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Theorem List for Metamath Proof Explorer - 33201-33300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembj-cbvalimt 33201 A lemma in closed form used to prove bj-cbval 33208 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) Do not use 19.35 1924 since only the direction of the biconditional used here holds in intuitionistic logic. (Proof modification is discouraged.)
(∀𝑦𝑥𝜒 → (∀𝑦𝑥(𝜒 → (𝜑𝜓)) → ((∀𝑥𝜑 → ∀𝑦𝑥𝜑) → (∀𝑦(∃𝑥𝜓𝜓) → (∀𝑥𝜑 → ∀𝑦𝜓)))))

Theorembj-cbveximt 33202 A lemma in closed form used to prove bj-cbvex in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) Do not use 19.35 1924 since only the direction of the biconditional used here holds in intuitionistic logic. (Proof modification is discouraged.)
(∀𝑥𝑦𝜒 → (∀𝑥𝑦(𝜒 → (𝜑𝜓)) → (∀𝑥(𝜑 → ∀𝑦𝜑) → ((∃𝑥𝑦𝜓 → ∃𝑦𝜓) → (∃𝑥𝜑 → ∃𝑦𝜓)))))

Theorembj-ax12wlem 33203* A lemma used to prove a weak version of the axiom of substitution ax-12 2163. (Temporary comment: The general statement that ax12wlem 2126 proves.) (Contributed by BJ, 20-Mar-2020.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥(𝜑𝜓)))

Theorembj-cbvalim 33204* A lemma used to prove a justification of df-bj-mo 33211 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
(∀𝑦𝑥𝜒 → (∀𝑦𝑥(𝜒 → (𝜑𝜓)) → (∀𝑥𝜑 → ∀𝑦𝜓)))

Theorembj-cbvexim 33205* A lemma used to prove justification theorems in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
(∀𝑥𝑦𝜒 → (∀𝑥𝑦(𝜒 → (𝜑𝜓)) → (∃𝑥𝜑 → ∃𝑦𝜓)))

Theorembj-cbvalimi 33206* An equality-free general instance of one half of a precise form of bj-cbval 33208. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
(𝜒 → (𝜑𝜓))    &   𝑦𝑥𝜒       (∀𝑥𝜑 → ∀𝑦𝜓)

Theorembj-cbveximi 33207* An equality-free general instance of one half of a precise form of bj-cbvex 33209. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
(𝜒 → (𝜑𝜓))    &   𝑥𝑦𝜒       (∃𝑥𝜑 → ∃𝑦𝜓)

Theorembj-cbval 33208* 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.)
𝑦𝑥 𝑥 = 𝑦    &   𝑥𝑦 𝑦 = 𝑥    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑦 = 𝑥𝑥 = 𝑦)       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Theorembj-cbvex 33209* 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.)
𝑦𝑥 𝑥 = 𝑦    &   𝑥𝑦 𝑦 = 𝑥    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑦 = 𝑥𝑥 = 𝑦)       (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Syntaxwmoo 33210 Syntax for BJ's version of the uniqueness quantifier.
wff ∃**𝑥𝜑

Definitiondf-bj-mo 33211* 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 ↔ ∀𝑧𝑦𝑥(𝜑𝑥 = 𝑦))

20.14.4.4  Equality and substitution

Theorembj-ssbjustlem 33212* Lemma for bj-ssbjust 33213. (Contributed by BJ, 9-Nov-2021.)
(∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧𝜑)))

Theorembj-ssbjust 33213* Justification theorem for df-ssb 33215 from Tarski's FOL. (Contributed by BJ, 22-Jan-2023.)
(∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧𝜑)))

Syntaxwssb 33214 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 33215* 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 2250 which is valid for disjoint variables, so we introduce a dummy variable 𝑦 to isolate 𝑥 from 𝑡, as in dfsb7 2535 with respect to sb5 2251.

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 disjoint variable 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 33213, 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 2167, of the substitution axiom ax-12 2163, and of commutation of quantifiers ax-11 2150; that is, ax-13 2334 will often be avoided. (Contributed by BJ, 22-Dec-2020.)

([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theorembj-ssbim 33216 Distribute substitution over implication, closed form. Specialization of implication. Uses only ax-1--5. Compare spsbim 2470. (Contributed by BJ, 22-Dec-2020.)
(∀𝑥(𝜑𝜓) → ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓))

Theorembj-ssbbi 33217 Biconditional property for substitution, closed form. Specialization of biconditional. Uses only ax-1--5. Compare spsbbi 2478. (Contributed by BJ, 22-Dec-2020.)
(∀𝑥(𝜑𝜓) → ([𝑡/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜓))

Theorembj-ssbimi 33218 Distribute substitution over implication. Uses only ax-1--5. (Contributed by BJ, 22-Dec-2020.)
(𝜑𝜓)       ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓)

Theorembj-ssbbii 33219 Biconditional property for substitution. Uses only ax-1--5. (Contributed by BJ, 22-Dec-2020.)
(𝜑𝜓)       ([𝑡/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜓)

Theorembj-alsb 33220 If a proposition is true for all instances, then it is true for any specific one. Uses only ax-1--5. Compare stdpc4 2428 which uses auxiliary axioms. (Contributed by BJ, 22-Dec-2020.)
(∀𝑥𝜑 → [𝑡/𝑥]b𝜑)

Theorembj-sbex 33221 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 2015 which, due to the specific form of df-sb 2012, uses fewer axioms. (Contributed by BJ, 22-Dec-2020.)
([𝑡/𝑥]b𝜑 → ∃𝑥𝜑)

Theorembj-ssbeq 33222* 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 33222 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 33223* Substitution for a variable not occurring in a proposition. See sbf 2456. (Contributed by BJ, 22-Dec-2020.)
([𝑡/𝑥]b𝜑𝜑)

Theorembj-ssbequ 33224 Equality property for substitution, from Tarski's system. Compare sbequ 2452. (Contributed by BJ, 30-Dec-2020.)
(𝑠 = 𝑡 → ([𝑠/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜑))

Theorembj-ssblem1 33225* A lemma for the definiens of df-sb 2012. An instance of sp 2167 proved without it. Note: it has a common subproof with bj-ssbjust 33213. (Contributed by BJ, 22-Dec-2020.)
(∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theorembj-ssblem2 33226* An instance of ax-11 2150 proved without it. The converse may not be provable without ax-11 2150 (since using alcomiw 2088 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 33227* One direction of a simplified definition of substitution in case of disjoint variables. See bj-ssb1 33228 for the biconditional, which requires ax-11 2150. (Contributed by BJ, 22-Dec-2020.)
(∀𝑥(𝑥 = 𝑡𝜑) → [𝑡/𝑥]b𝜑)

Theorembj-ssb1 33228* A simplified definition of substitution in case of disjoint variables. See bj-ssb1a 33227 for the backward implication, which does not require ax-11 2150 (note that here, the version of ax-11 2150 with disjoint setvar metavariables would suffice). Compare sb6 2250. (Contributed by BJ, 22-Dec-2020.)
([𝑡/𝑥]b𝜑 ↔ ∀𝑥(𝑥 = 𝑡𝜑))

Theorembj-ax12 33229* A weaker form of ax-12 2163 and ax12v2 2165, 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 33230* The axiom bj-ax12 33229 expressed using substitution. (Contributed by BJ, 26-Dec-2020.)
[𝑡/𝑥]b(𝜑 → [𝑡/𝑥]b𝜑)

Theorembj-19.41al 33231 Special case of 19.41 2221 proved from Tarski, ax-10 2135 (modal5) and hba1 2268 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
(∃𝑥(𝜑 ∧ ∀𝑥𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓))

Theorembj-equsexval 33232* Special case of equsexv 2242 proved from Tarski, ax-10 2135 (modal5) and hba1 2268 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥𝜓)

Theorembj-sb56 33233* Proof of sb56 2248 from Tarski, ax-10 2135 (modal5) and bj-ax12 33229. (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
(∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theorembj-dfssb2 33234* An alternate definition of df-ssb 33215. Note that the use of a dummy variable in the definition df-ssb 33215 allows to use bj-sb56 33233 instead of equs45f 2425 and hence to avoid dependency on ax-13 2334 and to use ax-12 2163 only through bj-ax12 33229. Compare dfsb7 2535. (Contributed by BJ, 25-Dec-2020.)
([𝑡/𝑥]b𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))

Theorembj-ssbn 33235 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 2135, bj-ax12 33229. Compare sbn 2467. (Contributed by BJ, 25-Dec-2020.)
([𝑡/𝑥]b ¬ 𝜑 ↔ ¬ [𝑡/𝑥]b𝜑)

Theorembj-ssbft 33236 See sbft 2455. This proof is from Tarski's FOL together with sp 2167 (and its dual). (Contributed by BJ, 22-Dec-2020.)
(Ⅎ𝑥𝜑 → ([𝑡/𝑥]b𝜑𝜑))

Theorembj-ssbequ2 33237 Note that ax-12 2163 is used only via sp 2167. See sbequ2 2013 and stdpc7 2076. (Contributed by BJ, 22-Dec-2020.)
(𝑥 = 𝑡 → ([𝑡/𝑥]b𝜑𝜑))

Theorembj-ssbequ1 33238 This uses ax-12 2163 with a direct reference to ax12v 2164. Therefore, compared to bj-ax12 33229, there is a hidden use of sp 2167. Note that with ax-12 2163, it can be proved with disjoint variable condition on 𝑥, 𝑡. See sbequ1 2228. (Contributed by BJ, 22-Dec-2020.)
(𝑥 = 𝑡 → (𝜑 → [𝑡/𝑥]b𝜑))

Theorembj-ssbid2 33239 A special case of bj-ssbequ2 33237. (Contributed by BJ, 22-Dec-2020.)
([𝑥/𝑥]b𝜑𝜑)

Theorembj-ssbid2ALT 33240 Alternate proof of bj-ssbid2 33239, not using bj-ssbequ2 33237. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑥/𝑥]b𝜑𝜑)

Theorembj-ssbid1 33241 A special case of bj-ssbequ1 33238. (Contributed by BJ, 22-Dec-2020.)
(𝜑 → [𝑥/𝑥]b𝜑)

Theorembj-ssbid1ALT 33242 Alternate proof of bj-ssbid1 33241, not using bj-ssbequ1 33238. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → [𝑥/𝑥]b𝜑)

Theorembj-ssbssblem 33243* 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 33244* 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 33245* Lemma for bj-ax6e 33247. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theorembj-ax6elem2 33246* Lemma for bj-ax6e 33247. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
(∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)

Theorembj-ax6e 33247 Proof of ax6e 2347 (hence ax6 2348) from Tarski's system, ax-c9 35049, ax-c16 35051. Remark: ax-6 2021 is used only via its principal (unbundled) instance ax6v 2022. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥 𝑥 = 𝑦

Theorembj-alequexv 33248* Version of bj-alequex 33300 with DV(x,y), requiring fewer axioms. (Contributed by BJ, 9-Nov-2021.)
(∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)

Theorembj-spimvwt 33249* Closed form of spimvw 2045. See also spimt 2350. (Contributed by BJ, 8-Nov-2021.)
(∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∀𝑥𝜑𝜓))

Theorembj-spimevw 33250* Existential introduction, using implicit substitution. This is to spimeh 2043 what spimvw 2045 is to spimw 2044. (Contributed by BJ, 17-Mar-2020.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)

Theorembj-spnfw 33251 Theorem close to a closed form of spnfw 2046. (Contributed by BJ, 12-May-2019.)
((∃𝑥𝜑𝜓) → (∀𝑥𝜑𝜓))

Theorembj-cbvexiw 33252* Change bound variable. This is to cbvexvw 2087 what cbvaliw 2053 is to cbvalvw 2086. [TODO: move after cbvalivw 2054]. (Contributed by BJ, 17-Mar-2020.)
(∃𝑥𝑦𝜓 → ∃𝑦𝜓)    &   (𝜑 → ∀𝑦𝜑)    &   (𝑦 = 𝑥 → (𝜑𝜓))       (∃𝑥𝜑 → ∃𝑦𝜓)

Theorembj-cbvexivw 33253* Change bound variable. This is to cbvexvw 2087 what cbvalivw 2054 is to cbvalvw 2086. [TODO: move after cbvalivw 2054]. (Contributed by BJ, 17-Mar-2020.)
(𝑦 = 𝑥 → (𝜑𝜓))       (∃𝑥𝜑 → ∃𝑦𝜓)

Theorembj-modald 33254 A short form of the axiom D of modal logic. (Contributed by BJ, 4-Apr-2021.)
(∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)

Theorembj-denot 33255* A weakening of ax-6 2021 and ax6v 2022. (Contributed by BJ, 4-Apr-2021.) (New usage is discouraged.)
(𝑥 = 𝑥 → ¬ ∀𝑦 ¬ 𝑦 = 𝑥)

Theorembj-eqs 33256* A lemma for substitutions, proved from Tarski's FOL. The version without DV(x,y) is true but requires ax-13 2334. The disjoint variable condition DV(x,ph) is necessary for both directions: consider substituting 𝑥 = 𝑧 for 𝜑. (Contributed by BJ, 25-May-2021.)
(𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theorembj-cbvexw 33257* Change bound variable. This is to cbvexvw 2087 what cbvalw 2085 is to cbvalvw 2086. (Contributed by BJ, 17-Mar-2020.)
(∃𝑥𝑦𝜓 → ∃𝑦𝜓)    &   (𝜑 → ∀𝑦𝜑)    &   (∃𝑦𝑥𝜑 → ∃𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Theorembj-ax12w 33258* The general statement that ax12w 2127 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 33259* A form of elequ2 2121 with a universal quantifier. Its converse is ax-ext 2754. (TODO: move to main part, minimize axext4 2758--- as of 4-Nov-2020, minimizes only axext4 2758, by 13 bytes; and link to it in the comment of ax-ext 2754.) (Contributed by BJ, 3-Oct-2019.)
(𝑥 = 𝑦 → ∀𝑧(𝑧𝑥𝑧𝑦))

Theorembj-ax89 33260 A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 2109 and ax-9 2116. Indeed, it is implied over propositional calculus by the conjunction of ax-8 2109 and ax-9 2116, as proved here. In the other direction, one can prove ax-8 2109 (respectively ax-9 2116) from bj-ax89 33260 by using mpan2 681 ( respectively mpan 680) and equid 2059. (TODO: move to main part.) (Contributed by BJ, 3-Oct-2019.)
((𝑥 = 𝑦𝑧 = 𝑡) → (𝑥𝑧𝑦𝑡))

Theorembj-elequ12 33261 An identity law for the non-logical predicate, which combines elequ1 2114 and elequ2 2121. For the analogous theorems for class terms, see eleq1 2847, eleq2 2848 and eleq12 2849. (TODO: move to main part.) (Contributed by BJ, 29-Sep-2019.)
((𝑥 = 𝑦𝑧 = 𝑡) → (𝑥𝑧𝑦𝑡))

Theorembj-cleljusti 33262* One direction of cleljust 2115, requiring only ax-1 6-- ax-5 1953 and ax8v1 2111. (Contributed by BJ, 31-Dec-2020.) (Proof modification is discouraged.)
(∃𝑧(𝑧 = 𝑥𝑧𝑦) → 𝑥𝑦)

Theorembj-alcomexcom 33263 Commutation of universal quantifiers implies commutation of existential quantifiers. Can be placed in the ax-4 1853 section, soon after 2nexaln 1873, and used to prove excom 2155. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.)
((∀𝑥𝑦 ¬ 𝜑 → ∀𝑦𝑥 ¬ 𝜑) → (∃𝑦𝑥𝜑 → ∃𝑥𝑦𝜑))

Theorembj-hbalt 33264 Closed form of hbal 2160. When in main part, prove hbal 2160 and hbald 2161 from it. (Contributed by BJ, 2-May-2019.)
(∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥𝑦𝜑))

Theoremaxc11n11 33265 Proof of axc11n 2392 from { ax-1 6-- ax-7 2055, axc11 2396 } . Almost identical to axc11nfromc11 35085. (Contributed by NM, 6-Jul-2021.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremaxc11n11r 33266 Proof of axc11n 2392 from { ax-1 6-- ax-7 2055, axc9 2346, axc11r 2333 } (note that axc16 2234 is provable from { ax-1 6-- ax-7 2055, axc11r 2333 }).

Note that axc11n 2392 proves (over minimal calculus) that axc11 2396 and axc11r 2333 are equivalent. Therefore, axc11n11 33265 and axc11n11r 33266 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2396 appears slightly stronger since axc11n11r 33266 requires axc9 2346 while axc11n11 33265 does not).

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

(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theorembj-axc16g16 33267* Proof of axc16g 2233 from { ax-1 6-- ax-7 2055, axc16 2234 }. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))

Theorembj-ax12v3 33268* A weak version of ax-12 2163 which is stronger than ax12v 2164. Note that if one assumes reflexivity of equality 𝑥 = 𝑥 (equid 2059), then bj-ax12v3 33268 implies ax-5 1953 over modal logic K (substitute 𝑥 for 𝑦). See also bj-ax12v3ALT 33269. (Contributed by BJ, 6-Jul-2021.)
(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theorembj-ax12v3ALT 33269* Alternate proof of bj-ax12v3 33268. Uses axc11r 2333 and axc15 2387 instead of ax-12 2163. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theorembj-sb 33270* A weak variant of sbid2 2489 not requiring ax-13 2334 nor ax-10 2135. On top of Tarski's FOL, one implication requires only ax12v 2164, and the other requires only sp 2167. (Contributed by BJ, 25-May-2021.)
(𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theorembj-modalbe 33271 The predicate-calculus version of the axiom (B) of modal logic. See also modal-b 2294. (Contributed by BJ, 20-Oct-2019.)
(𝜑 → ∀𝑥𝑥𝜑)

Theorembj-spst 33272 Closed form of sps 2169. Once in main part, prove sps 2169 and spsd 2171 from it. (Contributed by BJ, 20-Oct-2019.)
((𝜑𝜓) → (∀𝑥𝜑𝜓))

Theorembj-19.21bit 33273 Closed form of 19.21bi 2173. (Contributed by BJ, 20-Oct-2019.)
((𝜑 → ∀𝑥𝜓) → (𝜑𝜓))

Theorembj-19.23bit 33274 Closed form of 19.23bi 2175. (Contributed by BJ, 20-Oct-2019.)
((∃𝑥𝜑𝜓) → (𝜑𝜓))

Theorembj-nexrt 33275 Closed form of nexr 2176. Contrapositive of 19.8a 2166. (Contributed by BJ, 20-Oct-2019.)
(¬ ∃𝑥𝜑 → ¬ 𝜑)

Theorembj-alrim 33276 Closed form of alrimi 2199. (Contributed by BJ, 2-May-2019.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))

Theorembj-alrim2 33277 Uncurried (imported) form of bj-alrim 33276. (Contributed by BJ, 2-May-2019.)
((Ⅎ𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → (𝜑 → ∀𝑥𝜓))

Theorembj-nfdt0 33278 A theorem close to a closed form of nf5d 2258 and nf5dh 2141. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → (∀𝑥𝜑 → Ⅎ𝑥𝜓))

Theorembj-nfdt 33279 Closed form of nf5d 2258 and nf5dh 2141. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → ((𝜑 → ∀𝑥𝜑) → (𝜑 → Ⅎ𝑥𝜓)))

Theorembj-nexdt 33280 Closed form of nexd 2207. (Contributed by BJ, 20-Oct-2019.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓)))

Theorembj-nexdvt 33281* Closed form of nexdv 1979. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓))

Theorembj-alexbiex 33282 Adding a second quantifier is a tranparent operation, (∀∃ case). (Contributed by BJ, 20-Oct-2019.)
(∀𝑥𝑥𝜑 ↔ ∃𝑥𝜑)

Theorembj-exexbiex 33283 Adding a second quantifier is a tranparent operation, (∃∃ case). (Contributed by BJ, 20-Oct-2019.)
(∃𝑥𝑥𝜑 ↔ ∃𝑥𝜑)

Theorembj-alalbial 33284 Adding a second quantifier is a tranparent operation, (∀∀ case). (Contributed by BJ, 20-Oct-2019.)
(∀𝑥𝑥𝜑 ↔ ∀𝑥𝜑)

Theorembj-exalbial 33285 Adding a second quantifier is a tranparent operation, (∃∀ case). (Contributed by BJ, 20-Oct-2019.)
(∃𝑥𝑥𝜑 ↔ ∀𝑥𝜑)

Theorembj-19.9htbi 33286 Strengthening 19.9ht 2295 by replacing its succedent with a biconditional (19.9t 2189 does have a biconditional succedent). This propagates. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))

Theorembj-hbntbi 33287 Strengthening hbnt 2269 by replacing its succedent with a biconditional. See also hbntg 32303 and hbntal 39723. (Contributed by BJ, 20-Oct-2019.) Proved from bj-19.9htbi 33286. (Proof modification is discouraged.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑))

Theorembj-biexal1 33288 A general FOL biconditional that generalizes 19.9ht 2295 among others. For this and the following theorems, see also 19.35 1924, 19.21 2192, 19.23 2197. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))

Theorembj-biexal2 33289 A general FOL biconditional. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(∃𝑥𝜑𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))

Theorembj-biexal3 33290 A general FOL biconditional. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(∃𝑥𝜑𝜓))

Theorembj-bialal 33291 A general FOL biconditional. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(∀𝑥𝜑𝜓) ↔ (∀𝑥𝜑 → ∀𝑥𝜓))

Theorembj-biexex 33292 A general FOL biconditional. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ∃𝑥𝜓) ↔ (∃𝑥𝜑 → ∃𝑥𝜓))

Theorembj-hbext 33293 Closed form of hbex 2301. (Contributed by BJ, 10-Oct-2019.)
(∀𝑦𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑦𝜑 → ∀𝑥𝑦𝜑))

Theorembj-nfalt 33294 Closed form of nfal 2299. (Contributed by BJ, 2-May-2019.)
(∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)

Theorembj-nfext 33295 Closed form of nfex 2300. (Contributed by BJ, 10-Oct-2019.)
(∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)

Theorembj-eeanvw 33296* Version of exdistrv 1998 with a disjoint variable condition on 𝑥, 𝑦 not requiring ax-11 2150. (The same can be done with eeeanv 2319 and ee4anv 2320.) (Contributed by BJ, 29-Sep-2019.) (Proof modification is discouraged.)
(∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Theorembj-modal4e 33297 Dual statement of hba1 2268 (which is modal-4 ). (Contributed by BJ, 21-Dec-2020.)
(∃𝑥𝑥𝜑 → ∃𝑥𝜑)

Theorembj-modalb 33298 A short form of the axiom B of modal logic. (Contributed by BJ, 4-Apr-2021.)
𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)