Axiom | ax-prv3 33938 | Third property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.) |
⊢ (Prv 𝜑 → Prv Prv 𝜑) | ||
Theorem | prvlem1 33939 | An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (Prv 𝜑 → Prv 𝜓) | ||
Theorem | prvlem2 33940 | An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (Prv 𝜑 → (Prv 𝜓 → Prv 𝜒)) | ||
Theorem | bj-babygodel 33941 |
See the section header comments for the context.
The first hypothesis reads "𝜑 is true if and only if it is not provable in T" (and having this first hypothesis means that we can prove this fact in T). The wff 𝜑 is a formal version of the sentence "This sentence is not provable". The hard part of the proof of Gödel's theorem is to construct such a 𝜑, called a "Gödel–Rosser sentence", for a first-order theory T which is effectively axiomatizable and contains Robinson arithmetic, through Gödel diagonalization (this can be done in primitive recursive arithmetic). The second hypothesis means that ⊥ is not provable in T, that is, that the theory T is consistent (and having this second hypothesis means that we can prove in T that the theory T is consistent). The conclusion is the falsity, so having the conclusion means that T can prove the falsity, that is, T is inconsistent. Therefore, taking the contrapositive, this theorem expresses that if a first-order theory is consistent (and one can prove in it that some formula is true if and only if it is not provable in it), then this theory does not prove its own consistency. This proof is due to George Boolos, Gödel's Second Incompleteness Theorem Explained in Words of One Syllable, Mind, New Series, Vol. 103, No. 409 (January 1994), pp. 1--3. (Contributed by BJ, 3-Apr-2019.) |
⊢ (𝜑 ↔ ¬ Prv 𝜑) & ⊢ ¬ Prv ⊥ ⇒ ⊢ ⊥ | ||
Theorem | bj-babylob 33942 |
See the section header comments for the context, as well as the comments
for bj-babygodel 33941.
Löb's theorem when the Löb sentence is given as a hypothesis (the hard part of the proof of Löb's theorem is to construct this Löb sentence; this can be done, using Gödel diagonalization, for any first-order effectively axiomatizable theory containing Robinson arithmetic). More precisely, the present theorem states that if a first-order theory proves that the provability of a given sentence entails its truth (and if one can construct in this theory a provability predicate and a Löb sentence, given here as the first hypothesis), then the theory actually proves that sentence. See for instance, Eliezer Yudkowsky, The Cartoon Guide to Löb's Theorem (available at http://yudkowsky.net/rational/lobs-theorem/ 33941). (Contributed by BJ, 20-Apr-2019.) |
⊢ (𝜓 ↔ (Prv 𝜓 → 𝜑)) & ⊢ (Prv 𝜑 → 𝜑) ⇒ ⊢ 𝜑 | ||
Theorem | bj-godellob 33943 | Proof of Gödel's theorem from Löb's theorem (see comments at bj-babygodel 33941 and bj-babylob 33942 for details). (Contributed by BJ, 20-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 ↔ ¬ Prv 𝜑) & ⊢ ¬ Prv ⊥ ⇒ ⊢ ⊥ | ||
Utility lemmas or strengthenings of theorems in the main part (biconditional or closed forms, or fewer disjoint variable conditions, or disjoint variable conditions replaced with nonfreeness hypotheses...). Sorted in the same order as in the main part. | ||
Theorem | bj-genr 33944 | Generalization rule on the right conjunct. See 19.28 2229. (Contributed by BJ, 7-Jul-2021.) |
⊢ (𝜑 ∧ 𝜓) ⇒ ⊢ (𝜑 ∧ ∀𝑥𝜓) | ||
Theorem | bj-genl 33945 | Generalization rule on the left conjunct. See 19.27 2228. (Contributed by BJ, 7-Jul-2021.) |
⊢ (𝜑 ∧ 𝜓) ⇒ ⊢ (∀𝑥𝜑 ∧ 𝜓) | ||
Theorem | bj-genan 33946 | Generalization rule on a conjunction. Forward inference associated with 19.26 1870. (Contributed by BJ, 7-Jul-2021.) |
⊢ (𝜑 ∧ 𝜓) ⇒ ⊢ (∀𝑥𝜑 ∧ ∀𝑥𝜓) | ||
Theorem | bj-mpgs 33947 | From a closed form theorem (the major premise) with an antecedent in the "strong necessity" modality (in the language of modal logic), deduce the inference ⊢ 𝜑 ⇒ ⊢ 𝜓. Strong necessity is stronger than necessity, and equivalent to it when sp 2181 (modal T) is available. Therefore, this theorem is stronger than mpg 1797 when sp 2181 is not available. (Contributed by BJ, 1-Nov-2023.) |
⊢ 𝜑 & ⊢ ((𝜑 ∧ ∀𝑥𝜑) → 𝜓) ⇒ ⊢ 𝜓 | ||
Theorem | bj-2alim 33948 | Closed form of 2alimi 1812. (Contributed by BJ, 6-May-2019.) |
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓)) | ||
Theorem | bj-2exim 33949 | Closed form of 2eximi 1835. (Contributed by BJ, 6-May-2019.) |
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (∃𝑥∃𝑦𝜑 → ∃𝑥∃𝑦𝜓)) | ||
Theorem | bj-alanim 33950 | Closed form of alanimi 1816. (Contributed by BJ, 6-May-2019.) |
⊢ (∀𝑥((𝜑 ∧ 𝜓) → 𝜒) → ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)) | ||
Theorem | bj-2albi 33951 | Closed form of 2albii 1820. (Contributed by BJ, 6-May-2019.) |
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∀𝑥∀𝑦𝜑 ↔ ∀𝑥∀𝑦𝜓)) | ||
Theorem | bj-notalbii 33952 | Equivalence of universal quantification of negation of equivalent formulas. Shortens ab0 4336 (103>94), ballotlem2 31750 (2655>2648), bnj1143 32066 (522>519), hausdiag 22256 (2119>2104). (Contributed by BJ, 17-Jul-2021.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑥 ¬ 𝜓) | ||
Theorem | bj-2exbi 33953 | Closed form of 2exbii 1848. (Contributed by BJ, 6-May-2019.) |
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓)) | ||
Theorem | bj-3exbi 33954 | Closed form of 3exbii 1849. (Contributed by BJ, 6-May-2019.) |
⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) → (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧𝜓)) | ||
Theorem | bj-sylgt2 33955 | Uncurried (imported) form of sylgt 1821. (Contributed by BJ, 2-May-2019.) |
⊢ ((∀𝑥(𝜓 → 𝜒) ∧ (𝜑 → ∀𝑥𝜓)) → (𝜑 → ∀𝑥𝜒)) | ||
Theorem | bj-alrimg 33956 | The general form of the *alrim* family of theorems: if 𝜑 is substituted for 𝜓, then the antecedent expresses a form of nonfreeness of 𝑥 in 𝜑, so the theorem means that under a nonfreeness condition in an antecedent, one can deduce from the universally quantified implication an implication where the consequent is universally quantified. Dual of bj-exlimg 33960. (Contributed by BJ, 9-Dec-2023.) |
⊢ ((𝜑 → ∀𝑥𝜓) → (∀𝑥(𝜓 → 𝜒) → (𝜑 → ∀𝑥𝜒))) | ||
Theorem | bj-alrimd 33957 | A slightly more general alrimd 2214. A common usage will have 𝜑 substituted for 𝜓 and 𝜒 substituted for 𝜃, giving a form closer to alrimd 2214. (Contributed by BJ, 25-Dec-2023.) |
⊢ (𝜑 → ∀𝑥𝜓) & ⊢ (𝜑 → (𝜒 → ∀𝑥𝜃)) & ⊢ (𝜓 → (𝜃 → 𝜏)) ⇒ ⊢ (𝜑 → (𝜒 → ∀𝑥𝜏)) | ||
Theorem | bj-sylget 33958 | Dual statement of sylgt 1821. Closed form of bj-sylge 33961. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥(𝜒 → 𝜑) → ((∃𝑥𝜑 → 𝜓) → (∃𝑥𝜒 → 𝜓))) | ||
Theorem | bj-sylget2 33959 | Uncurried (imported) form of bj-sylget 33958. (Contributed by BJ, 2-May-2019.) |
⊢ ((∀𝑥(𝜑 → 𝜓) ∧ (∃𝑥𝜓 → 𝜒)) → (∃𝑥𝜑 → 𝜒)) | ||
Theorem | bj-exlimg 33960 | The general form of the *exlim* family of theorems: if 𝜑 is substituted for 𝜓, then the antecedent expresses a form of nonfreeness of 𝑥 in 𝜑, so the theorem means that under a nonfreeness condition in a consequent, one can deduce from the universally quantified implication an implication where the antecedent is existentially quantified. Dual of bj-alrimg 33956. (Contributed by BJ, 9-Dec-2023.) |
⊢ ((∃𝑥𝜑 → 𝜓) → (∀𝑥(𝜒 → 𝜑) → (∃𝑥𝜒 → 𝜓))) | ||
Theorem | bj-sylge 33961 | Dual statement of sylg 1822 (the final "e" in the label stands for "existential (version of sylg 1822)". Variant of exlimih 2296. (Contributed by BJ, 25-Dec-2023.) |
⊢ (∃𝑥𝜑 → 𝜓) & ⊢ (𝜒 → 𝜑) ⇒ ⊢ (∃𝑥𝜒 → 𝜓) | ||
Theorem | bj-exlimd 33962 | A slightly more general exlimd 2217. A common usage will have 𝜑 substituted for 𝜓 and 𝜃 substituted for 𝜏, giving a form closer to exlimd 2217. (Contributed by BJ, 25-Dec-2023.) |
⊢ (𝜑 → ∀𝑥𝜓) & ⊢ (𝜑 → (∃𝑥𝜃 → 𝜏)) & ⊢ (𝜓 → (𝜒 → 𝜃)) ⇒ ⊢ (𝜑 → (∃𝑥𝜒 → 𝜏)) | ||
Theorem | bj-nfimexal 33963 | A weak from of nonfreeness in either an antecedent or a consequent implies that a universally quantified implication is equivalent to the associated implication where the antecedent is existentially quantified and the consequent is universally quantified. The forward implication always holds (this is 19.38 1838) and the converse implication is the join of instances of bj-alrimg 33956 and bj-exlimg 33960 (see 19.38a 1839 and 19.38b 1840). TODO: prove a version where the antecedents use the nonfreeness quantifier. (Contributed by BJ, 9-Dec-2023.) |
⊢ (((∃𝑥𝜑 → ∀𝑥𝜑) ∨ (∃𝑥𝜓 → ∀𝑥𝜓)) → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓))) | ||
Theorem | bj-alexim 33964 | Closed form of aleximi 1831. Note: this proof is shorter, so aleximi 1831 could be deduced from it (exim 1833 would have to be proved first, see bj-eximALT 33978 but its proof is shorter (currently almost a subproof of aleximi 1831)). (Contributed by BJ, 8-Nov-2021.) |
⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))) | ||
Theorem | bj-nexdh 33965 | Closed form of nexdh 1865 (actually, its general instance). (Contributed by BJ, 6-May-2019.) |
⊢ (∀𝑥(𝜑 → ¬ 𝜓) → ((𝜒 → ∀𝑥𝜑) → (𝜒 → ¬ ∃𝑥𝜓))) | ||
Theorem | bj-nexdh2 33966 | Uncurried (imported) form of bj-nexdh 33965. (Contributed by BJ, 6-May-2019.) |
⊢ ((∀𝑥(𝜑 → ¬ 𝜓) ∧ (𝜒 → ∀𝑥𝜑)) → (𝜒 → ¬ ∃𝑥𝜓)) | ||
Theorem | bj-hbxfrbi 33967 | Closed form of hbxfrbi 1824. Note: it is less important than nfbiit 1850. The antecedent is in the "strong necessity" modality of modal logic (see also bj-nnftht 34074) in order not to require sp 2181 (modal T). See bj-hbyfrbi 33968 for its version with existential quantifiers. (Contributed by BJ, 6-May-2019.) |
⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓))) | ||
Theorem | bj-hbyfrbi 33968 | Version of bj-hbxfrbi 33967 with existential quantifiers. (Contributed by BJ, 23-Aug-2023.) |
⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → ((∃𝑥𝜑 → 𝜑) ↔ (∃𝑥𝜓 → 𝜓))) | ||
Theorem | bj-exalim 33969 |
Distribute quantifiers over a nested implication.
This and the following theorems are the general instances of already proved theorems. They could be moved to the main part, before ax-5 1910. I propose to move to the main part: bj-exalim 33969, bj-exalimi 33970, bj-exalims 33971, bj-exalimsi 33972, bj-ax12i 33974, bj-ax12wlem 33981, bj-ax12w 34014, and remove equs3OLD 1964. A new label is needed for bj-ax12i 33974 and label suggestions are welcome for the others. I also propose to change ¬ ∀𝑥¬ to ∃𝑥 in speimfw 1965 and spimfw 1967 (other spim* theorems use ∃𝑥 and very few theorems in set.mm use ¬ ∀𝑥¬). (Contributed by BJ, 8-Nov-2021.) |
⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))) | ||
Theorem | bj-exalimi 33970 | An inference for distributing quantifiers over a nested implication. The canonical derivation from its closed form bj-exalim 33969 (using mpg 1797) has fewer essential steps, but more steps in total (yielding a longer compressed proof). (Almost) the general statement that speimfw 1965 proves. (Contributed by BJ, 29-Sep-2019.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒)) | ||
Theorem | bj-exalims 33971 | Distributing quantifiers over a nested implication. (Almost) the general statement that spimfw 1967 proves. (Contributed by BJ, 29-Sep-2019.) |
⊢ (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒)) ⇒ ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒))) | ||
Theorem | bj-exalimsi 33972 | An inference for distributing quantifiers over a nested implication. (Almost) the general statement that spimfw 1967 proves. (Contributed by BJ, 29-Sep-2019.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒)) ⇒ ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒)) | ||
Theorem | bj-ax12ig 33973 | A lemma used to prove a weak form of the axiom of substitution. A generalization of bj-ax12i 33974. (Contributed by BJ, 19-Dec-2020.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓))) | ||
Theorem | bj-ax12i 33974 | A weakening of bj-ax12ig 33973 that is sufficient to prove a weak form of the axiom of substitution ax-12 2176. The general statement of which ax12i 1968 is an instance. (Contributed by BJ, 29-Sep-2019.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜒 → ∀𝑥𝜒) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓))) | ||
Theorem | bj-nfimt 33975 | Closed form of nfim 1896 and curried (exported) form of nfimt 1895. (Contributed by BJ, 20-Oct-2021.) |
⊢ (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → Ⅎ𝑥(𝜑 → 𝜓))) | ||
Theorem | bj-cbvalimt 33976 | A lemma in closed form used to prove bj-cbval 33986 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) Do not use 19.35 1877 since only the direction of the biconditional used here holds in intuitionistic logic. (Proof modification is discouraged.) |
⊢ (∀𝑦∃𝑥𝜒 → (∀𝑦∀𝑥(𝜒 → (𝜑 → 𝜓)) → ((∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) → (∀𝑦(∃𝑥𝜓 → 𝜓) → (∀𝑥𝜑 → ∀𝑦𝜓))))) | ||
Theorem | bj-cbveximt 33977 | A lemma in closed form used to prove bj-cbvex 33987 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) Do not use 19.35 1877 since only the direction of the biconditional used here holds in intuitionistic logic. (Proof modification is discouraged.) |
⊢ (∀𝑥∃𝑦𝜒 → (∀𝑥∀𝑦(𝜒 → (𝜑 → 𝜓)) → (∀𝑥(𝜑 → ∀𝑦𝜑) → ((∃𝑥∃𝑦𝜓 → ∃𝑦𝜓) → (∃𝑥𝜑 → ∃𝑦𝜓))))) | ||
Theorem | bj-eximALT 33978 | Alternate proof of exim 1833 directly from alim 1810 by using df-ex 1780 (using duality of ∀ and ∃. (Contributed by BJ, 9-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓)) | ||
Theorem | bj-aleximiALT 33979 | Alternate proof of aleximi 1831 from exim 1833, which is sometimes used as an axiom in instuitionistic modal logic. (Contributed by BJ, 9-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)) | ||
Theorem | bj-eximcom 33980 | A commuted form of exim 1833 which is sometimes posited as an axiom in instuitionistic modal logic. (Contributed by BJ, 9-Dec-2023.) |
⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓)) | ||
Theorem | bj-ax12wlem 33981* | A lemma used to prove a weak version of the axiom of substitution ax-12 2176. (Temporary comment: The general statement that ax12wlem 2135 proves.) (Contributed by BJ, 20-Mar-2020.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓))) | ||
Theorem | bj-cbvalim 33982* | A lemma used to prove bj-cbval 33986 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.) |
⊢ (∀𝑦∃𝑥𝜒 → (∀𝑦∀𝑥(𝜒 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → ∀𝑦𝜓))) | ||
Theorem | bj-cbvexim 33983* | A lemma used to prove bj-cbvex 33987 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.) |
⊢ (∀𝑥∃𝑦𝜒 → (∀𝑥∀𝑦(𝜒 → (𝜑 → 𝜓)) → (∃𝑥𝜑 → ∃𝑦𝜓))) | ||
Theorem | bj-cbvalimi 33984* | An equality-free general instance of one half of a precise form of bj-cbval 33986. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.) |
⊢ (𝜒 → (𝜑 → 𝜓)) & ⊢ ∀𝑦∃𝑥𝜒 ⇒ ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) | ||
Theorem | bj-cbveximi 33985* | An equality-free general instance of one half of a precise form of bj-cbvex 33987. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.) |
⊢ (𝜒 → (𝜑 → 𝜓)) & ⊢ ∀𝑥∃𝑦𝜒 ⇒ ⊢ (∃𝑥𝜑 → ∃𝑦𝜓) | ||
Theorem | bj-cbval 33986* | Changing a bound variable (universal quantification case) in a weak axiomatization, assuming that all variables denote (which is valid in inclusive free logic) and that equality is symmetric. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.) |
⊢ ∀𝑦∃𝑥 𝑥 = 𝑦 & ⊢ ∀𝑥∃𝑦 𝑦 = 𝑥 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦) ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) | ||
Theorem | bj-cbvex 33987* | Changing a bound variable (existential quantification case) in a weak axiomatization, assuming that all variables denote (which is valid in inclusive free logic) and that equality is symmetric. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.) |
⊢ ∀𝑦∃𝑥 𝑥 = 𝑦 & ⊢ ∀𝑥∃𝑦 𝑦 = 𝑥 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦) ⇒ ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓) | ||
Syntax | wmoo 33988 | Syntax for BJ's version of the uniqueness quantifier. |
wff ∃**𝑥𝜑 | ||
Definition | df-bj-mo 33989* | Definition of the uniqueness quantifier which is correct on the empty domain. Instead of the fresh variable 𝑧, one could save a dummy variable by using 𝑥 or 𝑦 at the cost of having nested quantifiers on the same variable. (Contributed by BJ, 12-Mar-2023.) |
⊢ (∃**𝑥𝜑 ↔ ∀𝑧∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | ||
Theorem | bj-ssbeq 33990* | Substitution in an equality, disjoint variables case. Uses only ax-1 6 through ax-6 1969. It might be shorter to prove the result about composition of two substitutions and prove bj-ssbeq 33990 first with a DV condition on 𝑥, 𝑡, and then in the general case. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) |
⊢ ([𝑡 / 𝑥]𝑦 = 𝑧 ↔ 𝑦 = 𝑧) | ||
Theorem | bj-ssblem1 33991* | A lemma for the definiens of df-sb 2069. An instance of sp 2181 proved without it. Note: it has a common subproof with sbjust 2067. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) |
⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | bj-ssblem2 33992* | An instance of ax-11 2160 proved without it. The converse may not be provable without ax-11 2160 (since using alcomiw 2049 would require a DV on 𝜑, 𝑥, which defeats the purpose). (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) |
⊢ (∀𝑥∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) → ∀𝑦∀𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑))) | ||
Theorem | bj-ax12v 33993* | A weaker form of ax-12 2176 and ax12v 2177, namely the generalization over 𝑥 of the latter. In this statement, all occurrences of 𝑥 are bound. (Contributed by BJ, 26-Dec-2020.) (Proof modification is discouraged.) |
⊢ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))) | ||
Theorem | bj-ax12 33994* | Remove a DV condition from bj-ax12v 33993 (using core axioms only). (Contributed by BJ, 26-Dec-2020.) (Proof modification is discouraged.) |
⊢ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))) | ||
Theorem | bj-ax12ssb 33995* | The axiom bj-ax12 33994 expressed using substitution. (Contributed by BJ, 26-Dec-2020.) (Proof modification is discouraged.) |
⊢ [𝑡 / 𝑥](𝜑 → [𝑡 / 𝑥]𝜑) | ||
Theorem | bj-19.41al 33996 | Special case of 19.41 2236 proved from Tarski, ax-10 2144 (modal5) and hba1 2300 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.) |
⊢ (∃𝑥(𝜑 ∧ ∀𝑥𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓)) | ||
Theorem | bj-equsexval 33997* | Special case of equsexv 2268 proved from Tarski, ax-10 2144 (modal5) and hba1 2300 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥𝜓)) ⇒ ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥𝜓) | ||
Theorem | bj-sb56 33998* | Proof of sb56 2276 from Tarski, ax-10 2144 (modal5) and bj-ax12 33994. (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.) |
⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | bj-ssbid2 33999 | A special case of sbequ2 2249. (Contributed by BJ, 22-Dec-2020.) |
⊢ ([𝑥 / 𝑥]𝜑 → 𝜑) | ||
Theorem | bj-ssbid2ALT 34000 | Alternate proof of bj-ssbid2 33999, not using sbequ2 2249. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ([𝑥 / 𝑥]𝜑 → 𝜑) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |