HomeHome Metamath Proof Explorer
Theorem List (p. 356 of 479)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-30171)
  Hilbert Space Explorer  Hilbert Space Explorer
(30172-31694)
  Users' Mathboxes  Users' Mathboxes
(31695-47852)
 

Theorem List for Metamath Proof Explorer - 35501-35600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembj-sylge 35501 Dual statement of sylg 1826 (the final "e" in the label stands for "existential (version of sylg 1826)". Variant of exlimih 2286. (Contributed by BJ, 25-Dec-2023.)
(∃𝑥𝜑𝜓)    &   (𝜒𝜑)       (∃𝑥𝜒𝜓)
 
Theorembj-exlimd 35502 A slightly more general exlimd 2212. A common usage will have 𝜑 substituted for 𝜓 and 𝜃 substituted for 𝜏, giving a form closer to exlimd 2212. (Contributed by BJ, 25-Dec-2023.)
(𝜑 → ∀𝑥𝜓)    &   (𝜑 → (∃𝑥𝜃𝜏))    &   (𝜓 → (𝜒𝜃))       (𝜑 → (∃𝑥𝜒𝜏))
 
Theorembj-nfimexal 35503 A weak from of nonfreeness in either an antecedent or a consequent implies that a universally quantified implication is equivalent to the associated implication where the antecedent is existentially quantified and the consequent is universally quantified. The forward implication always holds (this is 19.38 1842) and the converse implication is the join of instances of bj-alrimg 35496 and bj-exlimg 35500 (see 19.38a 1843 and 19.38b 1844). TODO: prove a version where the antecedents use the nonfreeness quantifier. (Contributed by BJ, 9-Dec-2023.)
(((∃𝑥𝜑 → ∀𝑥𝜑) ∨ (∃𝑥𝜓 → ∀𝑥𝜓)) → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))
 
Theorembj-alexim 35504 Closed form of aleximi 1835. Note: this proof is shorter, so aleximi 1835 could be deduced from it (exim 1837 would have to be proved first, see bj-eximALT 35518 but its proof is shorter (currently almost a subproof of aleximi 1835)). (Contributed by BJ, 8-Nov-2021.)
(∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)))
 
Theorembj-nexdh 35505 Closed form of nexdh 1869 (actually, its general instance). (Contributed by BJ, 6-May-2019.)
(∀𝑥(𝜑 → ¬ 𝜓) → ((𝜒 → ∀𝑥𝜑) → (𝜒 → ¬ ∃𝑥𝜓)))
 
Theorembj-nexdh2 35506 Uncurried (imported) form of bj-nexdh 35505. (Contributed by BJ, 6-May-2019.)
((∀𝑥(𝜑 → ¬ 𝜓) ∧ (𝜒 → ∀𝑥𝜑)) → (𝜒 → ¬ ∃𝑥𝜓))
 
Theorembj-hbxfrbi 35507 Closed form of hbxfrbi 1828. Note: it is less important than nfbiit 1854. The antecedent is in the "strong necessity" modality of modal logic (see also bj-nnftht 35619) in order not to require sp 2177 (modal T). See bj-hbyfrbi 35508 for its version with existential quantifiers. (Contributed by BJ, 6-May-2019.)
(((𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)))
 
Theorembj-hbyfrbi 35508 Version of bj-hbxfrbi 35507 with existential quantifiers. (Contributed by BJ, 23-Aug-2023.)
(((𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) → ((∃𝑥𝜑𝜑) ↔ (∃𝑥𝜓𝜓)))
 
Theorembj-exalim 35509 Distribute quantifiers over a nested implication.

This and the following theorems are the general instances of already proved theorems. They could be moved to the main part, before ax-5 1914. I propose to move to the main part: bj-exalim 35509, bj-exalimi 35510, bj-exalims 35511, bj-exalimsi 35512, bj-ax12i 35514, bj-ax12wlem 35521, bj-ax12w 35554. A new label is needed for bj-ax12i 35514 and label suggestions are welcome for the others. I also propose to change ¬ ∀𝑥¬ to 𝑥 in speimfw 1968 and spimfw 1970 (other spim* theorems use 𝑥 and very few theorems in set.mm use ¬ ∀𝑥¬). (Contributed by BJ, 8-Nov-2021.)

(∀𝑥(𝜑 → (𝜓𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒)))
 
Theorembj-exalimi 35510 An inference for distributing quantifiers over a nested implication. The canonical derivation from its closed form bj-exalim 35509 (using mpg 1800) has fewer essential steps, but more steps in total (yielding a longer compressed proof). (Almost) the general statement that speimfw 1968 proves. (Contributed by BJ, 29-Sep-2019.)
(𝜑 → (𝜓𝜒))       (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))
 
Theorembj-exalims 35511 Distributing quantifiers over a nested implication. (Almost) the general statement that spimfw 1970 proves. (Contributed by BJ, 29-Sep-2019.)
(∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))       (∀𝑥(𝜑 → (𝜓𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓𝜒)))
 
Theorembj-exalimsi 35512 An inference for distributing quantifiers over a nested implication. (Almost) the general statement that spimfw 1970 proves. (Contributed by BJ, 29-Sep-2019.)
(𝜑 → (𝜓𝜒))    &   (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))       (∃𝑥𝜑 → (∀𝑥𝜓𝜒))
 
Theorembj-ax12ig 35513 A lemma used to prove a weak form of the axiom of substitution. A generalization of bj-ax12i 35514. (Contributed by BJ, 19-Dec-2020.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒 → ∀𝑥𝜒))       (𝜑 → (𝜓 → ∀𝑥(𝜑𝜓)))
 
Theorembj-ax12i 35514 A weakening of bj-ax12ig 35513 that is sufficient to prove a weak form of the axiom of substitution ax-12 2172. The general statement of which ax12i 1971 is an instance. (Contributed by BJ, 29-Sep-2019.)
(𝜑 → (𝜓𝜒))    &   (𝜒 → ∀𝑥𝜒)       (𝜑 → (𝜓 → ∀𝑥(𝜑𝜓)))
 
Theorembj-nfimt 35515 Closed form of nfim 1900 and curried (exported) form of nfimt 1899. (Contributed by BJ, 20-Oct-2021.)
(Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → Ⅎ𝑥(𝜑𝜓)))
 
Theorembj-cbvalimt 35516 A lemma in closed form used to prove bj-cbval 35526 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) Do not use 19.35 1881 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 35517 A lemma in closed form used to prove bj-cbvex 35527 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) Do not use 19.35 1881 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-eximALT 35518 Alternate proof of exim 1837 directly from alim 1813 by using df-ex 1783 (using duality of and . (Contributed by BJ, 9-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
 
Theorembj-aleximiALT 35519 Alternate proof of aleximi 1835 from exim 1837, 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 → (𝜓𝜒))       (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
 
Theorembj-eximcom 35520 A commuted form of exim 1837 which is sometimes posited as an axiom in instuitionistic modal logic. (Contributed by BJ, 9-Dec-2023.)
(∃𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
 
21.17.4.3  Adding ax-5
 
Theorembj-ax12wlem 35521* A lemma used to prove a weak version of the axiom of substitution ax-12 2172. (Temporary comment: The general statement that ax12wlem 2129 proves.) (Contributed by BJ, 20-Mar-2020.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥(𝜑𝜓)))
 
Theorembj-cbvalim 35522* A lemma used to prove bj-cbval 35526 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
(∀𝑦𝑥𝜒 → (∀𝑦𝑥(𝜒 → (𝜑𝜓)) → (∀𝑥𝜑 → ∀𝑦𝜓)))
 
Theorembj-cbvexim 35523* A lemma used to prove bj-cbvex 35527 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
(∀𝑥𝑦𝜒 → (∀𝑥𝑦(𝜒 → (𝜑𝜓)) → (∃𝑥𝜑 → ∃𝑦𝜓)))
 
Theorembj-cbvalimi 35524* An equality-free general instance of one half of a precise form of bj-cbval 35526. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
(𝜒 → (𝜑𝜓))    &   𝑦𝑥𝜒       (∀𝑥𝜑 → ∀𝑦𝜓)
 
Theorembj-cbveximi 35525* An equality-free general instance of one half of a precise form of bj-cbvex 35527. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
(𝜒 → (𝜑𝜓))    &   𝑥𝑦𝜒       (∃𝑥𝜑 → ∃𝑦𝜓)
 
Theorembj-cbval 35526* 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 35527* 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 35528 Syntax for BJ's version of the uniqueness quantifier.
wff ∃**𝑥𝜑
 
Definitiondf-bj-mo 35529* 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))
 
21.17.4.4  Equality and substitution
 
Theorembj-ssbeq 35530* Substitution in an equality, disjoint variables case. Uses only ax-1 6 through ax-6 1972. It might be shorter to prove the result about composition of two substitutions and prove bj-ssbeq 35530 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 = 𝑧𝑦 = 𝑧)
 
Theorembj-ssblem1 35531* A lemma for the definiens of df-sb 2069. An instance of sp 2177 proved without it. Note: it has a common subproof with sbjust 2067. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
(∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theorembj-ssblem2 35532* An instance of ax-11 2155 proved without it. The converse may not be provable without ax-11 2155 (since using alcomiw 2047 would require a DV on 𝜑, 𝑥, which defeats the purpose). (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
(∀𝑥𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) → ∀𝑦𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)))
 
Theorembj-ax12v 35533* A weaker form of ax-12 2172 and ax12v 2173, 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 = 𝑡𝜑)))
 
Theorembj-ax12 35534* Remove a DV condition from bj-ax12v 35533 (using core axioms only). (Contributed by BJ, 26-Dec-2020.) (Proof modification is discouraged.)
𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))
 
Theorembj-ax12ssb 35535* Axiom bj-ax12 35534 expressed using substitution. (Contributed by BJ, 26-Dec-2020.) (Proof modification is discouraged.)
[𝑡 / 𝑥](𝜑 → [𝑡 / 𝑥]𝜑)
 
Theorembj-19.41al 35536 Special case of 19.41 2229 proved from core axioms, ax-10 2138 (modal5), and hba1 2290 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
(∃𝑥(𝜑 ∧ ∀𝑥𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓))
 
Theorembj-equsexval 35537* Special case of equsexv 2260 proved from core axioms, ax-10 2138 (modal5), and hba1 2290 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥𝜓)
 
Theorembj-subst 35538* Proof of sbalex 2236 from core axioms, ax-10 2138 (modal5), and bj-ax12 35534. (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
(∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
 
Theorembj-ssbid2 35539 A special case of sbequ2 2242. (Contributed by BJ, 22-Dec-2020.)
([𝑥 / 𝑥]𝜑𝜑)
 
Theorembj-ssbid2ALT 35540 Alternate proof of bj-ssbid2 35539, not using sbequ2 2242. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑥 / 𝑥]𝜑𝜑)
 
Theorembj-ssbid1 35541 A special case of sbequ1 2241. (Contributed by BJ, 22-Dec-2020.)
(𝜑 → [𝑥 / 𝑥]𝜑)
 
Theorembj-ssbid1ALT 35542 Alternate proof of bj-ssbid1 35541, not using sbequ1 2241. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → [𝑥 / 𝑥]𝜑)
 
Theorembj-ax6elem1 35543* Lemma for bj-ax6e 35545. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
 
Theorembj-ax6elem2 35544* Lemma for bj-ax6e 35545. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
(∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)
 
Theorembj-ax6e 35545 Proof of ax6e 2383 (hence ax6 2384) from Tarski's system, ax-c9 37760, ax-c16 37762. Remark: ax-6 1972 is used only via its principal (unbundled) instance ax6v 1973. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥 𝑥 = 𝑦
 
21.17.4.5  Adding ax-6
 
Theorembj-spimvwt 35546* Closed form of spimvw 2000. See also spimt 2386. (Contributed by BJ, 8-Nov-2021.)
(∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∀𝑥𝜑𝜓))
 
Theorembj-spnfw 35547 Theorem close to a closed form of spnfw 1984. (Contributed by BJ, 12-May-2019.)
((∃𝑥𝜑𝜓) → (∀𝑥𝜑𝜓))
 
Theorembj-cbvexiw 35548* Change bound variable. This is to cbvexvw 2041 what cbvaliw 2010 is to cbvalvw 2040. TODO: move after cbvalivw 2011. (Contributed by BJ, 17-Mar-2020.)
(∃𝑥𝑦𝜓 → ∃𝑦𝜓)    &   (𝜑 → ∀𝑦𝜑)    &   (𝑦 = 𝑥 → (𝜑𝜓))       (∃𝑥𝜑 → ∃𝑦𝜓)
 
Theorembj-cbvexivw 35549* Change bound variable. This is to cbvexvw 2041 what cbvalivw 2011 is to cbvalvw 2040. TODO: move after cbvalivw 2011. (Contributed by BJ, 17-Mar-2020.)
(𝑦 = 𝑥 → (𝜑𝜓))       (∃𝑥𝜑 → ∃𝑦𝜓)
 
Theorembj-modald 35550 A short form of the axiom D of modal logic. (Contributed by BJ, 4-Apr-2021.)
(∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)
 
Theorembj-denot 35551* A weakening of ax-6 1972 and ax6v 1973. (Contributed by BJ, 4-Apr-2021.) (New usage is discouraged.)
(𝑥 = 𝑥 → ¬ ∀𝑦 ¬ 𝑦 = 𝑥)
 
Theorembj-eqs 35552* A lemma for substitutions, proved from Tarski's FOL. The version without DV (𝑥, 𝑦) is true but requires ax-13 2372. 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 = 𝑦𝜑))
 
21.17.4.6  Adding ax-7
 
Theorembj-cbvexw 35553* Change bound variable. This is to cbvexvw 2041 what cbvalw 2039 is to cbvalvw 2040. (Contributed by BJ, 17-Mar-2020.)
(∃𝑥𝑦𝜓 → ∃𝑦𝜓)    &   (𝜑 → ∀𝑦𝜑)    &   (∃𝑦𝑥𝜑 → ∃𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)
 
Theorembj-ax12w 35554* The general statement that ax12w 2130 proves. (Contributed by BJ, 20-Mar-2020.)
(𝜑 → (𝜓𝜒))    &   (𝑦 = 𝑧 → (𝜓𝜃))       (𝜑 → (∀𝑦𝜓 → ∀𝑥(𝜑𝜓)))
 
21.17.4.7  Membership predicate, ax-8 and ax-9
 
Theorembj-ax89 35555 A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 2109 and ax-9 2117. Indeed, it is implied over propositional calculus by the conjunction of ax-8 2109 and ax-9 2117, as proved here. In the other direction, one can prove ax-8 2109 (respectively ax-9 2117) from bj-ax89 35555 by using mpan2 690 (respectively mpan 689) and equid 2016. TODO: move to main part. (Contributed by BJ, 3-Oct-2019.)
((𝑥 = 𝑦𝑧 = 𝑡) → (𝑥𝑧𝑦𝑡))
 
Theorembj-elequ12 35556 An identity law for the non-logical predicate, which combines elequ1 2114 and elequ2 2122. For the analogous theorems for class terms, see eleq1 2822, eleq2 2823 and eleq12 2824. TODO: move to main part. (Contributed by BJ, 29-Sep-2019.)
((𝑥 = 𝑦𝑧 = 𝑡) → (𝑥𝑧𝑦𝑡))
 
Theorembj-cleljusti 35557* One direction of cleljust 2116, requiring only ax-1 6-- ax-5 1914 and ax8v1 2111. (Contributed by BJ, 31-Dec-2020.) (Proof modification is discouraged.)
(∃𝑧(𝑧 = 𝑥𝑧𝑦) → 𝑥𝑦)
 
21.17.4.8  Adding ax-11
 
Theorembj-alcomexcom 35558 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 1812 section, soon after 2nexaln 1833, and used to prove excom 2163. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.)
((∀𝑥𝑦 ¬ 𝜑 → ∀𝑦𝑥 ¬ 𝜑) ↔ (∃𝑦𝑥𝜑 → ∃𝑥𝑦𝜑))
 
Theorembj-hbalt 35559 Closed form of hbal 2168. When in main part, prove hbal 2168 and hbald 2169 from it. (Contributed by BJ, 2-May-2019.)
(∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥𝑦𝜑))
 
21.17.4.9  Adding ax-12
 
Theoremaxc11n11 35560 Proof of axc11n 2426 from { ax-1 6-- ax-7 2012, axc11 2430 } . Almost identical to axc11nfromc11 37796. (Contributed by NM, 6-Jul-2021.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
 
Theoremaxc11n11r 35561 Proof of axc11n 2426 from { ax-1 6-- ax-7 2012, axc9 2382, axc11r 2366 } (note that axc16 2253 is provable from { ax-1 6-- ax-7 2012, axc11r 2366 }).

Note that axc11n 2426 proves (over minimal calculus) that axc11 2430 and axc11r 2366 are equivalent. Therefore, axc11n11 35560 and axc11n11r 35561 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2430 appears slightly stronger since axc11n11r 35561 requires axc9 2382 while axc11n11 35560 does not).

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

(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
 
Theorembj-axc16g16 35562* Proof of axc16g 2252 from { ax-1 6-- ax-7 2012, axc16 2253 }. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
 
Theorembj-ax12v3 35563* A weak version of ax-12 2172 which is stronger than ax12v 2173. Note that if one assumes reflexivity of equality 𝑥 = 𝑥 (equid 2016), then bj-ax12v3 35563 implies ax-5 1914 over modal logic K (substitute 𝑥 for 𝑦). See also bj-ax12v3ALT 35564. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theorembj-ax12v3ALT 35564* Alternate proof of bj-ax12v3 35563. Uses axc11r 2366 and axc15 2422 instead of ax-12 2172. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theorembj-sb 35565* A weak variant of sbid2 2508 not requiring ax-13 2372 nor ax-10 2138. On top of Tarski's FOL, one implication requires only ax12v 2173, and the other requires only sp 2177. (Contributed by BJ, 25-May-2021.)
(𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theorembj-modalbe 35566 The predicate-calculus version of the axiom (B) of modal logic. See also modal-b 2313. (Contributed by BJ, 20-Oct-2019.)
(𝜑 → ∀𝑥𝑥𝜑)
 
Theorembj-spst 35567 Closed form of sps 2179. Once in main part, prove sps 2179 and spsd 2181 from it. (Contributed by BJ, 20-Oct-2019.)
((𝜑𝜓) → (∀𝑥𝜑𝜓))
 
Theorembj-19.21bit 35568 Closed form of 19.21bi 2183. (Contributed by BJ, 20-Oct-2019.)
((𝜑 → ∀𝑥𝜓) → (𝜑𝜓))
 
Theorembj-19.23bit 35569 Closed form of 19.23bi 2185. (Contributed by BJ, 20-Oct-2019.)
((∃𝑥𝜑𝜓) → (𝜑𝜓))
 
Theorembj-nexrt 35570 Closed form of nexr 2186. Contrapositive of 19.8a 2175. (Contributed by BJ, 20-Oct-2019.)
(¬ ∃𝑥𝜑 → ¬ 𝜑)
 
Theorembj-alrim 35571 Closed form of alrimi 2207. (Contributed by BJ, 2-May-2019.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))
 
Theorembj-alrim2 35572 Uncurried (imported) form of bj-alrim 35571. (Contributed by BJ, 2-May-2019.)
((Ⅎ𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → (𝜑 → ∀𝑥𝜓))
 
Theorembj-nfdt0 35573 A theorem close to a closed form of nf5d 2281 and nf5dh 2144. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → (∀𝑥𝜑 → Ⅎ𝑥𝜓))
 
Theorembj-nfdt 35574 Closed form of nf5d 2281 and nf5dh 2144. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → ((𝜑 → ∀𝑥𝜑) → (𝜑 → Ⅎ𝑥𝜓)))
 
Theorembj-nexdt 35575 Closed form of nexd 2215. (Contributed by BJ, 20-Oct-2019.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓)))
 
Theorembj-nexdvt 35576* Closed form of nexdv 1940. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓))
 
Theorembj-alexbiex 35577 Adding a second quantifier over the same variable is a transparent operation, (∀∃ case). (Contributed by BJ, 20-Oct-2019.)
(∀𝑥𝑥𝜑 ↔ ∃𝑥𝜑)
 
Theorembj-exexbiex 35578 Adding a second quantifier over the same variable is a transparent operation, (∃∃ case). (Contributed by BJ, 20-Oct-2019.)
(∃𝑥𝑥𝜑 ↔ ∃𝑥𝜑)
 
Theorembj-alalbial 35579 Adding a second quantifier over the same variable is a transparent operation, (∀∀ case). (Contributed by BJ, 20-Oct-2019.)
(∀𝑥𝑥𝜑 ↔ ∀𝑥𝜑)
 
Theorembj-exalbial 35580 Adding a second quantifier over the same variable is a transparent operation, (∃∀ case). (Contributed by BJ, 20-Oct-2019.)
(∃𝑥𝑥𝜑 ↔ ∀𝑥𝜑)
 
Theorembj-19.9htbi 35581 Strengthening 19.9ht 2314 by replacing its consequent with a biconditional (19.9t 2198 does have a biconditional consequent). This propagates. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))
 
Theorembj-hbntbi 35582 Strengthening hbnt 2291 by replacing its consequent with a biconditional. See also hbntg 34777 and hbntal 43314. (Contributed by BJ, 20-Oct-2019.) Proved from bj-19.9htbi 35581. (Proof modification is discouraged.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑))
 
Theorembj-biexal1 35583 A general FOL biconditional that generalizes 19.9ht 2314 among others. For this and the following theorems, see also 19.35 1881, 19.21 2201, 19.23 2205. When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))
 
Theorembj-biexal2 35584 When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(∃𝑥𝜑𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))
 
Theorembj-biexal3 35585 When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(∃𝑥𝜑𝜓))
 
Theorembj-bialal 35586 When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(∀𝑥𝜑𝜓) ↔ (∀𝑥𝜑 → ∀𝑥𝜓))
 
Theorembj-biexex 35587 When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ∃𝑥𝜓) ↔ (∃𝑥𝜑 → ∃𝑥𝜓))
 
Theorembj-hbext 35588 Closed form of hbex 2319. (Contributed by BJ, 10-Oct-2019.)
(∀𝑦𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑦𝜑 → ∀𝑥𝑦𝜑))
 
Theorembj-nfalt 35589 Closed form of nfal 2317. (Contributed by BJ, 2-May-2019.)
(∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)
 
Theorembj-nfext 35590 Closed form of nfex 2318. (Contributed by BJ, 10-Oct-2019.)
(∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)
 
Theorembj-eeanvw 35591* Version of exdistrv 1960 with a disjoint variable condition on 𝑥, 𝑦 not requiring ax-11 2155. (The same can be done with eeeanv 2347 and ee4anv 2348.) (Contributed by BJ, 29-Sep-2019.) (Proof modification is discouraged.)
(∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
 
Theorembj-modal4 35592 First-order logic form of the modal axiom (4). See hba1 2290. This is the standard proof of the implication in modal logic (B5 4). Its dual statement is bj-modal4e 35593. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
(∀𝑥𝜑 → ∀𝑥𝑥𝜑)
 
Theorembj-modal4e 35593 First-order logic form of the modal axiom (4) using existential quantifiers. Dual statement of bj-modal4 35592 (hba1 2290). (Contributed by BJ, 21-Dec-2020.) (Proof modification is discouraged.)
(∃𝑥𝑥𝜑 → ∃𝑥𝜑)
 
Theorembj-modalb 35594 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)
 
Theorembj-wnf1 35595 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))
 
Theorembj-wnf2 35596 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))
 
Theorembj-wnfanf 35597 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))
 
Theorembj-wnfenf 35598 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(∃𝑥𝜑𝜓))
 
Theorembj-substax12 35599 Equivalent form of the axiom of substitution bj-ax12 35534. Although both sides need a DV condition on 𝑥, 𝑡 (or as in bj-ax12v3 35563 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 35600. Note that in the LHS, the reverse implication holds by equs4 2416 (or equs4v 2004 if a DV condition is added on 𝑥, 𝑡 as in bj-ax12 35534), and the forward implication is sbalex 2236.

The LHS can be read as saying that if there exists a setvar equal to a given term witnessing 𝜑, then all setvars equal to that term also witness 𝜑. An equivalent suggestive form for the LHS is ¬ (∃𝑥(𝑥 = 𝑡𝜑) ∧ ∃𝑥(𝑥 = 𝑡 ∧ ¬ 𝜑)), which expresses that there can be no two variables both equal to a given term, one witnessing 𝜑 and the other witnessing ¬ 𝜑. (Contributed by BJ, 21-May-2024.) (Proof modification is discouraged.)

((∃𝑥(𝑥 = 𝑡𝜑) → ∀𝑥(𝑥 = 𝑡𝜑)) ↔ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑))))
 
Theorembj-substw 35600* Weak form of the LHS of bj-substax12 35599 proved from the core axiom schemes. Compare ax12w 2130. (Contributed by BJ, 26-May-2024.) (Proof modification is discouraged.)
(𝑥 = 𝑡 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑡𝜑) → ∀𝑥(𝑥 = 𝑡𝜑))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47852
  Copyright terms: Public domain < Previous  Next >