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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | bj-nimn 33901 | 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 33902 | Inference associated with bj-nimn 33901. (Contributed by BJ, 19-Mar-2020.) |
⊢ 𝜑 ⇒ ⊢ ¬ (𝜑 → ¬ 𝜑) | ||
Theorem | bj-peircei 33903 | Inference associated with peirce 204. (Contributed by BJ, 30-Mar-2020.) |
⊢ ((𝜑 → 𝜓) → 𝜑) ⇒ ⊢ 𝜑 | ||
Theorem | bj-looinvi 33904 | Inference associated with looinv 205. Its associated inference is bj-looinvii 33905. (Contributed by BJ, 30-Mar-2020.) |
⊢ ((𝜑 → 𝜓) → 𝜓) ⇒ ⊢ ((𝜓 → 𝜑) → 𝜑) | ||
Theorem | bj-looinvii 33905 | Inference associated with bj-looinvi 33904. (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 33906 | Shortens orfa2 35391 (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 33907 | Shortens consensus 1047 (110>106), elnn0z 11976 (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 33908 | Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.) |
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓))) | ||
Theorem | bj-dfbi5 33909 | Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.) |
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) → (𝜑 ∧ 𝜓))) | ||
Theorem | bj-dfbi6 33910 | Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.) |
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ↔ (𝜑 ∧ 𝜓))) | ||
Theorem | bj-bijust0ALT 33911 | 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 33912 | 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 33911 are therefore also instances of it). (Contributed by BJ, 7-Sep-2022.) |
⊢ ¬ ((𝜑 → 𝜑) → ¬ (𝜓 → 𝜓)) | ||
Theorem | bj-consensus 33913 | 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 33914 | Alternate proof of bj-consensus 33913. (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 33915* | 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 2799. We reprove the current df-if 4449 from it in bj-dfif 33916. (Contributed by BJ, 20-Sep-2019.) (Proof modification is discouraged.) |
⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)} | ||
Theorem | bj-dfif 33916* | 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 33917 | 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 33918 | Uncurried (imported) form of imbi12 349. (Contributed by BJ, 6-May-2019.) |
⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))) | ||
Theorem | bj-biorfi 33919 | 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 33920 | Dual of truan 1548 (which has biconditional reversed). (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 ↔ (⊥ ∨ 𝜑)) | ||
Theorem | bj-falor2 33921 | Dual of truan 1548. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.) |
⊢ ((⊥ ∨ 𝜑) ↔ 𝜑) | ||
Theorem | bj-bibibi 33922 | A property of the biconditional. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 ↔ (𝜓 ↔ (𝜑 ↔ 𝜓))) | ||
Theorem | bj-imn3ani 33923 | Duplication of bnj1224 32075. 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 33924 | 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 33925 | 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 33926 | 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 33927 | 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 1796 corresponds to the necessitation rule of modal logic, and ax-4 1810 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/ 1810. A basic result in this logic is bj-gl4 33931. | ||
Theorem | bj-axdd2 33928 | This implication, proved using only ax-gen 1796 and ax-4 1810 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 33929. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜓)) | ||
Theorem | bj-axd2d 33929 | This implication, proved using only ax-gen 1796 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 33928. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((∀𝑥⊤ → ∃𝑥⊤) → ∃𝑥⊤) | ||
Theorem | bj-axtd 33930 | 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 33928 and bj-axd2d 33929. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((∀𝑥 ¬ 𝜑 → ¬ 𝜑) → ((∀𝑥𝜑 → 𝜑) → (∀𝑥𝜑 → ∃𝑥𝜑))) | ||
Theorem | bj-gl4 33931 | 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 33931 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 33932 | Over minimal calculus, the modal axiom (4) (hba1 2301) and the modal axiom (K) (ax-4 1810) together imply axc4 2340. (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 33934 and ax-prv2 33935 and ax-prv3 33936. Note the similarity with ax-gen 1796, ax-4 1810 and hba1 2301 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/ 2301. 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 33939) and Löb's theorem (bj-babylob 33940). See the comments of these theorems for details. | ||
Syntax | cprvb 33933 | Syntax for the provability predicate. |
wff Prv 𝜑 | ||
Axiom | ax-prv1 33934 | First property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.) |
⊢ 𝜑 ⇒ ⊢ Prv 𝜑 | ||
Axiom | ax-prv2 33935 | Second property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.) |
⊢ (Prv (𝜑 → 𝜓) → (Prv 𝜑 → Prv 𝜓)) | ||
Axiom | ax-prv3 33936 | Third property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.) |
⊢ (Prv 𝜑 → Prv Prv 𝜑) | ||
Theorem | prvlem1 33937 | An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (Prv 𝜑 → Prv 𝜓) | ||
Theorem | prvlem2 33938 | 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 33939 |
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 33940 |
See the section header comments for the context, as well as the comments
for bj-babygodel 33939.
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/ 33939). (Contributed by BJ, 20-Apr-2019.) |
⊢ (𝜓 ↔ (Prv 𝜓 → 𝜑)) & ⊢ (Prv 𝜑 → 𝜑) ⇒ ⊢ 𝜑 | ||
Theorem | bj-godellob 33941 | Proof of Gödel's theorem from Löb's theorem (see comments at bj-babygodel 33939 and bj-babylob 33940 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 33942 | Generalization rule on the right conjunct. See 19.28 2230. (Contributed by BJ, 7-Jul-2021.) |
⊢ (𝜑 ∧ 𝜓) ⇒ ⊢ (𝜑 ∧ ∀𝑥𝜓) | ||
Theorem | bj-genl 33943 | Generalization rule on the left conjunct. See 19.27 2229. (Contributed by BJ, 7-Jul-2021.) |
⊢ (𝜑 ∧ 𝜓) ⇒ ⊢ (∀𝑥𝜑 ∧ 𝜓) | ||
Theorem | bj-genan 33944 | Generalization rule on a conjunction. Forward inference associated with 19.26 1871. (Contributed by BJ, 7-Jul-2021.) |
⊢ (𝜑 ∧ 𝜓) ⇒ ⊢ (∀𝑥𝜑 ∧ ∀𝑥𝜓) | ||
Theorem | bj-mpgs 33945 | 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 2182 (modal T) is available. Therefore, this theorem is stronger than mpg 1798 when sp 2182 is not available. (Contributed by BJ, 1-Nov-2023.) |
⊢ 𝜑 & ⊢ ((𝜑 ∧ ∀𝑥𝜑) → 𝜓) ⇒ ⊢ 𝜓 | ||
Theorem | bj-2alim 33946 | Closed form of 2alimi 1813. (Contributed by BJ, 6-May-2019.) |
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓)) | ||
Theorem | bj-2exim 33947 | Closed form of 2eximi 1836. (Contributed by BJ, 6-May-2019.) |
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (∃𝑥∃𝑦𝜑 → ∃𝑥∃𝑦𝜓)) | ||
Theorem | bj-alanim 33948 | Closed form of alanimi 1817. (Contributed by BJ, 6-May-2019.) |
⊢ (∀𝑥((𝜑 ∧ 𝜓) → 𝜒) → ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)) | ||
Theorem | bj-2albi 33949 | Closed form of 2albii 1821. (Contributed by BJ, 6-May-2019.) |
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∀𝑥∀𝑦𝜑 ↔ ∀𝑥∀𝑦𝜓)) | ||
Theorem | bj-notalbii 33950 | Equivalence of universal quantification of negation of equivalent formulas. Shortens ab0 4314 (103>94), ballotlem2 31748 (2655>2648), bnj1143 32064 (522>519), hausdiag 22231 (2119>2104). (Contributed by BJ, 17-Jul-2021.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑥 ¬ 𝜓) | ||
Theorem | bj-2exbi 33951 | Closed form of 2exbii 1849. (Contributed by BJ, 6-May-2019.) |
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓)) | ||
Theorem | bj-3exbi 33952 | Closed form of 3exbii 1850. (Contributed by BJ, 6-May-2019.) |
⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) → (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧𝜓)) | ||
Theorem | bj-sylgt2 33953 | Uncurried (imported) form of sylgt 1822. (Contributed by BJ, 2-May-2019.) |
⊢ ((∀𝑥(𝜓 → 𝜒) ∧ (𝜑 → ∀𝑥𝜓)) → (𝜑 → ∀𝑥𝜒)) | ||
Theorem | bj-alrimg 33954 | 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 33958. (Contributed by BJ, 9-Dec-2023.) |
⊢ ((𝜑 → ∀𝑥𝜓) → (∀𝑥(𝜓 → 𝜒) → (𝜑 → ∀𝑥𝜒))) | ||
Theorem | bj-alrimd 33955 | A slightly more general alrimd 2215. A common usage will have 𝜑 substituted for 𝜓 and 𝜒 substituted for 𝜃, giving a form closer to alrimd 2215. (Contributed by BJ, 25-Dec-2023.) |
⊢ (𝜑 → ∀𝑥𝜓) & ⊢ (𝜑 → (𝜒 → ∀𝑥𝜃)) & ⊢ (𝜓 → (𝜃 → 𝜏)) ⇒ ⊢ (𝜑 → (𝜒 → ∀𝑥𝜏)) | ||
Theorem | bj-sylget 33956 | Dual statement of sylgt 1822. Closed form of bj-sylge 33959. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥(𝜒 → 𝜑) → ((∃𝑥𝜑 → 𝜓) → (∃𝑥𝜒 → 𝜓))) | ||
Theorem | bj-sylget2 33957 | Uncurried (imported) form of bj-sylget 33956. (Contributed by BJ, 2-May-2019.) |
⊢ ((∀𝑥(𝜑 → 𝜓) ∧ (∃𝑥𝜓 → 𝜒)) → (∃𝑥𝜑 → 𝜒)) | ||
Theorem | bj-exlimg 33958 | 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 33954. (Contributed by BJ, 9-Dec-2023.) |
⊢ ((∃𝑥𝜑 → 𝜓) → (∀𝑥(𝜒 → 𝜑) → (∃𝑥𝜒 → 𝜓))) | ||
Theorem | bj-sylge 33959 | Dual statement of sylg 1823 (the final "e" in the label stands for "existential (version of sylg 1823)". Variant of exlimih 2297. (Contributed by BJ, 25-Dec-2023.) |
⊢ (∃𝑥𝜑 → 𝜓) & ⊢ (𝜒 → 𝜑) ⇒ ⊢ (∃𝑥𝜒 → 𝜓) | ||
Theorem | bj-exlimd 33960 | A slightly more general exlimd 2218. A common usage will have 𝜑 substituted for 𝜓 and 𝜃 substituted for 𝜏, giving a form closer to exlimd 2218. (Contributed by BJ, 25-Dec-2023.) |
⊢ (𝜑 → ∀𝑥𝜓) & ⊢ (𝜑 → (∃𝑥𝜃 → 𝜏)) & ⊢ (𝜓 → (𝜒 → 𝜃)) ⇒ ⊢ (𝜑 → (∃𝑥𝜒 → 𝜏)) | ||
Theorem | bj-nfimexal 33961 | 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 1839) and the converse implication is the join of instances of bj-alrimg 33954 and bj-exlimg 33958 (see 19.38a 1840 and 19.38b 1841). 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 33962 | Closed form of aleximi 1832. Note: this proof is shorter, so aleximi 1832 could be deduced from it (exim 1834 would have to be proved first, see bj-eximALT 33976 but its proof is shorter (currently almost a subproof of aleximi 1832)). (Contributed by BJ, 8-Nov-2021.) |
⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))) | ||
Theorem | bj-nexdh 33963 | Closed form of nexdh 1866 (actually, its general instance). (Contributed by BJ, 6-May-2019.) |
⊢ (∀𝑥(𝜑 → ¬ 𝜓) → ((𝜒 → ∀𝑥𝜑) → (𝜒 → ¬ ∃𝑥𝜓))) | ||
Theorem | bj-nexdh2 33964 | Uncurried (imported) form of bj-nexdh 33963. (Contributed by BJ, 6-May-2019.) |
⊢ ((∀𝑥(𝜑 → ¬ 𝜓) ∧ (𝜒 → ∀𝑥𝜑)) → (𝜒 → ¬ ∃𝑥𝜓)) | ||
Theorem | bj-hbxfrbi 33965 | Closed form of hbxfrbi 1825. Note: it is less important than nfbiit 1851. The antecedent is in the "strong necessity" modality of modal logic (see also bj-nnftht 34072) in order not to require sp 2182 (modal T). See bj-hbyfrbi 33966 for its version with existential quantifiers. (Contributed by BJ, 6-May-2019.) |
⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓))) | ||
Theorem | bj-hbyfrbi 33966 | Version of bj-hbxfrbi 33965 with existential quantifiers. (Contributed by BJ, 23-Aug-2023.) |
⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → ((∃𝑥𝜑 → 𝜑) ↔ (∃𝑥𝜓 → 𝜓))) | ||
Theorem | bj-exalim 33967 |
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 1911. I propose to move to the main part: bj-exalim 33967, bj-exalimi 33968, bj-exalims 33969, bj-exalimsi 33970, bj-ax12i 33972, bj-ax12wlem 33979, bj-ax12w 34012, and remove equs3OLD 1965. A new label is needed for bj-ax12i 33972 and label suggestions are welcome for the others. I also propose to change ¬ ∀𝑥¬ to ∃𝑥 in speimfw 1966 and spimfw 1968 (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 33968 | An inference for distributing quantifiers over a nested implication. The canonical derivation from its closed form bj-exalim 33967 (using mpg 1798) has fewer essential steps, but more steps in total (yielding a longer compressed proof). (Almost) the general statement that speimfw 1966 proves. (Contributed by BJ, 29-Sep-2019.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒)) | ||
Theorem | bj-exalims 33969 | Distributing quantifiers over a nested implication. (Almost) the general statement that spimfw 1968 proves. (Contributed by BJ, 29-Sep-2019.) |
⊢ (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒)) ⇒ ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒))) | ||
Theorem | bj-exalimsi 33970 | An inference for distributing quantifiers over a nested implication. (Almost) the general statement that spimfw 1968 proves. (Contributed by BJ, 29-Sep-2019.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒)) ⇒ ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒)) | ||
Theorem | bj-ax12ig 33971 | A lemma used to prove a weak form of the axiom of substitution. A generalization of bj-ax12i 33972. (Contributed by BJ, 19-Dec-2020.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓))) | ||
Theorem | bj-ax12i 33972 | A weakening of bj-ax12ig 33971 that is sufficient to prove a weak form of the axiom of substitution ax-12 2177. The general statement of which ax12i 1969 is an instance. (Contributed by BJ, 29-Sep-2019.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜒 → ∀𝑥𝜒) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓))) | ||
Theorem | bj-nfimt 33973 | Closed form of nfim 1897 and curried (exported) form of nfimt 1896. (Contributed by BJ, 20-Oct-2021.) |
⊢ (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → Ⅎ𝑥(𝜑 → 𝜓))) | ||
Theorem | bj-cbvalimt 33974 | A lemma in closed form used to prove bj-cbval 33984 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) Do not use 19.35 1878 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 33975 | A lemma in closed form used to prove bj-cbvex 33985 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) Do not use 19.35 1878 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 33976 | Alternate proof of exim 1834 directly from alim 1811 by using df-ex 1781 (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 33977 | Alternate proof of aleximi 1832 from exim 1834, 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 33978 | A commuted form of exim 1834 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 33979* | A lemma used to prove a weak version of the axiom of substitution ax-12 2177. (Temporary comment: The general statement that ax12wlem 2136 proves.) (Contributed by BJ, 20-Mar-2020.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓))) | ||
Theorem | bj-cbvalim 33980* | A lemma used to prove bj-cbval 33984 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 33981* | A lemma used to prove bj-cbvex 33985 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 33982* | An equality-free general instance of one half of a precise form of bj-cbval 33984. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.) |
⊢ (𝜒 → (𝜑 → 𝜓)) & ⊢ ∀𝑦∃𝑥𝜒 ⇒ ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) | ||
Theorem | bj-cbveximi 33983* | An equality-free general instance of one half of a precise form of bj-cbvex 33985. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.) |
⊢ (𝜒 → (𝜑 → 𝜓)) & ⊢ ∀𝑥∃𝑦𝜒 ⇒ ⊢ (∃𝑥𝜑 → ∃𝑦𝜓) | ||
Theorem | bj-cbval 33984* | 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 33985* | 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 33986 | Syntax for BJ's version of the uniqueness quantifier. |
wff ∃**𝑥𝜑 | ||
Definition | df-bj-mo 33987* | 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 33988* | Substitution in an equality, disjoint variables case. Uses only ax-1 6 through ax-6 1970. It might be shorter to prove the result about composition of two substitutions and prove bj-ssbeq 33988 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 33989* | A lemma for the definiens of df-sb 2070. An instance of sp 2182 proved without it. Note: it has a common subproof with sbjust 2068. (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 33990* | An instance of ax-11 2161 proved without it. The converse may not be provable without ax-11 2161 (since using alcomiw 2050 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 33991* | A weaker form of ax-12 2177 and ax12v 2178, 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 33992* | Remove a DV condition from bj-ax12v 33991 (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 33993* | The axiom bj-ax12 33992 expressed using substitution. (Contributed by BJ, 26-Dec-2020.) (Proof modification is discouraged.) |
⊢ [𝑡 / 𝑥](𝜑 → [𝑡 / 𝑥]𝜑) | ||
Theorem | bj-19.41al 33994 | Special case of 19.41 2237 proved from Tarski, ax-10 2145 (modal5) and hba1 2301 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.) |
⊢ (∃𝑥(𝜑 ∧ ∀𝑥𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓)) | ||
Theorem | bj-equsexval 33995* | Special case of equsexv 2269 proved from Tarski, ax-10 2145 (modal5) and hba1 2301 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥𝜓)) ⇒ ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥𝜓) | ||
Theorem | bj-sb56 33996* | Proof of sb56 2277 from Tarski, ax-10 2145 (modal5) and bj-ax12 33992. (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.) |
⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | bj-ssbid2 33997 | A special case of sbequ2 2250. (Contributed by BJ, 22-Dec-2020.) |
⊢ ([𝑥 / 𝑥]𝜑 → 𝜑) | ||
Theorem | bj-ssbid2ALT 33998 | Alternate proof of bj-ssbid2 33997, not using sbequ2 2250. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ([𝑥 / 𝑥]𝜑 → 𝜑) | ||
Theorem | bj-ssbid1 33999 | A special case of sbequ1 2249. (Contributed by BJ, 22-Dec-2020.) |
⊢ (𝜑 → [𝑥 / 𝑥]𝜑) | ||
Theorem | bj-ssbid1ALT 34000 | Alternate proof of bj-ssbid1 33999, not using sbequ1 2249. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → [𝑥 / 𝑥]𝜑) |
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