P. 367 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36601-36700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-ax12i 36601	A weakening of bj-ax12ig 36600 that is sufficient to prove a weak form of the axiom of substitution ax-12 2177. The general statement of which ax12i 1966 is an instance. (Contributed by BJ, 29-Sep-2019.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜒 → ∀𝑥𝜒)    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))
Theorem	bj-nfimt 36602	Closed form of nfim 1896 and curried (exported) form of nfimt 1895. (Contributed by BJ, 20-Oct-2021.)
⊢ (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → Ⅎ𝑥(𝜑 → 𝜓)))
Theorem	bj-cbvalimt 36603	A lemma in closed form used to prove bj-cbval 36613 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) Do not use 19.35 1877 since only the direction of the biconditional used here holds in intuitionistic logic. (Proof modification is discouraged.)
⊢ (∀𝑦∃𝑥𝜒 → (∀𝑦∀𝑥(𝜒 → (𝜑 → 𝜓)) → ((∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) → (∀𝑦(∃𝑥𝜓 → 𝜓) → (∀𝑥𝜑 → ∀𝑦𝜓)))))
Theorem	bj-cbveximt 36604	A lemma in closed form used to prove bj-cbvex 36614 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) Do not use 19.35 1877 since only the direction of the biconditional used here holds in intuitionistic logic. (Proof modification is discouraged.)
⊢ (∀𝑥∃𝑦𝜒 → (∀𝑥∀𝑦(𝜒 → (𝜑 → 𝜓)) → (∀𝑥(𝜑 → ∀𝑦𝜑) → ((∃𝑥∃𝑦𝜓 → ∃𝑦𝜓) → (∃𝑥𝜑 → ∃𝑦𝜓)))))
Theorem	bj-eximALT 36605	Alternate proof of exim 1834 directly from alim 1810 by using df-ex 1780 (using duality of ∀ and ∃. (Contributed by BJ, 9-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
Theorem	bj-aleximiALT 36606	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 36607	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))
21.19.4.3  Adding ax-5
Theorem	bj-ax12wlem 36608*	A lemma used to prove a weak version of the axiom of substitution ax-12 2177. (Temporary comment: The general statement that ax12wlem 2132 proves.) (Contributed by BJ, 20-Mar-2020.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))
Theorem	bj-cbvalim 36609*	A lemma used to prove bj-cbval 36613 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 36610*	A lemma used to prove bj-cbvex 36614 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 36611*	An equality-free general instance of one half of a precise form of bj-cbval 36613. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
⊢ (𝜒 → (𝜑 → 𝜓))    &   ⊢ ∀𝑦∃𝑥𝜒    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
Theorem	bj-cbveximi 36612*	An equality-free general instance of one half of a precise form of bj-cbvex 36614. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
⊢ (𝜒 → (𝜑 → 𝜓))    &   ⊢ ∀𝑥∃𝑦𝜒    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
Theorem	bj-cbval 36613*	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 36614*	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 36615	Syntax for BJ's version of the uniqueness quantifier.
wff ∃**𝑥𝜑
Definition	df-bj-mo 36616*	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))
21.19.4.4  Equality and substitution
Theorem	bj-ssbeq 36617*	Substitution in an equality, disjoint variables case. Uses only ax-1 6 through ax-6 1967. It might be shorter to prove the result about composition of two substitutions and prove bj-ssbeq 36617 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 36618*	A lemma for the definiens of df-sb 2065. An instance of sp 2183 proved without it. Note: it has a common subproof with sbjust 2063. (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 36619*	An instance of ax-11 2157 proved without it. The converse may not be provable without ax-11 2157 (since using alcomimw 2042 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 36620*	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 36621*	Remove a DV condition from bj-ax12v 36620 (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 36622*	Axiom bj-ax12 36621 expressed using substitution. (Contributed by BJ, 26-Dec-2020.) (Proof modification is discouraged.)
⊢ [𝑡 / 𝑥](𝜑 → [𝑡 / 𝑥]𝜑)
Theorem	bj-19.41al 36623	Special case of 19.41 2235 proved from core axioms, ax-10 2141 (modal5), and hba1 2293 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
⊢ (∃𝑥(𝜑 ∧ ∀𝑥𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓))
Theorem	bj-equsexval 36624*	Special case of equsexv 2268 proved from core axioms, ax-10 2141 (modal5), and hba1 2293 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥𝜓))    ⇒   ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥𝜓)
Theorem	bj-subst 36625*	Proof of sbalex 2242 from core axioms, ax-10 2141 (modal5), and bj-ax12 36621. (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	bj-ssbid2 36626	A special case of sbequ2 2249. (Contributed by BJ, 22-Dec-2020.)
⊢ ([𝑥 / 𝑥]𝜑 → 𝜑)
Theorem	bj-ssbid2ALT 36627	Alternate proof of bj-ssbid2 36626, not using sbequ2 2249. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ([𝑥 / 𝑥]𝜑 → 𝜑)
Theorem	bj-ssbid1 36628	A special case of sbequ1 2248. (Contributed by BJ, 22-Dec-2020.)
⊢ (𝜑 → [𝑥 / 𝑥]𝜑)
Theorem	bj-ssbid1ALT 36629	Alternate proof of bj-ssbid1 36628, not using sbequ1 2248. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → [𝑥 / 𝑥]𝜑)
Theorem	bj-ax6elem1 36630*	Lemma for bj-ax6e 36632. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Theorem	bj-ax6elem2 36631*	Lemma for bj-ax6e 36632. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)
Theorem	bj-ax6e 36632	Proof of ax6e 2387 (hence ax6 2388) from Tarski's system, ax-c9 38854, ax-c16 38856. Remark: ax-6 1967 is used only via its principal (unbundled) instance ax6v 1968. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑥 𝑥 = 𝑦
21.19.4.5  Adding ax-6
Theorem	bj-spimvwt 36633*	Closed form of spimvw 1995. See also spimt 2390. (Contributed by BJ, 8-Nov-2021.)
⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-spnfw 36634	Theorem close to a closed form of spnfw 1979. (Contributed by BJ, 12-May-2019.)
⊢ ((∃𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-cbvexiw 36635*	Change bound variable. This is to cbvexvw 2036 what cbvaliw 2005 is to cbvalvw 2035. TODO: move after cbvalivw 2006. (Contributed by BJ, 17-Mar-2020.)
⊢ (∃𝑥∃𝑦𝜓 → ∃𝑦𝜓)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
Theorem	bj-cbvexivw 36636*	Change bound variable. This is to cbvexvw 2036 what cbvalivw 2006 is to cbvalvw 2035. TODO: move after cbvalivw 2006. (Contributed by BJ, 17-Mar-2020.)
⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
Theorem	bj-modald 36637	A short form of the axiom D of modal logic. (Contributed by BJ, 4-Apr-2021.)
⊢ (∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)
Theorem	bj-denot 36638*	A weakening of ax-6 1967 and ax6v 1968. (Contributed by BJ, 4-Apr-2021.) (New usage is discouraged.)
⊢ (𝑥 = 𝑥 → ¬ ∀𝑦 ¬ 𝑦 = 𝑥)
Theorem	bj-eqs 36639*	A lemma for substitutions, proved from Tarski's FOL. The version without DV (𝑥, 𝑦) is true but requires ax-13 2376. The disjoint variable condition DV (𝑥, 𝜑) is necessary for both directions: consider substituting 𝑥 = 𝑧 for 𝜑. (Contributed by BJ, 25-May-2021.)
⊢ (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
21.19.4.6  Adding ax-7
Theorem	bj-cbvexw 36640*	Change bound variable. This is to cbvexvw 2036 what cbvalw 2034 is to cbvalvw 2035. (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 36641*	The general statement that ax12w 2133 proves. (Contributed by BJ, 20-Mar-2020.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑦 = 𝑧 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑥(𝜑 → 𝜓)))
21.19.4.7  Membership predicate, ax-8 and ax-9
Theorem	bj-ax89 36642	A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 2110 and ax-9 2118. Indeed, it is implied over propositional calculus by the conjunction of ax-8 2110 and ax-9 2118, as proved here. In the other direction, one can prove ax-8 2110 (respectively ax-9 2118) from bj-ax89 36642 by using mpan2 691 (respectively mpan 690) and equid 2011. TODO: move to main part. (Contributed by BJ, 3-Oct-2019.)
⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑡))
Theorem	bj-cleljusti 36643*	One direction of cleljust 2117, requiring only ax-1 6-- ax-5 1910 and ax8v1 2112. (Contributed by BJ, 31-Dec-2020.) (Proof modification is discouraged.)
⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦)
21.19.4.8  Adding ax-11
Theorem	bj-alcomexcom 36644	Commutation of two existential quantifiers on a formula is equivalent to commutation of two universal quantifiers over the same variables on the negation of that formula. Can be placed in the ax-4 1809 section, soon after 2nexaln 1830, and used to prove excom 2162. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.)
⊢ ((∀𝑥∀𝑦 ¬ 𝜑 → ∀𝑦∀𝑥 ¬ 𝜑) ↔ (∃𝑦∃𝑥𝜑 → ∃𝑥∃𝑦𝜑))
Theorem	bj-hbalt 36645	Closed form of hbal 2167. When in main part, prove hbal 2167 and hbald 2168 from it. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))
21.19.4.9  Adding ax-12
Theorem	axc11n11 36646	Proof of axc11n 2430 from { ax-1 6-- ax-7 2007, axc11 2434 } . Almost identical to axc11nfromc11 38890. (Contributed by NM, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	axc11n11r 36647	Proof of axc11n 2430 from { ax-1 6-- ax-7 2007, axc9 2386, axc11r 2370 } (note that axc16 2261 is provable from { ax-1 6-- ax-7 2007, axc11r 2370 }). Note that axc11n 2430 proves (over minimal calculus) that axc11 2434 and axc11r 2370 are equivalent. Therefore, axc11n11 36646 and axc11n11r 36647 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2434 appears slightly stronger since axc11n11r 36647 requires axc9 2386 while axc11n11 36646 does not). (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	bj-axc16g16 36648*	Proof of axc16g 2260 from { ax-1 6-- ax-7 2007, axc16 2261 }. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
Theorem	bj-ax12v3 36649*	A weak version of ax-12 2177 which is stronger than ax12v 2178. Note that if one assumes reflexivity of equality ⊢ 𝑥 = 𝑥 (equid 2011), then bj-ax12v3 36649 implies ax-5 1910 over modal logic K (substitute 𝑥 for 𝑦). See also bj-ax12v3ALT 36650. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-ax12v3ALT 36650*	Alternate proof of bj-ax12v3 36649. Uses axc11r 2370 and axc15 2426 instead of ax-12 2177. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-sb 36651*	A weak variant of sbid2 2512 not requiring ax-13 2376 nor ax-10 2141. On top of Tarski's FOL, one implication requires only ax12v 2178, and the other requires only sp 2183. (Contributed by BJ, 25-May-2021.)
⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-modalbe 36652	The predicate-calculus version of the axiom (B) of modal logic. See also modal-b 2319. (Contributed by BJ, 20-Oct-2019.)
⊢ (𝜑 → ∀𝑥∃𝑥𝜑)
Theorem	bj-spst 36653	Closed form of sps 2185. Once in main part, prove sps 2185 and spsd 2187 from it. (Contributed by BJ, 20-Oct-2019.)
⊢ ((𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-19.21bit 36654	Closed form of 19.21bi 2189. (Contributed by BJ, 20-Oct-2019.)
⊢ ((𝜑 → ∀𝑥𝜓) → (𝜑 → 𝜓))
Theorem	bj-19.23bit 36655	Closed form of 19.23bi 2191. (Contributed by BJ, 20-Oct-2019.)
⊢ ((∃𝑥𝜑 → 𝜓) → (𝜑 → 𝜓))
Theorem	bj-nexrt 36656	Closed form of nexr 2192. Contrapositive of 19.8a 2181. (Contributed by BJ, 20-Oct-2019.)
⊢ (¬ ∃𝑥𝜑 → ¬ 𝜑)
Theorem	bj-alrim 36657	Closed form of alrimi 2213. (Contributed by BJ, 2-May-2019.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
Theorem	bj-alrim2 36658	Uncurried (imported) form of bj-alrim 36657. (Contributed by BJ, 2-May-2019.)
⊢ ((Ⅎ𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → (𝜑 → ∀𝑥𝜓))
Theorem	bj-nfdt0 36659	A theorem close to a closed form of nf5d 2284 and nf5dh 2147. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → (∀𝑥𝜑 → Ⅎ𝑥𝜓))
Theorem	bj-nfdt 36660	Closed form of nf5d 2284 and nf5dh 2147. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → ((𝜑 → ∀𝑥𝜑) → (𝜑 → Ⅎ𝑥𝜓)))
Theorem	bj-nexdt 36661	Closed form of nexd 2221. (Contributed by BJ, 20-Oct-2019.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓)))
Theorem	bj-nexdvt 36662*	Closed form of nexdv 1936. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓))
Theorem	bj-alexbiex 36663	Adding a second quantifier over the same variable is a transparent operation, (∀∃ case). (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥∃𝑥𝜑 ↔ ∃𝑥𝜑)
Theorem	bj-exexbiex 36664	Adding a second quantifier over the same variable is a transparent operation, (∃∃ case). (Contributed by BJ, 20-Oct-2019.)
⊢ (∃𝑥∃𝑥𝜑 ↔ ∃𝑥𝜑)
Theorem	bj-alalbial 36665	Adding a second quantifier over the same variable is a transparent operation, (∀∀ case). (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥∀𝑥𝜑 ↔ ∀𝑥𝜑)
Theorem	bj-exalbial 36666	Adding a second quantifier over the same variable is a transparent operation, (∃∀ case). (Contributed by BJ, 20-Oct-2019.)
⊢ (∃𝑥∀𝑥𝜑 ↔ ∀𝑥𝜑)
Theorem	bj-19.9htbi 36667	Strengthening 19.9ht 2320 by replacing its consequent with a biconditional (19.9t 2204 does have a biconditional consequent). This propagates. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 ↔ 𝜑))
Theorem	bj-hbntbi 36668	Strengthening hbnt 2294 by replacing its consequent with a biconditional. See also hbntg 35769 and hbntal 44526. (Contributed by BJ, 20-Oct-2019.) Proved from bj-19.9htbi 36667. (Proof modification is discouraged.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑))
Theorem	bj-biexal1 36669	A general FOL biconditional that generalizes 19.9ht 2320 among others. For this and the following theorems, see also 19.35 1877, 19.21 2207, 19.23 2211. When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))
Theorem	bj-biexal2 36670	When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(∃𝑥𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))
Theorem	bj-biexal3 36671	When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(∃𝑥𝜑 → 𝜓))
Theorem	bj-bialal 36672	When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∀𝑥𝜓))
Theorem	bj-biexex 36673	When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(𝜑 → ∃𝑥𝜓) ↔ (∃𝑥𝜑 → ∃𝑥𝜓))
Theorem	bj-hbext 36674	Closed form of hbex 2325. (Contributed by BJ, 10-Oct-2019.)
⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑))
Theorem	bj-nfalt 36675	Closed form of nfal 2323. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑)
Theorem	bj-nfext 36676	Closed form of nfex 2324. (Contributed by BJ, 10-Oct-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∃𝑥𝜑)
Theorem	bj-eeanvw 36677*	Version of exdistrv 1955 with a disjoint variable condition on 𝑥, 𝑦 not requiring ax-11 2157. (The same can be done with eeeanv 2351 and ee4anv 2352.) (Contributed by BJ, 29-Sep-2019.) (Proof modification is discouraged.)
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
Theorem	bj-modal4 36678	First-order logic form of the modal axiom (4). See hba1 2293. This is the standard proof of the implication in modal logic (B5 ⇒ 4). Its dual statement is bj-modal4e 36679. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
Theorem	bj-modal4e 36679	First-order logic form of the modal axiom (4) using existential quantifiers. Dual statement of bj-modal4 36678 (hba1 2293). (Contributed by BJ, 21-Dec-2020.) (Proof modification is discouraged.)
⊢ (∃𝑥∃𝑥𝜑 → ∃𝑥𝜑)
Theorem	bj-modalb 36680	A short form of the axiom B of modal logic using only primitive symbols (→ , ¬ , ∀). (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.)
⊢ (¬ 𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
Theorem	bj-wnf1 36681	When 𝜑 is substituted for 𝜓, this is the first half of nonfreness (. → ∀) of the weak form of nonfreeness (∃ → ∀). (Contributed by BJ, 9-Dec-2023.)
⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))
Theorem	bj-wnf2 36682	When 𝜑 is substituted for 𝜓, this is the first half of nonfreness (. → ∀) of the weak form of nonfreeness (∃ → ∀). (Contributed by BJ, 9-Dec-2023.)
⊢ (∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))
Theorem	bj-wnfanf 36683	When 𝜑 is substituted for 𝜓, this statement expresses that weak nonfreeness implies the universal form of nonfreeness. (Contributed by BJ, 9-Dec-2023.)
⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → ∀𝑥𝜓))
Theorem	bj-wnfenf 36684	When 𝜑 is substituted for 𝜓, this statement expresses that weak nonfreeness implies the existential form of nonfreeness. (Contributed by BJ, 9-Dec-2023.)
⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → 𝜓))
Theorem	bj-substax12 36685	Equivalent form of the axiom of substitution bj-ax12 36621. Although both sides need a DV condition on 𝑥, 𝑡 (or as in bj-ax12v3 36649 on 𝑡, 𝜑) to hold, their equivalence holds without DV conditions. The forward implication is proved in modal (K4) while the reverse implication is proved in modal (T5). The LHS has the advantage of not involving nested quantifiers on the same variable. Its metaweakening is proved from the core axiom schemes in bj-substw 36686. Note that in the LHS, the reverse implication holds by equs4 2420 (or equs4v 1999 if a DV condition is added on 𝑥, 𝑡 as in bj-ax12 36621), and the forward implication is sbalex 2242. The LHS can be read as saying that if there exists a variable equal to a given term witnessing a given formula, then all variables equal to that term also witness that formula. The equivalent form of the LHS using only primitive symbols is (∀𝑥(𝑥 = 𝑡 → 𝜑) ∨ ∀𝑥(𝑥 = 𝑡 → ¬ 𝜑)), which expresses that a given formula is true at all variables equal to a given term, or false at all these variables. An equivalent form of the LHS using only the existential quantifier is ¬ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ∧ ∃𝑥(𝑥 = 𝑡 ∧ ¬ 𝜑)), which expresses that there can be no two variables both equal to a given term, one witnessing a formula and the other witnessing its negation. These equivalences do not hold in intuitionistic logic. The LHS should be the preferred form, and has the advantage of having no negation nor nested quantifiers. (Contributed by BJ, 21-May-2024.) (Proof modification is discouraged.)
⊢ ((∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) ↔ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))))
Theorem	bj-substw 36686*	Weak form of the LHS of bj-substax12 36685 proved from the core axiom schemes. Compare ax12w 2133. (Contributed by BJ, 26-May-2024.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑡 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑))
21.19.4.10  Nonfreeness
Syntax	wnnf 36687	Syntax for the nonfreeness quantifier.
wff Ⅎ'𝑥𝜑
Definition	df-bj-nnf 36688	Definition of the nonfreeness quantifier. The formula Ⅎ'𝑥𝜑 has the intended meaning that the variable 𝑥 is semantically nonfree in the formula 𝜑. The motivation for this quantifier is to have a condition expressible in the logic which is as close as possible to the non-occurrence condition DV (𝑥, 𝜑) (in Metamath files, "$d x ph $."), which belongs to the metalogic. The standard syntactic nonfreeness condition, also expressed in the metalogic, is intermediate between these two notions: semantic nonfreeness implies syntactic nonfreeness, which implies non-occurrence. Both implications are strict; for the first, note that ⊢ Ⅎ'𝑥𝑥 = 𝑥, that is, 𝑥 is semantically (but not syntactically) nonfree in the formula 𝑥 = 𝑥; for the second, note that 𝑥 is syntactically nonfree in the formula ∀𝑥𝑥 = 𝑥 although it occurs in it. We now prove two metatheorems which make precise the above fact that, as far as proving power is concerned, the nonfreeness condition Ⅎ'𝑥𝜑 is very close to the non-occurrence condition DV (𝑥, 𝜑). Let S be a Metamath system with the FOL-syntax of (i)set.mm, containing intuitionistic positive propositional calculus and ax-5 1910 and ax5e 1912. Theorem 1. If the scheme (Ⅎ'𝑥𝜑 & PHI1 & ... & PHIn ⇒ PHI0, DV) is provable in S, then so is the scheme (PHI1 & ... & PHIn ⇒ PHI0, DV ∪ {{𝑥, 𝜑}}). Proof: By bj-nnfv 36718, we can prove (Ⅎ'𝑥𝜑, {{𝑥, 𝜑}}), from which the theorem follows. QED Theorem 2. Suppose that S also contains (the FOL version of) modal logic KB and commutation of quantifiers alcom 2159 and excom 2162 (possibly weakened by a DV condition on the quantifying variables), and that S can be axiomatized such that the only axioms with a DV condition involving a formula variable are among ax-5 1910, ax5e 1912, ax5ea 1913. If the scheme (PHI1 & ... & PHIn ⇒ PHI0, DV) is provable in S, then so is the scheme (Ⅎ'𝑥𝜑 & PHI1 & ... & PHIn ⇒ PHI0, DV ∖ {{𝑥, 𝜑}}). More precisely, if S contains modal 45 and if the variables quantified over in PHI0, ..., PHIn are among 𝑥1, ..., 𝑥m, then the scheme (PHI1 & ... & PHIn ⇒ (antecedent → PHI0), DV ∖ {{𝑥, 𝜑}}) is provable in S, where the antecedent is a finite conjunction of formulas of the form ∀𝑥i1 ...∀𝑥ip Ⅎ'𝑥𝜑 where the 𝑥ij's are among the 𝑥i's. Lemma: If 𝑥 ∉ OC(PHI), then S proves the scheme (Ⅎ'𝑥𝜑 ⇒ Ⅎ'𝑥 PHI, {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PHI) ∖ {𝜑}}). More precisely, if the variables quantified over in PHI are among 𝑥1, ..., 𝑥m, then ((antecedent → Ⅎ'𝑥 PHI), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PHI) ∖ {𝜑}}) is provable in S, with the same form of antecedent as above. Proof: By induction on the height of PHI. We first note that by bj-nnfbi 36689 we can assume that PHI contains only primitive (as opposed to defined) symbols. For the base case, atomic formulas are either 𝜑, in which case the scheme to prove is an instance of id 22, or have variables all in OC(PHI) ∖ {𝜑}, so (Ⅎ'𝑥 PHI, {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PHI) ∖ {𝜑}}) by bj-nnfv 36718, hence ((Ⅎ'𝑥𝜑 → Ⅎ'𝑥 PHI), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PHI) ∖ {𝜑}}) by a1i 11. For the induction step, PHI is either an implication, a negation, a conjunction, a disjunction, a biconditional, a universal or an existential quantification of formulas where 𝑥 does not occur. We use respectively bj-nnfim 36710, bj-nnfnt 36704, bj-nnfan 36712, bj-nnfor 36714, bj-nnfbit 36716, bj-nnfalt 36730, bj-nnfext 36731. For instance, in the implication case, if we have by induction hypothesis ((∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 → Ⅎ'𝑥 PHI), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PHI) ∖ {𝜑}}) and ((∀𝑦1 ...∀𝑦n Ⅎ'𝑥𝜑 → Ⅎ'𝑥 PSI), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PSI) ∖ {𝜑}}), then bj-nnfim 36710 yields (((∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 ∧ ∀𝑦1 ...∀𝑦n Ⅎ'𝑥𝜑) → Ⅎ'𝑥 (PHI → PSI)), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PHI → PSI) ∖ {𝜑}}) and similarly for antecedents which are conjunctions as in the statement of the lemma. In the universal quantification case, say quantification over 𝑦, if we have by induction hypothesis ((∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 → Ⅎ'𝑥 PHI), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PHI) ∖ {𝜑}}), then bj-nnfalt 36730 yields ((∀𝑦∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 → Ⅎ'𝑥∀𝑦 PHI), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(∀𝑦 PHI) ∖ {𝜑}}) and similarly for antecedents which are conjunctions as in the statement of the lemma. Note bj-nnfalt 36730 and bj-nnfext 36731 are proved from positive propositional calculus with alcom 2159 and excom 2162 (possibly weakened by a DV condition on the quantifying variables), and modalB (via bj-19.12 36725). QED Proof of the theorem: Consider a proof of that scheme directly from the axioms. Consider a step where a DV condition involving 𝜑 is used. By hypothesis, that step is an instance of ax-5 1910 or ax5e 1912 or ax5ea 1913. It has the form (PSI → ∀𝑥 PSI) where PSI has the form of the lemma and the DV conditions of the proof contain {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PSI) }. Therefore, one has ((∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 → Ⅎ'𝑥 PSI), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PSI) ∖ {𝜑}}) for appropriate 𝑥i's, and by bj-nnfa 36692 we obtain ((∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 → (PSI → ∀𝑥 PSI)), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PSI) ∖ {𝜑}}) and similarly for antecedents which are conjunctions as in the statement of the theorem. Similarly if the step is using ax5e 1912 or ax5ea 1913, we would use bj-nnfe 36695 or bj-nnfea 36698 respectively. Therefore, taking as antecedent of the theorem to prove the conjunction of all the antecedents at each of these steps, we obtain a proof by "carrying the context over", which is possible, as in the deduction theorem when the step uses ax-mp 5, and when the step uses ax-gen 1795, by bj-nnf-alrim 36719 and bj-nnfa1 36723 (which requires modal 45). The condition DV (𝑥, 𝜑) is not required by the resulting proof. Finally, there may be in the global antecedent thus constructed some dummy variables, which can be removed by spvw 1980. QED Compared with df-nf 1784, the present definition is stricter on positive propositional calculus (bj-nnfnfTEMP 36702) and equivalent on core FOL plus sp 2183 (bj-nfnnfTEMP 36722). While being stricter, it still holds for non-occurring variables (bj-nnfv 36718), which is the basic requirement for this quantifier. In particular, it translates more closely the associated variable disjointness condition. Since the nonfreeness quantifier is a means to translate a variable disjointness condition from the metalogic to the logic, it seems preferable. Also, since nonfreeness is mainly used as a hypothesis, this definition would allow more theorems, notably the 19.xx theorems, to be proved from the core axioms, without needing a 19.xxv variant. One can devise infinitely many definitions increasingly close to the non-occurring condition, like ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)) ∧ ∀𝑥((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)) ∧ ∀𝑥∀𝑥... and each stronger definition would permit more theorems to be proved from the core axioms. A reasonable rule seems to be to stop before nested quantifiers appear (since they typically require ax-10 2141 to work with), and also not to have redundant conjuncts when full metacomplete FOL= is developed. (Contributed by BJ, 28-Jul-2023.)
⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
Theorem	bj-nnfbi 36689	If two formulas are equivalent for all 𝑥, then nonfreeness of 𝑥 in one of them is equivalent to nonfreeness in the other. Compare nfbiit 1851. From this and bj-nnfim 36710 and bj-nnfnt 36704, one can prove analogous nonfreeness conservation results for other propositional operators. The antecedent is in the "strong necessity" modality of modal logic (see also bj-nnftht 36705) in order not to require sp 2183 (modal T). (Contributed by BJ, 27-Aug-2023.)
⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓))
Theorem	bj-nnfbd 36690*	If two formulas are equivalent for all 𝑥, then nonfreeness of 𝑥 in one of them is equivalent to nonfreeness in the other, deduction form. See bj-nnfbi 36689. (Contributed by BJ, 27-Aug-2023.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒))
Theorem	bj-nnfbii 36691	If two formulas are equivalent for all 𝑥, then nonfreeness of 𝑥 in one of them is equivalent to nonfreeness in the other, inference form. See bj-nnfbi 36689. (Contributed by BJ, 18-Nov-2023.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓)
Theorem	bj-nnfa 36692	Nonfreeness implies the equivalent of ax-5 1910. See nf5r 2194. (Contributed by BJ, 28-Jul-2023.)
⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
Theorem	bj-nnfad 36693	Nonfreeness implies the equivalent of ax-5 1910, deduction form. See nf5rd 2196. (Contributed by BJ, 2-Dec-2023.)
⊢ (𝜑 → Ⅎ'𝑥𝜓)    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
Theorem	bj-nnfai 36694	Nonfreeness implies the equivalent of ax-5 1910, inference form. See nf5ri 2195. (Contributed by BJ, 22-Sep-2024.)
⊢ Ⅎ'𝑥𝜑    ⇒   ⊢ (𝜑 → ∀𝑥𝜑)
Theorem	bj-nnfe 36695	Nonfreeness implies the equivalent of ax5e 1912. (Contributed by BJ, 28-Jul-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑))
Theorem	bj-nnfed 36696	Nonfreeness implies the equivalent of ax5e 1912, deduction form. (Contributed by BJ, 2-Dec-2023.)
⊢ (𝜑 → Ⅎ'𝑥𝜓)    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → 𝜓))
Theorem	bj-nnfei 36697	Nonfreeness implies the equivalent of ax5e 1912, inference form. (Contributed by BJ, 22-Sep-2024.)
⊢ Ⅎ'𝑥𝜑    ⇒   ⊢ (∃𝑥𝜑 → 𝜑)
Theorem	bj-nnfea 36698	Nonfreeness implies the equivalent of ax5ea 1913. (Contributed by BJ, 28-Jul-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
Theorem	bj-nnfead 36699	Nonfreeness implies the equivalent of ax5ea 1913, deduction form. (Contributed by BJ, 2-Dec-2023.)
⊢ (𝜑 → Ⅎ'𝑥𝜓)    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))
Theorem	bj-nnfeai 36700	Nonfreeness implies the equivalent of ax5ea 1913, inference form. (Contributed by BJ, 22-Sep-2024.)
⊢ Ⅎ'𝑥𝜑    ⇒   ⊢ (∃𝑥𝜑 → ∀𝑥𝜑)