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Theorem List for Metamath Proof Explorer - 32701-32800   *Has distinct variable group(s)
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20.14.2  Modal logic

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 dv conditions between 𝑥 and any other metavariables appearing in the statements.)

For instance, ax-gen 1762 corresponds to the necessitation rule of modal logic, and ax-4 1777 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/. A basic result in this logic is bj-gl4 32705.

Theorembj-axdd2 32701 This implication, proved using only ax-gen 1762 and ax-4 1777 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 32702. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜓))

Theorembj-axd2d 32702 This implication, proved using only ax-gen 1762 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 32701. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥⊤ → ∃𝑥⊤) → ∃𝑥⊤)

Theorembj-axtd 32703 This implication, proved from propositional calculus only (hence holding, up to the standard interpretation, in any modal logic), shows that the axiom scheme (∀𝑥𝜑𝜑) (modal T) implies the axiom scheme (∀𝑥𝜑 → ∃𝑥𝜑) (modal D). See also bj-axdd2 32701 and bj-axd2d 32702. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥 ¬ 𝜑 → ¬ 𝜑) → ((∀𝑥𝜑𝜑) → (∀𝑥𝜑 → ∃𝑥𝜑)))

Theorembj-gl4lem 32704 Lemma for bj-gl4 32705. Note that this proof holds in the modal logic (K). (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝜑 → ∀𝑥(∀𝑥(∀𝑥𝜑𝜑) → (∀𝑥𝜑𝜑)))

Theorembj-gl4 32705 In a normal modal logic, the modal axiom GL implies the modal axiom (4). Note that the antecedent of bj-gl4 32705 is an instance of the axiom GL, with 𝜑 replaced by (∀𝑥𝜑𝜑), sometimes called the "strong necessity" of 𝜑. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥(∀𝑥(∀𝑥𝜑𝜑) → (∀𝑥𝜑𝜑)) → ∀𝑥(∀𝑥𝜑𝜑)) → (∀𝑥𝜑 → ∀𝑥𝑥𝜑))

Theorembj-axc4 32706 Over minimal calculus, the modal axiom (4) (hba1 2189) and the modal axiom (K) (ax-4 1777) together imply axc4 2168. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥𝜑 → ∀𝑥𝑥𝜑) → ((∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝑥𝜑 → ∀𝑥𝜓)) → (∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))))

20.14.3  Provability logic

In this section, we assume that, on top of propositional calculus, there is given a provability predicate Prv satisfying the three axioms ax-prv1 32708 and ax-prv2 32709 and ax-prv3 32710. Note the similarity with ax-gen 1762, ax-4 1777 and hba1 2189 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/.

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 32713) and Löb's theorem (bj-babylob 32714). See the comments of these theorems for details.

Syntaxcprvb 32707 Syntax for the provability predicate.
wff Prv 𝜑

Axiomax-prv1 32708 First property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
𝜑       Prv 𝜑

Axiomax-prv2 32709 Second property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
(Prv (𝜑𝜓) → (Prv 𝜑 → Prv 𝜓))

Axiomax-prv3 32710 Third property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
(Prv 𝜑 → Prv Prv 𝜑)

Theoremprvlem1 32711 An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
(𝜑𝜓)       (Prv 𝜑 → Prv 𝜓)

Theoremprvlem2 32712 An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
(𝜑 → (𝜓𝜒))       (Prv 𝜑 → (Prv 𝜓 → Prv 𝜒))

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 ⊥

Theorembj-babylob 32714 See the section header comments for the context, as well as the comments for bj-babygodel 32713.

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/).

(Contributed by BJ, 20-Apr-2019.)

(𝜓 ↔ (Prv 𝜓𝜑))    &   (Prv 𝜑𝜑)       𝜑

Theorembj-godellob 32715 Proof of Gödel's theorem from Löb's theorem (see comments at bj-babygodel 32713 and bj-babylob 32714 for details). (Contributed by BJ, 20-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ↔ ¬ Prv 𝜑)    &    ¬ Prv ⊥

20.14.4  First-order logic

Utility lemmas or strengthenings of theorems in the main part (biconditional or closed forms, or fewer dv conditions, or dv conditions replaced with non-freeness hypotheses...). Sorted in the same order as in the main part.

Theorembj-genr 32716 Generalization rule on the right conjunct. See 19.28 2134. (Contributed by BJ, 7-Jul-2021.)
(𝜑𝜓)       (𝜑 ∧ ∀𝑥𝜓)

Theorembj-genl 32717 Generalization rule on the left conjunct. See 19.27 2133. (Contributed by BJ, 7-Jul-2021.)
(𝜑𝜓)       (∀𝑥𝜑𝜓)

Theorembj-genan 32718 Generalization rule on a conjunction. Forward inference associated with 19.26 1838. (Contributed by BJ, 7-Jul-2021.)
(𝜑𝜓)       (∀𝑥𝜑 ∧ ∀𝑥𝜓)

Theorembj-2alim 32719 Closed form of 2alimi 1780. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦(𝜑𝜓) → (∀𝑥𝑦𝜑 → ∀𝑥𝑦𝜓))

Theorembj-2exim 32720 Closed form of 2eximi 1803. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦(𝜑𝜓) → (∃𝑥𝑦𝜑 → ∃𝑥𝑦𝜓))

Theorembj-alanim 32721 Closed form of alanimi 1784. (Contributed by BJ, 6-May-2019.)
(∀𝑥((𝜑𝜓) → 𝜒) → ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒))

Theorembj-2albi 32722 Closed form of 2albii 1788. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦(𝜑𝜓) → (∀𝑥𝑦𝜑 ↔ ∀𝑥𝑦𝜓))

Theorembj-notalbii 32723 Equivalence of universal quantification of negation of equivalent formulas. Shortens ab0 3984 (103>94), ballotlem2 30678 (2655>2648), bnj1143 30987 (522>519), hausdiag 21496 (2119>2104). (Contributed by BJ, 17-Jul-2021.)
(𝜑𝜓)       (∀𝑥 ¬ 𝜑 ↔ ∀𝑥 ¬ 𝜓)

Theorembj-2exbi 32724 Closed form of 2exbii 1815. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦(𝜑𝜓) → (∃𝑥𝑦𝜑 ↔ ∃𝑥𝑦𝜓))

Theorembj-3exbi 32725 Closed form of 3exbii 1816. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦𝑧(𝜑𝜓) → (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑥𝑦𝑧𝜓))

Theorembj-sylgt2 32726 Uncurried (imported) form of sylgt 1789. (Contributed by BJ, 2-May-2019.)
((∀𝑥(𝜓𝜒) ∧ (𝜑 → ∀𝑥𝜓)) → (𝜑 → ∀𝑥𝜒))

Theorembj-exlimh 32727 Closed form of close to exlimih 2186. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜑𝜓) → ((∃𝑥𝜓𝜒) → (∃𝑥𝜑𝜒)))

Theorembj-exlimh2 32728 Uncurried (imported) form of bj-exlimh 32727. (Contributed by BJ, 2-May-2019.)
((∀𝑥(𝜑𝜓) ∧ (∃𝑥𝜓𝜒)) → (∃𝑥𝜑𝜒))

Theorembj-alrimhi 32729 An inference associated with sylgt 1789 and bj-exlimh 32727. (Contributed by BJ, 12-May-2019.)
(𝜑𝜓)       (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜓))

Theorembj-alexim 32730 Closed form of aleximi 1799 (with a shorter proof, so aleximi 1799 could be deduced from it -- exim 1801 would have to be proved first, but its proof is shorter (currently almost a subproof of aleximi 1799)). (Contributed by BJ, 8-Nov-2021.)
(∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)))

Theorembj-nexdh 32731 Closed form of nexdh 1832 (actually, its general instance). (Contributed by BJ, 6-May-2019.)
(∀𝑥(𝜑 → ¬ 𝜓) → ((𝜒 → ∀𝑥𝜑) → (𝜒 → ¬ ∃𝑥𝜓)))

Theorembj-nexdh2 32732 Uncurried (imported) form of bj-nexdh 32731. (Contributed by BJ, 6-May-2019.)
((∀𝑥(𝜑 → ¬ 𝜓) ∧ (𝜒 → ∀𝑥𝜑)) → (𝜒 → ¬ ∃𝑥𝜓))

Theorembj-hbxfrbi 32733 Closed form of hbxfrbi 1792. Notes: it is less important than nfbiit 1817; it requires sp 2091 (unlike nfbiit 1817); there is an obvious version with (∃𝑥𝜑𝜑) instead. (Contributed by BJ, 6-May-2019.)
(∀𝑥(𝜑𝜓) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)))

Theorembj-exlime 32734 Variant of exlimih 2186 where the non-freeness of 𝑥 in 𝜓 is expressed using an existential quantifier, thus requiring fewer axioms. (Contributed by BJ, 17-Mar-2020.)
(∃𝑥𝜓𝜓)    &   (𝜑𝜓)       (∃𝑥𝜑𝜓)

Theorembj-exnalimn 32735 A transformation of quantifiers and logical connectives. The general statement that equs3 1932 proves.

This and the following theorems are the general instances of already proved theorems. They could be moved to the main part, before ax-5 1879. I propose to move to the main part: bj-exnalimn 32735, bj-exalim 32736, bj-exalimi 32737, bj-exalims 32738, bj-exalimsi 32739, bj-ax12i 32741, bj-ax12wlem 32742, bj-ax12w 32790, and remove equs3 1932. A new label is needed for bj-ax12i 32741 and label suggestions are welcome for the others. I also propose to change ¬ ∀𝑥¬ to 𝑥 in speimfw 1933 and spimfw 1935 (other spim* theorems use 𝑥 and very few theorems in set.mm use ¬ ∀𝑥¬). (Contributed by BJ, 29-Sep-2019.)

(∃𝑥(𝜑𝜓) ↔ ¬ ∀𝑥(𝜑 → ¬ 𝜓))

Theorembj-exalim 32736 Distributing quantifiers over a double implication. (Contributed by BJ, 8-Nov-2021.)
(∀𝑥(𝜑 → (𝜓𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒)))

Theorembj-exalimi 32737 An inference for distributing quantifiers over a double implication. (Almost) the general statement that speimfw 1933 proves. (Contributed by BJ, 29-Sep-2019.)
(𝜑 → (𝜓𝜒))       (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))

Theorembj-exalims 32738 Distributing quantifiers over a double implication. (Almost) the general statement that spimfw 1935 proves. (Contributed by BJ, 29-Sep-2019.)
(∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))       (∀𝑥(𝜑 → (𝜓𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓𝜒)))

Theorembj-exalimsi 32739 An inference for distributing quantifiers over a double implication. (Almost) the general statement that spimfw 1935 proves. (Contributed by BJ, 29-Sep-2019.)
(𝜑 → (𝜓𝜒))    &   (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))       (∃𝑥𝜑 → (∀𝑥𝜓𝜒))

Theorembj-ax12ig 32740 A lemma used to prove a weak form of the axiom of substitution. A generalization of bj-ax12i 32741. (Contributed by BJ, 19-Dec-2020.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒 → ∀𝑥𝜒))       (𝜑 → (𝜓 → ∀𝑥(𝜑𝜓)))

Theorembj-ax12i 32741 A weakening of bj-ax12ig 32740 that is sufficient to prove a weak form of the axiom of substitution ax-12 2087. The general statement of which ax12i 1936 is an instance. (Contributed by BJ, 29-Sep-2019.)
(𝜑 → (𝜓𝜒))    &   (𝜒 → ∀𝑥𝜒)       (𝜑 → (𝜓 → ∀𝑥(𝜑𝜓)))

Theorembj-ax12wlem 32742* A lemma used to prove a weak version of the axiom of substitution ax-12 2087. (Temporary comment: The general statement that ax12wlem 2049 proves.) (Contributed by BJ, 20-Mar-2020.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥(𝜑𝜓)))

20.14.4.4  Equality and substitution

Theorembj-ssbjust 32743* Justification theorem for df-ssb 32745 from Tarski's FOL. (Contributed by BJ, 9-Nov-2021.)
(∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧𝜑)))

Syntaxwssb 32744 Syntax for the substitution of a variable for a variable in a formula. (Contributed by BJ, 22-Dec-2020.)
wff [𝑡/𝑥]b𝜑

Definitiondf-ssb 32745* Alternate definition of proper substitution. Note that the occurrences of a given variable are either all bound (𝑥, 𝑦) or all free (𝑡). Also note that the definiens uses only primitive symbols. It is obtained by applying twice Tarski's definition sb6 2457 which is valid for disjoint variables, so we introduce a dummy variable 𝑦 to isolate 𝑥 from 𝑡, as in dfsb7 2483 with respect to sb5 2458.

This double level definition will make several proofs using it appear as doubled. Alternately, one could often first prove as a lemma the same theorem with a DV condition on the substitute and the substituted variables, and then prove the original theorem by applying this lemma twice in a row.

This definition uses a dummy variable, but the justification theorem, bj-ssbjust 32743, is provable from Tarski's FOL.

Once this is proved, more of the fundamental properties of proper substitution will be provable from Tarski's FOL system, sometimes with the help of specialization sp 2091, of the substitution axiom ax-12 2087, and of commutation of quantifiers ax-11 2074; that is, ax-13 2282 will often be avoided. (Contributed by BJ, 22-Dec-2020.)

([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theorembj-ssbim 32746 Distribute substitution over implication, closed form. Specialization of implication. Uses only ax-1--5. Compare spsbim 2422. (Contributed by BJ, 22-Dec-2020.)
(∀𝑥(𝜑𝜓) → ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓))

Theorembj-ssbbi 32747 Biconditional property for substitution, closed form. Specialization of biconditional. Uses only ax-1--5. Compare spsbbi 2430. (Contributed by BJ, 22-Dec-2020.)
(∀𝑥(𝜑𝜓) → ([𝑡/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜓))

Theorembj-ssbimi 32748 Distribute substitution over implication. Uses only ax-1--5. (Contributed by BJ, 22-Dec-2020.)
(𝜑𝜓)       ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓)

Theorembj-ssbbii 32749 Biconditional property for substitution. Uses only ax-1--5. (Contributed by BJ, 22-Dec-2020.)
(𝜑𝜓)       ([𝑡/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜓)

Theorembj-alsb 32750 If a proposition is true for all instances, then it is true for any specific one. Uses only ax-1--5. Compare stdpc4 2381 which uses auxiliary axioms. (Contributed by BJ, 22-Dec-2020.)
(∀𝑥𝜑 → [𝑡/𝑥]b𝜑)

Theorembj-sbex 32751 If a proposition is true for a specific instance, then there exists an instance such that it is true for it. Uses only ax-1--6. Compare spsbe 1941 which, due to the specific form of df-sb 1938, uses fewer axioms. (Contributed by BJ, 22-Dec-2020.)
([𝑡/𝑥]b𝜑 → ∃𝑥𝜑)

Theorembj-ssbeq 32752* Substitution in an equality, disjoint variables case. Uses only ax-1--6. It might be shorter to prove the result about composition of two substitutions and prove bj-ssbeq 32752 first with a DV on x,t, and then in the general case. (Contributed by BJ, 22-Dec-2020.)
([𝑡/𝑥]b𝑦 = 𝑧𝑦 = 𝑧)

Theorembj-ssb0 32753* Substitution for a variable not occurring in a proposition. See sbf 2408. (Contributed by BJ, 22-Dec-2020.)
([𝑡/𝑥]b𝜑𝜑)

Theorembj-ssbequ 32754 Equality property for substitution, from Tarski's system. Compare sbequ 2404. (Contributed by BJ, 30-Dec-2020.)
(𝑠 = 𝑡 → ([𝑠/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜑))

Theorembj-ssblem1 32755* A lemma for the definiens of df-sb 1938. An instance of sp 2091 proved without it. Note: it has a common subproof with bj-ssbjust 32743. (Contributed by BJ, 22-Dec-2020.)
(∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theorembj-ssblem2 32756* An instance of ax-11 2074 proved without it. The converse may not be provable without ax-11 2074 (since using alcomiw 2013 would require a DV on 𝜑, 𝑥, which defeats the purpose). (Contributed by BJ, 22-Dec-2020.)
(∀𝑥𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) → ∀𝑦𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)))

Theorembj-ssb1a 32757* One direction of a simplified definition of substitution in case of disjoint variables. See bj-ssb1 32758 for the biconditional, which requires ax-11 2074. (Contributed by BJ, 22-Dec-2020.)
(∀𝑥(𝑥 = 𝑡𝜑) → [𝑡/𝑥]b𝜑)

Theorembj-ssb1 32758* A simplified definition of substitution in case of disjoint variables. See bj-ssb1a 32757 for the backward implication, which does not require ax-11 2074 (note that here, the version of ax-11 2074 with disjoint setvar metavariables would suffice). Compare sb6 2457. (Contributed by BJ, 22-Dec-2020.)
([𝑡/𝑥]b𝜑 ↔ ∀𝑥(𝑥 = 𝑡𝜑))

Theorembj-ax12 32759* A weaker form of ax-12 2087 and ax12v2 2089, namely the generalization over 𝑥 of the latter. In this statement, all occurrences of 𝑥 are bound. (Contributed by BJ, 26-Dec-2020.)
𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))

Theorembj-ax12ssb 32760* The axiom bj-ax12 32759 expressed using substitution. (Contributed by BJ, 26-Dec-2020.)
[𝑡/𝑥]b(𝜑 → [𝑡/𝑥]b𝜑)

Theorembj-modal5e 32761 Dual statement of hbe1 2061 (which is the real modal-5 2072). See also axc7 2170 and axc7e 2171. (Contributed by BJ, 21-Dec-2020.)
(∃𝑥𝑥𝜑 → ∀𝑥𝜑)

Theorembj-19.41al 32762 Special case of 19.41 2141 proved from Tarski, ax-10 2059 (modal5) and hba1 2189 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
(∃𝑥(𝜑 ∧ ∀𝑥𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓))

Theorembj-equsexval 32763* Special case of equsexv 2147 proved from Tarski, ax-10 2059 (modal5) and hba1 2189 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥𝜓)

Theorembj-sb56 32764* Proof of sb56 2188 from Tarski, ax-10 2059 (modal5) and bj-ax12 32759. (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
(∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theorembj-dfssb2 32765* An alternate definition of df-ssb 32745. Note that the use of a dummy variable in the definition df-ssb 32745 allows to use bj-sb56 32764 instead of equs45f 2378 and hence to avoid dependency on ax-13 2282 and to use ax-12 2087 only through bj-ax12 32759. Compare dfsb7 2483. (Contributed by BJ, 25-Dec-2020.)
([𝑡/𝑥]b𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))

Theorembj-ssbn 32766 The result of a substitution in the negation of a formula is the negation of the result of the same substitution in that formula. Proved from Tarski, ax-10 2059, bj-ax12 32759. Compare sbn 2419. (Contributed by BJ, 25-Dec-2020.)
([𝑡/𝑥]b ¬ 𝜑 ↔ ¬ [𝑡/𝑥]b𝜑)

Theorembj-ssbft 32767 See sbft 2407. This proof is from Tarski's FOL together with sp 2091 (and its dual). (Contributed by BJ, 22-Dec-2020.)
(Ⅎ𝑥𝜑 → ([𝑡/𝑥]b𝜑𝜑))

Theorembj-ssbequ2 32768 Note that ax-12 2087 is used only via sp 2091. See sbequ2 1939 and stdpc7 2002. (Contributed by BJ, 22-Dec-2020.)
(𝑥 = 𝑡 → ([𝑡/𝑥]b𝜑𝜑))

Theorembj-ssbequ1 32769 This uses ax-12 2087 with a direct reference to ax12v 2088. Therefore, compared to bj-ax12 32759, there is a hidden use of sp 2091. Note that with ax-12 2087, it can be proved with dv condition on 𝑥, 𝑡. See sbequ1 2148. (Contributed by BJ, 22-Dec-2020.)
(𝑥 = 𝑡 → (𝜑 → [𝑡/𝑥]b𝜑))

Theorembj-ssbid2 32770 A special case of bj-ssbequ2 32768. (Contributed by BJ, 22-Dec-2020.)
([𝑥/𝑥]b𝜑𝜑)

Theorembj-ssbid2ALT 32771 Alternate proof of bj-ssbid2 32770, not using bj-ssbequ2 32768. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑥/𝑥]b𝜑𝜑)

Theorembj-ssbid1 32772 A special case of bj-ssbequ1 32769. (Contributed by BJ, 22-Dec-2020.)
(𝜑 → [𝑥/𝑥]b𝜑)

Theorembj-ssbid1ALT 32773 Alternate proof of bj-ssbid1 32772, not using bj-ssbequ1 32769. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → [𝑥/𝑥]b𝜑)

Theorembj-ssbssblem 32774* Composition of two substitutions with a fresh intermediate variable. Remark: does not seem useful. (Contributed by BJ, 22-Dec-2020.)
([𝑡/𝑦]b[𝑦/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜑)

Theorembj-ssbcom3lem 32775* Lemma for bj-ssbcom3 when setvar variables are disjoint. Remark: does not seem useful. (Contributed by BJ, 30-Dec-2020.)
([𝑡/𝑦]b[𝑦/𝑥]b𝜑 ↔ [𝑡/𝑦]b[𝑡/𝑥]b𝜑)

Theorembj-ax6elem1 32776* Lemma for bj-ax6e 32778. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theorembj-ax6elem2 32777* Lemma for bj-ax6e 32778. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
(∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)

Theorembj-ax6e 32778 Proof of ax6e 2286 (hence ax6 2287) from Tarski's system, ax-c9 34494, ax-c16 34496. Remark: ax-6 1945 is used only via its principal (unbundled) instance ax6v 1946. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥 𝑥 = 𝑦

Theorembj-extru 32779 There exists a variable such that holds; that is, there exists a variable. This corresponds under the standard translation to one of the formulations of the modal axiom (D), the other being 19.2 1949. (This is also extt 32528; propose to move to Main extt 32528 and allt 32525; relabel exiftru 1948 to "exgen", for "existential generalization", which is the standard name for that rule of inference ? ). (Contributed by BJ, 12-May-2019.) (Proof modification is discouraged.)
𝑥

Theorembj-alequexv 32780* Version of bj-alequex 32833 with DV(x,y), requiring fewer axioms. (Contributed by BJ, 9-Nov-2021.)
(∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)

Theorembj-spimvwt 32781* Closed form of spimvw 1973. See also spimt 2289. (Contributed by BJ, 8-Nov-2021.)
(∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∀𝑥𝜑𝜓))

Theorembj-spimevw 32782* Existential introduction, using implicit substitution. This is to spimeh 1971 what spimvw 1973 is to spimw 1972. (Contributed by BJ, 17-Mar-2020.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)

Theorembj-spnfw 32783 Theorem close to a closed form of spnfw 1974. (Contributed by BJ, 12-May-2019.)
((∃𝑥𝜑𝜓) → (∀𝑥𝜑𝜓))

Theorembj-cbvexiw 32784* Change bound variable. This is to cbvexvw 2012 what cbvaliw 1979 is to cbvalvw 2011. [TODO: move after cbvalivw 1980]. (Contributed by BJ, 17-Mar-2020.)
(∃𝑥𝑦𝜓 → ∃𝑦𝜓)    &   (𝜑 → ∀𝑦𝜑)    &   (𝑦 = 𝑥 → (𝜑𝜓))       (∃𝑥𝜑 → ∃𝑦𝜓)

Theorembj-cbvexivw 32785* Change bound variable. This is to cbvexvw 2012 what cbvalivw 1980 is to cbvalvw 2011. [TODO: move after cbvalivw 1980]. (Contributed by BJ, 17-Mar-2020.)
(𝑦 = 𝑥 → (𝜑𝜓))       (∃𝑥𝜑 → ∃𝑦𝜓)

Theorembj-modald 32786 A short form of the axiom D of modal logic. (Contributed by BJ, 4-Apr-2021.)
(∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)

Theorembj-denot 32787* A weakening of ax-6 1945 and ax6v 1946. (Contributed by BJ, 4-Apr-2021.) (New usage is discouraged.)
(𝑥 = 𝑥 → ¬ ∀𝑦 ¬ 𝑦 = 𝑥)

Theorembj-eqs 32788* A lemma for substitutions, proved from Tarski's FOL. The version without DV(𝑥, 𝑦) is true but requires ax-13 2282. The DV condition DV( 𝑥, 𝜑) is necessary for both directions: consider substituting 𝑥 = 𝑧 for 𝜑. (Contributed by BJ, 25-May-2021.)
(𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theorembj-cbvexw 32789* Change bound variable. This is to cbvexvw 2012 what cbvalw 2010 is to cbvalvw 2011. (Contributed by BJ, 17-Mar-2020.)
(∃𝑥𝑦𝜓 → ∃𝑦𝜓)    &   (𝜑 → ∀𝑦𝜑)    &   (∃𝑦𝑥𝜑 → ∃𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Theorembj-ax12w 32790* The general statement that ax12w 2050 proves. (Contributed by BJ, 20-Mar-2020.)
(𝜑 → (𝜓𝜒))    &   (𝑦 = 𝑧 → (𝜓𝜃))       (𝜑 → (∀𝑦𝜓 → ∀𝑥(𝜑𝜓)))

20.14.4.7  Membership predicate, ax-8 and ax-9

Theorembj-elequ2g 32791* A form of elequ2 2044 with a universal quantifier. Its converse is ax-ext 2631. (TODO: move to main part, minimize axext4 2635--- as of 4-Nov-2020, minimizes only axext4 2635, by 13 bytes; and link to it in the comment of ax-ext 2631.) (Contributed by BJ, 3-Oct-2019.)
(𝑥 = 𝑦 → ∀𝑧(𝑧𝑥𝑧𝑦))

Theorembj-ax89 32792 A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 2032 and ax-9 2039. Indeed, it is implied over propositional calculus by the conjunction of ax-8 2032 and ax-9 2039, as proved here. In the other direction, one can prove ax-8 2032 (respectively ax-9 2039) from bj-ax89 32792 by using mpan2 707 ( respectively mpan 706) and equid 1985. (TODO: move to main part.) (Contributed by BJ, 3-Oct-2019.)
((𝑥 = 𝑦𝑧 = 𝑡) → (𝑥𝑧𝑦𝑡))

Theorembj-elequ12 32793 An identity law for the non-logical predicate, which combines elequ1 2037 and elequ2 2044. For the analogous theorems for class terms, see eleq1 2718, eleq2 2719 and eleq12 2720. (TODO: move to main part.) (Contributed by BJ, 29-Sep-2019.)
((𝑥 = 𝑦𝑧 = 𝑡) → (𝑥𝑧𝑦𝑡))

Theorembj-cleljusti 32794* One direction of cleljust 2038, requiring only ax-1 6-- ax-5 1879 and ax8v1 2034. (Contributed by BJ, 31-Dec-2020.) (Proof modification is discouraged.)
(∃𝑧(𝑧 = 𝑥𝑧𝑦) → 𝑥𝑦)

Theorembj-alcomexcom 32795 Commutation of universal quantifiers implies commutation of existential quantifiers. Can be placed in the ax-4 1777 section, soon after 2nexaln 1797, and used to prove excom 2082. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.)
((∀𝑥𝑦 ¬ 𝜑 → ∀𝑦𝑥 ¬ 𝜑) → (∃𝑦𝑥𝜑 → ∃𝑥𝑦𝜑))

Theorembj-hbalt 32796 Closed form of hbal 2076. When in main part, prove hbal 2076 and hbald 2081 from it. (Contributed by BJ, 2-May-2019.)
(∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥𝑦𝜑))

Theoremaxc11n11 32797 Proof of axc11n 2342 from { ax-1 6-- ax-7 1981, axc11 2347 } . Almost identical to axc11nfromc11 34530. (Contributed by NM, 6-Jul-2021.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremaxc11n11r 32798 Proof of axc11n 2342 from { ax-1 6-- ax-7 1981, axc9 2338, axc11r 2223 } (note that axc16 2173 is provable from { ax-1 6-- ax-7 1981, axc11r 2223 }).

Note that axc11n 2342 proves (over minimal calculus) that axc11 2347 and axc11r 2223 are equivalent. Therefore, axc11n11 32797 and axc11n11r 32798 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2347 appears slightly stronger since axc11n11r 32798 requires axc9 2338 while axc11n11 32797 does not).

(Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)

(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theorembj-axc16g16 32799* Proof of axc16g 2172 from { ax-1 6-- ax-7 1981, axc16 2173 }. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))

Theorembj-ax12v3 32800* A weak version of ax-12 2087 which is stronger than ax12v 2088. Note that if one assumes reflexivity of equality 𝑥 = 𝑥 (equid 1985), then bj-ax12v3 32800 implies ax-5 1879 over modal logic K (substitute 𝑥 for 𝑦). See also bj-ax12v3ALT 32801. (Contributed by BJ, 6-Jul-2021.)
(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
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