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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | bj-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.) |
⊢ (∃𝑥𝜑 → 𝜓) & ⊢ (𝜒 → 𝜑) ⇒ ⊢ (∃𝑥𝜒 → 𝜓) | ||
Theorem | bj-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 → (∃𝑥𝜃 → 𝜏)) & ⊢ (𝜓 → (𝜒 → 𝜃)) ⇒ ⊢ (𝜑 → (∃𝑥𝜒 → 𝜏)) | ||
Theorem | bj-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(𝜑 → 𝜓))) | ||
Theorem | bj-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 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))) | ||
Theorem | bj-nexdh 35505 | Closed form of nexdh 1869 (actually, its general instance). (Contributed by BJ, 6-May-2019.) |
⊢ (∀𝑥(𝜑 → ¬ 𝜓) → ((𝜒 → ∀𝑥𝜑) → (𝜒 → ¬ ∃𝑥𝜓))) | ||
Theorem | bj-nexdh2 35506 | Uncurried (imported) form of bj-nexdh 35505. (Contributed by BJ, 6-May-2019.) |
⊢ ((∀𝑥(𝜑 → ¬ 𝜓) ∧ (𝜒 → ∀𝑥𝜑)) → (𝜒 → ¬ ∃𝑥𝜓)) | ||
Theorem | bj-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.) |
⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓))) | ||
Theorem | bj-hbyfrbi 35508 | Version of bj-hbxfrbi 35507 with existential quantifiers. (Contributed by BJ, 23-Aug-2023.) |
⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → ((∃𝑥𝜑 → 𝜑) ↔ (∃𝑥𝜓 → 𝜓))) | ||
Theorem | bj-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 → (𝜓 → 𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))) | ||
Theorem | bj-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 → (𝜓 → 𝜒)) ⇒ ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒)) | ||
Theorem | bj-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 → (𝜓 → 𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒))) | ||
Theorem | bj-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 → (𝜓 → 𝜒)) & ⊢ (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒)) ⇒ ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒)) | ||
Theorem | bj-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 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓))) | ||
Theorem | bj-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 → (𝜓 ↔ 𝜒)) & ⊢ (𝜒 → ∀𝑥𝜒) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓))) | ||
Theorem | bj-nfimt 35515 | Closed form of nfim 1900 and curried (exported) form of nfimt 1899. (Contributed by BJ, 20-Oct-2021.) |
⊢ (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → Ⅎ𝑥(𝜑 → 𝜓))) | ||
Theorem | bj-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.) |
⊢ (∀𝑦∃𝑥𝜒 → (∀𝑦∀𝑥(𝜒 → (𝜑 → 𝜓)) → ((∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) → (∀𝑦(∃𝑥𝜓 → 𝜓) → (∀𝑥𝜑 → ∀𝑦𝜓))))) | ||
Theorem | bj-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.) |
⊢ (∀𝑥∃𝑦𝜒 → (∀𝑥∀𝑦(𝜒 → (𝜑 → 𝜓)) → (∀𝑥(𝜑 → ∀𝑦𝜑) → ((∃𝑥∃𝑦𝜓 → ∃𝑦𝜓) → (∃𝑥𝜑 → ∃𝑦𝜓))))) | ||
Theorem | bj-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(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓)) | ||
Theorem | bj-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 → (𝜓 → 𝜒)) ⇒ ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)) | ||
Theorem | bj-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(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓)) | ||
Theorem | bj-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 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓))) | ||
Theorem | bj-cbvalim 35522* | A lemma used to prove bj-cbval 35526 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 35523* | A lemma used to prove bj-cbvex 35527 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 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 → (𝜑 → 𝜓)) & ⊢ ∀𝑦∃𝑥𝜒 ⇒ ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) | ||
Theorem | bj-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 → (𝜑 → 𝜓)) & ⊢ ∀𝑥∃𝑦𝜒 ⇒ ⊢ (∃𝑥𝜑 → ∃𝑦𝜓) | ||
Theorem | bj-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.) |
⊢ ∀𝑦∃𝑥 𝑥 = 𝑦 & ⊢ ∀𝑥∃𝑦 𝑦 = 𝑥 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦) ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) | ||
Theorem | bj-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.) |
⊢ ∀𝑦∃𝑥 𝑥 = 𝑦 & ⊢ ∀𝑥∃𝑦 𝑦 = 𝑥 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦) ⇒ ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓) | ||
Syntax | wmoo 35528 | Syntax for BJ's version of the uniqueness quantifier. |
wff ∃**𝑥𝜑 | ||
Definition | df-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 ↔ ∀𝑧∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | ||
Theorem | bj-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 = 𝑧 ↔ 𝑦 = 𝑧) | ||
Theorem | bj-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 = 𝑦 → 𝜑)) → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | bj-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.) |
⊢ (∀𝑥∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) → ∀𝑦∀𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑))) | ||
Theorem | bj-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 = 𝑡 → 𝜑))) | ||
Theorem | bj-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 = 𝑡 → 𝜑))) | ||
Theorem | bj-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) | ||
Theorem | bj-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)) | ||
Theorem | bj-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 = 𝑦 ∧ 𝜑) ↔ ∀𝑥𝜓) | ||
Theorem | bj-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 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | bj-ssbid2 35539 | A special case of sbequ2 2242. (Contributed by BJ, 22-Dec-2020.) |
⊢ ([𝑥 / 𝑥]𝜑 → 𝜑) | ||
Theorem | bj-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]𝜑 → 𝜑) | ||
Theorem | bj-ssbid1 35541 | A special case of sbequ1 2241. (Contributed by BJ, 22-Dec-2020.) |
⊢ (𝜑 → [𝑥 / 𝑥]𝜑) | ||
Theorem | bj-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) | ||
Theorem | bj-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)) | ||
Theorem | bj-ax6elem2 35544* | Lemma for bj-ax6e 35545. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) |
⊢ (∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦) | ||
Theorem | bj-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 | ||
Theorem | bj-spimvwt 35546* | Closed form of spimvw 2000. See also spimt 2386. (Contributed by BJ, 8-Nov-2021.) |
⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → 𝜓)) | ||
Theorem | bj-spnfw 35547 | Theorem close to a closed form of spnfw 1984. (Contributed by BJ, 12-May-2019.) |
⊢ ((∃𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)) | ||
Theorem | bj-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.) |
⊢ (∃𝑥∃𝑦𝜓 → ∃𝑦𝜓) & ⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓)) ⇒ ⊢ (∃𝑥𝜑 → ∃𝑦𝜓) | ||
Theorem | bj-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 → (𝜑 → 𝜓)) ⇒ ⊢ (∃𝑥𝜑 → ∃𝑦𝜓) | ||
Theorem | bj-modald 35550 | A short form of the axiom D of modal logic. (Contributed by BJ, 4-Apr-2021.) |
⊢ (∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑) | ||
Theorem | bj-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) | ||
Theorem | bj-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 = 𝑦 → 𝜑)) | ||
Theorem | bj-cbvexw 35553* | Change bound variable. This is to cbvexvw 2041 what cbvalw 2039 is to cbvalvw 2040. (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 35554* | The general statement that ax12w 2130 proves. (Contributed by BJ, 20-Mar-2020.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝑦 = 𝑧 → (𝜓 ↔ 𝜃)) ⇒ ⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑥(𝜑 → 𝜓))) | ||
Theorem | bj-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 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑡)) | ||
Theorem | bj-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 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑡)) | ||
Theorem | bj-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 = 𝑥 ∧ 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦) | ||
Theorem | bj-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.) |
⊢ ((∀𝑥∀𝑦 ¬ 𝜑 → ∀𝑦∀𝑥 ¬ 𝜑) ↔ (∃𝑦∃𝑥𝜑 → ∃𝑥∃𝑦𝜑)) | ||
Theorem | bj-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 → ∀𝑥∀𝑦𝜑)) | ||
Theorem | axc11n11 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) | ||
Theorem | axc11n11r 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) | ||
Theorem | bj-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)) | ||
Theorem | bj-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 = 𝑦 → 𝜑))) | ||
Theorem | bj-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 = 𝑦 → 𝜑))) | ||
Theorem | bj-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 = 𝑦 → 𝜑))) | ||
Theorem | bj-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 → ∀𝑥∃𝑥𝜑) | ||
Theorem | bj-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.) |
⊢ ((𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)) | ||
Theorem | bj-19.21bit 35568 | Closed form of 19.21bi 2183. (Contributed by BJ, 20-Oct-2019.) |
⊢ ((𝜑 → ∀𝑥𝜓) → (𝜑 → 𝜓)) | ||
Theorem | bj-19.23bit 35569 | Closed form of 19.23bi 2185. (Contributed by BJ, 20-Oct-2019.) |
⊢ ((∃𝑥𝜑 → 𝜓) → (𝜑 → 𝜓)) | ||
Theorem | bj-nexrt 35570 | Closed form of nexr 2186. Contrapositive of 19.8a 2175. (Contributed by BJ, 20-Oct-2019.) |
⊢ (¬ ∃𝑥𝜑 → ¬ 𝜑) | ||
Theorem | bj-alrim 35571 | Closed form of alrimi 2207. (Contributed by BJ, 2-May-2019.) |
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))) | ||
Theorem | bj-alrim2 35572 | Uncurried (imported) form of bj-alrim 35571. (Contributed by BJ, 2-May-2019.) |
⊢ ((Ⅎ𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → (𝜑 → ∀𝑥𝜓)) | ||
Theorem | bj-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)) | ||
Theorem | bj-nfdt 35574 | Closed form of nf5d 2281 and nf5dh 2144. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → ((𝜑 → ∀𝑥𝜑) → (𝜑 → Ⅎ𝑥𝜓))) | ||
Theorem | bj-nexdt 35575 | Closed form of nexd 2215. (Contributed by BJ, 20-Oct-2019.) |
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓))) | ||
Theorem | bj-nexdvt 35576* | Closed form of nexdv 1940. (Contributed by BJ, 20-Oct-2019.) |
⊢ (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓)) | ||
Theorem | bj-alexbiex 35577 | 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 35578 | 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 35579 | 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 35580 | 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 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) → (∃𝑥𝜑 ↔ 𝜑)) | ||
Theorem | bj-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)) | ||
Theorem | bj-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)) | ||
Theorem | bj-biexal2 35584 | 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 35585 | 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 35586 | 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 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)) | ||
Theorem | bj-hbext 35588 | Closed form of hbex 2319. (Contributed by BJ, 10-Oct-2019.) |
⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)) | ||
Theorem | bj-nfalt 35589 | Closed form of nfal 2317. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑) | ||
Theorem | bj-nfext 35590 | Closed form of nfex 2318. (Contributed by BJ, 10-Oct-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∃𝑥𝜑) | ||
Theorem | bj-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.) |
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) | ||
Theorem | bj-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 → ∀𝑥∀𝑥𝜑) | ||
Theorem | bj-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.) |
⊢ (∃𝑥∃𝑥𝜑 → ∃𝑥𝜑) | ||
Theorem | bj-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) | ||
Theorem | bj-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)) | ||
Theorem | bj-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)) | ||
Theorem | bj-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)) | ||
Theorem | bj-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(∃𝑥𝜑 → 𝜓)) | ||
Theorem | bj-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 = 𝑡 ∧ 𝜑) ∧ ∃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 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) ↔ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))) | ||
Theorem | bj-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 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) |
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