P. 333 of Theorem List - Metamath Proof Explorer

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

Type	Label	Description
Statement
Theorem	bj-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.)
⊢ (∀𝑦∃𝑥𝜒 → (∀𝑦∀𝑥(𝜒 → (𝜑 → 𝜓)) → ((∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) → (∀𝑦(∃𝑥𝜓 → 𝜓) → (∀𝑥𝜑 → ∀𝑦𝜓)))))
Theorem	bj-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.)
⊢ (∀𝑥∃𝑦𝜒 → (∀𝑥∀𝑦(𝜒 → (𝜑 → 𝜓)) → (∀𝑥(𝜑 → ∀𝑦𝜑) → ((∃𝑥∃𝑦𝜓 → ∃𝑦𝜓) → (∃𝑥𝜑 → ∃𝑦𝜓)))))
20.14.4.3  Adding ax-5
Theorem	bj-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 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))
Theorem	bj-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.)
⊢ (∀𝑦∃𝑥𝜒 → (∀𝑦∀𝑥(𝜒 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → ∀𝑦𝜓)))
Theorem	bj-cbvexim 33205*	A lemma used to prove justification theorems 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 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 → (𝜑 → 𝜓))    &   ⊢ ∀𝑦∃𝑥𝜒    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
Theorem	bj-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 → (𝜑 → 𝜓))    &   ⊢ ∀𝑥∃𝑦𝜒    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
Theorem	bj-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.)
⊢ ∀𝑦∃𝑥 𝑥 = 𝑦    &   ⊢ ∀𝑥∃𝑦 𝑦 = 𝑥    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)    ⇒   ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Theorem	bj-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.)
⊢ ∀𝑦∃𝑥 𝑥 = 𝑦    &   ⊢ ∀𝑥∃𝑦 𝑦 = 𝑥    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)    ⇒   ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Syntax	wmoo 33210	Syntax for BJ's version of the uniqueness quantifier.
wff ∃**𝑥𝜑
Definition	df-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 ↔ ∀𝑧∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
20.14.4.4  Equality and substitution
Theorem	bj-ssbjustlem 33212*	Lemma for bj-ssbjust 33213. (Contributed by BJ, 9-Nov-2021.)
⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
Theorem	bj-ssbjust 33213*	Justification theorem for df-ssb 33215 from Tarski's FOL. (Contributed by BJ, 22-Jan-2023.)
⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
Syntax	wssb 33214	Syntax for the substitution of a variable for a variable in a formula. (Contributed by BJ, 22-Dec-2020.)
wff [𝑡/𝑥]b𝜑
Definition	df-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 = 𝑦 → 𝜑)))
Theorem	bj-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(𝜑 → 𝜓) → ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓))
Theorem	bj-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(𝜑 ↔ 𝜓) → ([𝑡/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜓))
Theorem	bj-ssbimi 33218	Distribute substitution over implication. Uses only ax-1--5. (Contributed by BJ, 22-Dec-2020.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓)
Theorem	bj-ssbbii 33219	Biconditional property for substitution. Uses only ax-1--5. (Contributed by BJ, 22-Dec-2020.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ([𝑡/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜓)
Theorem	bj-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)
Theorem	bj-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)
Theorem	bj-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 = 𝑧 ↔ 𝑦 = 𝑧)
Theorem	bj-ssb0 33223*	Substitution for a variable not occurring in a proposition. See sbf 2456. (Contributed by BJ, 22-Dec-2020.)
⊢ ([𝑡/𝑥]b𝜑 ↔ 𝜑)
Theorem	bj-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))
Theorem	bj-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 = 𝑦 → 𝜑)) → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-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.)
⊢ (∀𝑥∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) → ∀𝑦∀𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)))
Theorem	bj-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 = 𝑡 → 𝜑) → [𝑡/𝑥]b𝜑)
Theorem	bj-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 = 𝑡 → 𝜑))
Theorem	bj-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 = 𝑡 → 𝜑)))
Theorem	bj-ax12ssb 33230*	The axiom bj-ax12 33229 expressed using substitution. (Contributed by BJ, 26-Dec-2020.)
⊢ [𝑡/𝑥]b(𝜑 → [𝑡/𝑥]b𝜑)
Theorem	bj-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))
Theorem	bj-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 = 𝑦 ∧ 𝜑) ↔ ∀𝑥𝜓)
Theorem	bj-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 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	bj-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 = 𝑦 ∧ 𝜑)))
Theorem	bj-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)
Theorem	bj-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]b𝜑 ↔ 𝜑))
Theorem	bj-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]b𝜑 → 𝜑))
Theorem	bj-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))
Theorem	bj-ssbid2 33239	A special case of bj-ssbequ2 33237. (Contributed by BJ, 22-Dec-2020.)
⊢ ([𝑥/𝑥]b𝜑 → 𝜑)
Theorem	bj-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]b𝜑 → 𝜑)
Theorem	bj-ssbid1 33241	A special case of bj-ssbequ1 33238. (Contributed by BJ, 22-Dec-2020.)
⊢ (𝜑 → [𝑥/𝑥]b𝜑)
Theorem	bj-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)
Theorem	bj-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)
Theorem	bj-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)
Theorem	bj-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))
Theorem	bj-ax6elem2 33246*	Lemma for bj-ax6e 33247. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)
Theorem	bj-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
20.14.4.5  Adding ax-6
Theorem	bj-alequexv 33248*	Version of bj-alequex 33300 with DV(x,y), requiring fewer axioms. (Contributed by BJ, 9-Nov-2021.)
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)
Theorem	bj-spimvwt 33249*	Closed form of spimvw 2045. See also spimt 2350. (Contributed by BJ, 8-Nov-2021.)
⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-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 → (𝜑 → 𝜓))    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	bj-spnfw 33251	Theorem close to a closed form of spnfw 2046. (Contributed by BJ, 12-May-2019.)
⊢ ((∃𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-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.)
⊢ (∃𝑥∃𝑦𝜓 → ∃𝑦𝜓)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
Theorem	bj-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 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
Theorem	bj-modald 33254	A short form of the axiom D of modal logic. (Contributed by BJ, 4-Apr-2021.)
⊢ (∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)
Theorem	bj-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)
Theorem	bj-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 = 𝑦 → 𝜑))
20.14.4.6  Adding ax-7
Theorem	bj-cbvexw 33257*	Change bound variable. This is to cbvexvw 2087 what cbvalw 2085 is to cbvalvw 2086. (Contributed by BJ, 17-Mar-2020.)
⊢ (∃𝑥∃𝑦𝜓 → ∃𝑦𝜓)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (∃𝑦∃𝑥𝜑 → ∃𝑥𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Theorem	bj-ax12w 33258*	The general statement that ax12w 2127 proves. (Contributed by BJ, 20-Mar-2020.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑦 = 𝑧 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑥(𝜑 → 𝜓)))
20.14.4.7  Membership predicate, ax-8 and ax-9
Theorem	bj-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(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
Theorem	bj-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 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑡))
Theorem	bj-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 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑡))
Theorem	bj-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 = 𝑥 ∧ 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦)
20.14.4.8  Adding ax-11
Theorem	bj-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.)
⊢ ((∀𝑥∀𝑦 ¬ 𝜑 → ∀𝑦∀𝑥 ¬ 𝜑) → (∃𝑦∃𝑥𝜑 → ∃𝑥∃𝑦𝜑))
Theorem	bj-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 → ∀𝑥∀𝑦𝜑))
20.14.4.9  Adding ax-12
Theorem	axc11n11 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)
Theorem	axc11n11r 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)
Theorem	bj-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))
Theorem	bj-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 = 𝑦 → 𝜑)))
Theorem	bj-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 = 𝑦 → 𝜑)))
Theorem	bj-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 = 𝑦 → 𝜑)))
Theorem	bj-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 → ∀𝑥∃𝑥𝜑)
Theorem	bj-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.)
⊢ ((𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-19.21bit 33273	Closed form of 19.21bi 2173. (Contributed by BJ, 20-Oct-2019.)
⊢ ((𝜑 → ∀𝑥𝜓) → (𝜑 → 𝜓))
Theorem	bj-19.23bit 33274	Closed form of 19.23bi 2175. (Contributed by BJ, 20-Oct-2019.)
⊢ ((∃𝑥𝜑 → 𝜓) → (𝜑 → 𝜓))
Theorem	bj-nexrt 33275	Closed form of nexr 2176. Contrapositive of 19.8a 2166. (Contributed by BJ, 20-Oct-2019.)
⊢ (¬ ∃𝑥𝜑 → ¬ 𝜑)
Theorem	bj-alrim 33276	Closed form of alrimi 2199. (Contributed by BJ, 2-May-2019.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
Theorem	bj-alrim2 33277	Uncurried (imported) form of bj-alrim 33276. (Contributed by BJ, 2-May-2019.)
⊢ ((Ⅎ𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → (𝜑 → ∀𝑥𝜓))
Theorem	bj-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))
Theorem	bj-nfdt 33279	Closed form of nf5d 2258 and nf5dh 2141. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → ((𝜑 → ∀𝑥𝜑) → (𝜑 → Ⅎ𝑥𝜓)))
Theorem	bj-nexdt 33280	Closed form of nexd 2207. (Contributed by BJ, 20-Oct-2019.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓)))
Theorem	bj-nexdvt 33281*	Closed form of nexdv 1979. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓))
Theorem	bj-alexbiex 33282	Adding a second quantifier is a tranparent operation, (∀∃ case). (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥∃𝑥𝜑 ↔ ∃𝑥𝜑)
Theorem	bj-exexbiex 33283	Adding a second quantifier is a tranparent operation, (∃∃ case). (Contributed by BJ, 20-Oct-2019.)
⊢ (∃𝑥∃𝑥𝜑 ↔ ∃𝑥𝜑)
Theorem	bj-alalbial 33284	Adding a second quantifier is a tranparent operation, (∀∀ case). (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥∀𝑥𝜑 ↔ ∀𝑥𝜑)
Theorem	bj-exalbial 33285	Adding a second quantifier is a tranparent operation, (∃∀ case). (Contributed by BJ, 20-Oct-2019.)
⊢ (∃𝑥∀𝑥𝜑 ↔ ∀𝑥𝜑)
Theorem	bj-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) → (∃𝑥𝜑 ↔ 𝜑))
Theorem	bj-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))
Theorem	bj-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))
Theorem	bj-biexal2 33289	A general FOL biconditional. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(∃𝑥𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))
Theorem	bj-biexal3 33290	A general FOL biconditional. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(∃𝑥𝜑 → 𝜓))
Theorem	bj-bialal 33291	A general FOL biconditional. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∀𝑥𝜓))
Theorem	bj-biexex 33292	A general FOL biconditional. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(𝜑 → ∃𝑥𝜓) ↔ (∃𝑥𝜑 → ∃𝑥𝜓))
Theorem	bj-hbext 33293	Closed form of hbex 2301. (Contributed by BJ, 10-Oct-2019.)
⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑))
Theorem	bj-nfalt 33294	Closed form of nfal 2299. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑)
Theorem	bj-nfext 33295	Closed form of nfex 2300. (Contributed by BJ, 10-Oct-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∃𝑥𝜑)
Theorem	bj-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.)
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
Theorem	bj-modal4e 33297	Dual statement of hba1 2268 (which is modal-4 ). (Contributed by BJ, 21-Dec-2020.)
⊢ (∃𝑥∃𝑥𝜑 → ∃𝑥𝜑)
Theorem	bj-modalb 33298	A short form of the axiom B of modal logic. (Contributed by BJ, 4-Apr-2021.)
⊢ (¬ 𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
20.14.4.10  Adding ax-13
Theorem	bj-axc10 33299	Alternate (shorter) proof of axc10 2349. One can prove a version with DV(x,y) without ax-13 2334, by using ax6ev 2023 instead of ax6e 2347. (Contributed by BJ, 31-Mar-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
Theorem	bj-alequex 33300	A fol lemma. See bj-alequexv 33248 for a version with a disjoint variable condition requiring fewer axioms. Can be used to reduce the proof of spimt 2350 from 133 to 112 bytes. (Contributed by BJ, 6-Oct-2018.)
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)