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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | bj-spnfw 33901 | Theorem close to a closed form of spnfw 1975. (Contributed by BJ, 12-May-2019.) |
⊢ ((∃𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)) | ||
Theorem | bj-cbvexiw 33902* | Change bound variable. This is to cbvexvw 2035 what cbvaliw 2004 is to cbvalvw 2034. TODO: move after cbvalivw 2005. (Contributed by BJ, 17-Mar-2020.) |
⊢ (∃𝑥∃𝑦𝜓 → ∃𝑦𝜓) & ⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓)) ⇒ ⊢ (∃𝑥𝜑 → ∃𝑦𝜓) | ||
Theorem | bj-cbvexivw 33903* | Change bound variable. This is to cbvexvw 2035 what cbvalivw 2005 is to cbvalvw 2034. TODO: move after cbvalivw 2005. (Contributed by BJ, 17-Mar-2020.) |
⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓)) ⇒ ⊢ (∃𝑥𝜑 → ∃𝑦𝜓) | ||
Theorem | bj-modald 33904 | A short form of the axiom D of modal logic. (Contributed by BJ, 4-Apr-2021.) |
⊢ (∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑) | ||
Theorem | bj-denot 33905* | A weakening of ax-6 1961 and ax6v 1962. (Contributed by BJ, 4-Apr-2021.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑥 → ¬ ∀𝑦 ¬ 𝑦 = 𝑥) | ||
Theorem | bj-eqs 33906* | A lemma for substitutions, proved from Tarski's FOL. The version without DV (𝑥, 𝑦) is true but requires ax-13 2383. 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 33907* | Change bound variable. This is to cbvexvw 2035 what cbvalw 2033 is to cbvalvw 2034. (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 33908* | The general statement that ax12w 2128 proves. (Contributed by BJ, 20-Mar-2020.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝑦 = 𝑧 → (𝜓 ↔ 𝜃)) ⇒ ⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑥(𝜑 → 𝜓))) | ||
Theorem | bj-ax89 33909 | A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 2107 and ax-9 2115. Indeed, it is implied over propositional calculus by the conjunction of ax-8 2107 and ax-9 2115, as proved here. In the other direction, one can prove ax-8 2107 (respectively ax-9 2115) from bj-ax89 33909 by using mpan2 687 (respectively mpan 686) and equid 2010. TODO: move to main part. (Contributed by BJ, 3-Oct-2019.) |
⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑡)) | ||
Theorem | bj-elequ12 33910 | An identity law for the non-logical predicate, which combines elequ1 2112 and elequ2 2120. For the analogous theorems for class terms, see eleq1 2900, eleq2 2901 and eleq12 2902. TODO: move to main part. (Contributed by BJ, 29-Sep-2019.) |
⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑡)) | ||
Theorem | bj-cleljusti 33911* | One direction of cleljust 2114, requiring only ax-1 6-- ax-5 1902 and ax8v1 2109. (Contributed by BJ, 31-Dec-2020.) (Proof modification is discouraged.) |
⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦) | ||
Theorem | bj-alcomexcom 33912 | Commutation of universal quantifiers implies commutation of existential quantifiers. Can be placed in the ax-4 1801 section, soon after 2nexaln 1821, and used to prove excom 2159. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.) |
⊢ ((∀𝑥∀𝑦 ¬ 𝜑 → ∀𝑦∀𝑥 ¬ 𝜑) → (∃𝑦∃𝑥𝜑 → ∃𝑥∃𝑦𝜑)) | ||
Theorem | bj-hbalt 33913 | Closed form of hbal 2164. When in main part, prove hbal 2164 and hbald 2165 from it. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)) | ||
Theorem | axc11n11 33914 | Proof of axc11n 2443 from { ax-1 6-- ax-7 2006, axc11 2447 } . Almost identical to axc11nfromc11 35944. (Contributed by NM, 6-Jul-2021.) (Proof modification is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) | ||
Theorem | axc11n11r 33915 |
Proof of axc11n 2443 from { ax-1 6--
ax-7 2006, axc9 2393, axc11r 2379 } (note
that axc16 2253 is provable from { ax-1 6--
ax-7 2006, axc11r 2379 }).
Note that axc11n 2443 proves (over minimal calculus) that axc11 2447 and axc11r 2379 are equivalent. Therefore, axc11n11 33914 and axc11n11r 33915 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2447 appears slightly stronger since axc11n11r 33915 requires axc9 2393 while axc11n11 33914 does not). (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) | ||
Theorem | bj-axc16g16 33916* | Proof of axc16g 2252 from { ax-1 6-- ax-7 2006, axc16 2253 }. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) | ||
Theorem | bj-ax12v3 33917* | A weak version of ax-12 2167 which is stronger than ax12v 2168. Note that if one assumes reflexivity of equality ⊢ 𝑥 = 𝑥 (equid 2010), then bj-ax12v3 33917 implies ax-5 1902 over modal logic K (substitute 𝑥 for 𝑦). See also bj-ax12v3ALT 33918. (Contributed by BJ, 6-Jul-2021.) |
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | bj-ax12v3ALT 33918* | Alternate proof of bj-ax12v3 33917. Uses axc11r 2379 and axc15 2438 instead of ax-12 2167. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | bj-sb 33919* | A weak variant of sbid2 2546 not requiring ax-13 2383 nor ax-10 2136. On top of Tarski's FOL, one implication requires only ax12v 2168, and the other requires only sp 2172. (Contributed by BJ, 25-May-2021.) |
⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | bj-modalbe 33920 | The predicate-calculus version of the axiom (B) of modal logic. See also modal-b 2330. (Contributed by BJ, 20-Oct-2019.) |
⊢ (𝜑 → ∀𝑥∃𝑥𝜑) | ||
Theorem | bj-spst 33921 | Closed form of sps 2174. Once in main part, prove sps 2174 and spsd 2176 from it. (Contributed by BJ, 20-Oct-2019.) |
⊢ ((𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)) | ||
Theorem | bj-19.21bit 33922 | Closed form of 19.21bi 2178. (Contributed by BJ, 20-Oct-2019.) |
⊢ ((𝜑 → ∀𝑥𝜓) → (𝜑 → 𝜓)) | ||
Theorem | bj-19.23bit 33923 | Closed form of 19.23bi 2180. (Contributed by BJ, 20-Oct-2019.) |
⊢ ((∃𝑥𝜑 → 𝜓) → (𝜑 → 𝜓)) | ||
Theorem | bj-nexrt 33924 | Closed form of nexr 2181. Contrapositive of 19.8a 2170. (Contributed by BJ, 20-Oct-2019.) |
⊢ (¬ ∃𝑥𝜑 → ¬ 𝜑) | ||
Theorem | bj-alrim 33925 | Closed form of alrimi 2204. (Contributed by BJ, 2-May-2019.) |
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))) | ||
Theorem | bj-alrim2 33926 | Uncurried (imported) form of bj-alrim 33925. (Contributed by BJ, 2-May-2019.) |
⊢ ((Ⅎ𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → (𝜑 → ∀𝑥𝜓)) | ||
Theorem | bj-nfdt0 33927 | A theorem close to a closed form of nf5d 2284 and nf5dh 2142. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → (∀𝑥𝜑 → Ⅎ𝑥𝜓)) | ||
Theorem | bj-nfdt 33928 | Closed form of nf5d 2284 and nf5dh 2142. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → ((𝜑 → ∀𝑥𝜑) → (𝜑 → Ⅎ𝑥𝜓))) | ||
Theorem | bj-nexdt 33929 | Closed form of nexd 2214. (Contributed by BJ, 20-Oct-2019.) |
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓))) | ||
Theorem | bj-nexdvt 33930* | Closed form of nexdv 1928. (Contributed by BJ, 20-Oct-2019.) |
⊢ (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓)) | ||
Theorem | bj-alexbiex 33931 | Adding a second quantifier is a tranparent operation, (∀∃ case). (Contributed by BJ, 20-Oct-2019.) |
⊢ (∀𝑥∃𝑥𝜑 ↔ ∃𝑥𝜑) | ||
Theorem | bj-exexbiex 33932 | Adding a second quantifier is a tranparent operation, (∃∃ case). (Contributed by BJ, 20-Oct-2019.) |
⊢ (∃𝑥∃𝑥𝜑 ↔ ∃𝑥𝜑) | ||
Theorem | bj-alalbial 33933 | Adding a second quantifier is a tranparent operation, (∀∀ case). (Contributed by BJ, 20-Oct-2019.) |
⊢ (∀𝑥∀𝑥𝜑 ↔ ∀𝑥𝜑) | ||
Theorem | bj-exalbial 33934 | Adding a second quantifier is a tranparent operation, (∃∀ case). (Contributed by BJ, 20-Oct-2019.) |
⊢ (∃𝑥∀𝑥𝜑 ↔ ∀𝑥𝜑) | ||
Theorem | bj-19.9htbi 33935 | Strengthening 19.9ht 2331 by replacing its succedent with a biconditional (19.9t 2195 does have a biconditional succedent). This propagates. (Contributed by BJ, 20-Oct-2019.) |
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 ↔ 𝜑)) | ||
Theorem | bj-hbntbi 33936 | Strengthening hbnt 2294 by replacing its succedent with a biconditional. See also hbntg 32948 and hbntal 40767. (Contributed by BJ, 20-Oct-2019.) Proved from bj-19.9htbi 33935. (Proof modification is discouraged.) |
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑)) | ||
Theorem | bj-biexal1 33937 | A general FOL biconditional that generalizes 19.9ht 2331 among others. For this and the following theorems, see also 19.35 1869, 19.21 2198, 19.23 2202. 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 33938 | 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 33939 | 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 33940 | 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 33941 | 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 33942 | Closed form of hbex 2336. (Contributed by BJ, 10-Oct-2019.) |
⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)) | ||
Theorem | bj-nfalt 33943 | Closed form of nfal 2334. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑) | ||
Theorem | bj-nfext 33944 | Closed form of nfex 2335. (Contributed by BJ, 10-Oct-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∃𝑥𝜑) | ||
Theorem | bj-eeanvw 33945* | Version of exdistrv 1947 with a disjoint variable condition on 𝑥, 𝑦 not requiring ax-11 2151. (The same can be done with eeeanv 2363 and ee4anv 2364.) (Contributed by BJ, 29-Sep-2019.) (Proof modification is discouraged.) |
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) | ||
Theorem | bj-modal4 33946 | 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 33947. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) | ||
Theorem | bj-modal4e 33947 | First-order logic form of the modal axiom (4) using existential quantifiers. Dual statement of bj-modal4 33946 (hba1 2293). (Contributed by BJ, 21-Dec-2020.) (Proof modification is discouraged.) |
⊢ (∃𝑥∃𝑥𝜑 → ∃𝑥𝜑) | ||
Theorem | bj-modalb 33948 | 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 33949 | 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 33950 | 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 33951 | When 𝜑 is substituted for 𝜓, this statement expresses that weak nonfreeness implies the "forall" form of nonfreeness. (Contributed by BJ, 9-Dec-2023.) |
⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → ∀𝑥𝜓)) | ||
Theorem | bj-wnfenf 33952 | When 𝜑 is substituted for 𝜓, this statement expresses that weak nonfreeness implies the "exists" form of nonfreeness. (Contributed by BJ, 9-Dec-2023.) |
⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → 𝜓)) | ||
Syntax | wnnf 33953 | Syntax for the nonfreeness quantifier. |
wff Ⅎ'𝑥𝜑 | ||
Definition | df-bj-nnf 33954 |
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 1902 and ax5e 1904. 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 33981, we can prove (Ⅎ'𝑥𝜑, {{𝑥, 𝜑}}), from which the theorem follows. QED 2. Suppose that S also contains (the FOL version of) modal logic KB and commutation of quantifiers alcom 2153 and excom 2159 (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 1902, ax5e 1904, ax5ea 1905. 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 33955 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 33981, 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 33973, bj-nnfnt 33967, bj-nnfan 33975, bj-nnfor 33977, bj-nnfbit 33979, bj-nnfalt 33993, bj-nnfext 33994. 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 33973 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 33993 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 33993 and bj-nnfext 33994 are proved from positive propositional calculus with alcom 2153 and excom 2159 (possibly weakened by a DV condition on the quantifying variables), and modalB (via bj-19.12 33988). 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 1902 or ax5e 1904 or ax5ea . 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 33958 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 1904 or ax5ea 1905, we would use bj-nnfe 33960 or bj-nnfea 33962 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 1787, by bj-nnf-alrim 33982 and bj-nnfa1 33986 (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 1976. QED Compared with df-nf 1776, the present definition is stricter on positive propositional calculus (bj-nnfnfTEMP 33965) and equivalent on core FOL plus sp 2172 (bj-nfnnfTEMP 33985). While being stricter, it still holds for non-occurring variables (bj-nnfv 33981), 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 2136 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 33955 | 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 1842. From this and bj-nnfim 33973 and bj-nnfnt 33967, 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 33968) in order not to require sp 2172 (modal T). (Contributed by BJ, 27-Aug-2023.) |
⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓)) | ||
Theorem | bj-nnfbd 33956* | 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 33955. (Contributed by BJ, 27-Aug-2023.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒)) | ||
Theorem | bj-nnfbii 33957 | 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 33955. (Contributed by BJ, 18-Nov-2023.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓) | ||
Theorem | bj-nnfa 33958 | Nonfreeness implies the equivalent of ax-5 1902. See nf5r 2183, nf5ri 2185. (Contributed by BJ, 28-Jul-2023.) |
⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) | ||
Theorem | bj-nnfad 33959 | Nonfreeness implies the equivalent of ax-5 1902, deduction form. See nf5rd 2187. (Contributed by BJ, 2-Dec-2024.) |
⊢ (𝜑 → Ⅎ'𝑥𝜓) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) | ||
Theorem | bj-nnfe 33960 | Nonfreeness implies the equivalent of ax5e 1904. (Contributed by BJ, 28-Jul-2023.) |
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑)) | ||
Theorem | bj-nnfed 33961 | Nonfreeness implies the equivalent of ax5e 1904, deduction form. (Contributed by BJ, 2-Dec-2024.) |
⊢ (𝜑 → Ⅎ'𝑥𝜓) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 → 𝜓)) | ||
Theorem | bj-nnfea 33962 | Nonfreeness implies the equivalent of ax5ea 1905. (Contributed by BJ, 28-Jul-2023.) |
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑)) | ||
Theorem | bj-nnfead 33963 | Nonfreeness implies the equivalent of ax5ea 1905, deduction form. (Contributed by BJ, 2-Dec-2024.) |
⊢ (𝜑 → Ⅎ'𝑥𝜓) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓)) | ||
Theorem | bj-dfnnf2 33964 | Alternate definition of df-bj-nnf 33954 using only primitive symbols (→, ¬, ∀). (Contributed by BJ, 20-Aug-2023.) |
⊢ (Ⅎ'𝑥𝜑 ↔ ((𝜑 → ∀𝑥𝜑) ∧ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))) | ||
Theorem | bj-nnfnfTEMP 33965 | New nonfreeness implies old nonfreeness on implicational calculus (the proof indicates it uses ax-3 8 because of set.mm's definition of the biconditional, but the proof actually holds in minimal implicational calculus). (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1776 except via df-nf 1776 directly. (Proof modification is discouraged.) |
⊢ (Ⅎ'𝑥𝜑 → Ⅎ𝑥𝜑) | ||
Theorem | bj-wnfnf 33966 | When 𝜑 is substituted for 𝜓, this statement expresses nonfreeness in the weak form of nonfreeness (∃ → ∀). Note that this could also be proved from bj-nnfim 33973, bj-nnfe1 33987 and bj-nnfa1 33986. (Contributed by BJ, 9-Dec-2023.) |
⊢ Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓) | ||
Theorem | bj-nnfnt 33967 | A variable is nonfree in a formula if and only if it is nonfree in its negation. The foward implication is intuitionistically valid (and that direction is sufficient for the purpose of recursively proving that some formulas have a given variable not free in them, like bj-nnfim 33973). Intuitionistically, ⊢ (Ⅎ'𝑥¬ 𝜑 ↔ Ⅎ'𝑥¬ ¬ 𝜑). See nfnt 1847. (Contributed by BJ, 28-Jul-2023.) |
⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥 ¬ 𝜑) | ||
Theorem | bj-nnftht 33968 | A variable is nonfree in a theorem. The antecedent is in the "strong necessity" modality of modal logic in order not to require sp 2172 (modal T), as in bj-nnfbi 33955. (Contributed by BJ, 28-Jul-2023.) |
⊢ ((𝜑 ∧ ∀𝑥𝜑) → Ⅎ'𝑥𝜑) | ||
Theorem | bj-nnfth 33969 | A variable is nonfree in a theorem, inference form. (Contributed by BJ, 28-Jul-2023.) |
⊢ 𝜑 ⇒ ⊢ Ⅎ'𝑥𝜑 | ||
Theorem | bj-nnfnth 33970 | A variable is nonfree in the negation of a theorem, inference form. (Contributed by BJ, 27-Aug-2023.) |
⊢ ¬ 𝜑 ⇒ ⊢ Ⅎ'𝑥𝜑 | ||
Theorem | bj-nnfim1 33971 | A consequence of nonfreeness in the antecedent and the consequent of an implication. (Contributed by BJ, 27-Aug-2023.) |
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))) | ||
Theorem | bj-nnfim2 33972 | A consequence of nonfreeness in the antecedent and the consequent of an implication. (Contributed by BJ, 27-Aug-2023.) |
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((∀𝑥𝜑 → ∃𝑥𝜓) → (𝜑 → 𝜓))) | ||
Theorem | bj-nnfim 33973 | Nonfreeness in the antecedent and the consequent of an implication implies nonfreeness in the implication. (Contributed by BJ, 27-Aug-2023.) |
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 → 𝜓)) | ||
Theorem | bj-nnfimd 33974 | Nonfreeness in the antecedent and the consequent of an implication implies nonfreeness in the implication, deduction form. (Contributed by BJ, 2-Dec-2023.) |
⊢ (𝜑 → Ⅎ'𝑥𝜓) & ⊢ (𝜑 → Ⅎ'𝑥𝜒) ⇒ ⊢ (𝜑 → Ⅎ'𝑥(𝜓 → 𝜒)) | ||
Theorem | bj-nnfan 33975 | Nonfreeness in both conjuncts implies nonfreeness in the conjunction. (Contributed by BJ, 19-Nov-2023.) In classical logic, there is a proof using the definition of conjunction in terms of implication and negation, so using bj-nnfim 33973, bj-nnfnt 33967 and bj-nnfbi 33955, but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.) |
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ∧ 𝜓)) | ||
Theorem | bj-nnfand 33976 | Nonfreeness in both conjuncts implies nonfreeness in the conjunction, deduction form. Note: compared with the proof of bj-nnfan 33975, it has two more essential steps but fewer total steps (since there are fewer intermediate formulas to build) and is easier to follow and understand. This statement is of intermediate complexity: for simpler statements, closed-style proofs like that of bj-nnfan 33975 will generally be shorter than deduction-style proofs while still easy to follow, while for more complex statements, the opposite will be true (and deduction-style proofs like that of bj-nnfand 33976 will generally be easier to understand). (Contributed by BJ, 19-Nov-2023.) (Proof modification is discouraged.) |
⊢ (𝜑 → Ⅎ'𝑥𝜓) & ⊢ (𝜑 → Ⅎ'𝑥𝜒) ⇒ ⊢ (𝜑 → Ⅎ'𝑥(𝜓 ∧ 𝜒)) | ||
Theorem | bj-nnfor 33977 | Nonfreeness in both disjuncts implies nonfreeness in the disjunction. (Contributed by BJ, 19-Nov-2023.) In classical logic, there is a proof using the definition of disjunction in terms of implication and negation, so using bj-nnfim 33973, bj-nnfnt 33967 and bj-nnfbi 33955, but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.) |
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ∨ 𝜓)) | ||
Theorem | bj-nnford 33978 | Nonfreeness in both disjuncts implies nonfreeness in the disjunction, deduction form. See comments for bj-nnfor 33977 and bj-nnfand 33976. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.) |
⊢ (𝜑 → Ⅎ'𝑥𝜓) & ⊢ (𝜑 → Ⅎ'𝑥𝜒) ⇒ ⊢ (𝜑 → Ⅎ'𝑥(𝜓 ∨ 𝜒)) | ||
Theorem | bj-nnfbit 33979 | Nonfreeness in both sides implies nonfreeness in the biconditional. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.) |
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ↔ 𝜓)) | ||
Theorem | bj-nnfbid 33980 | Nonfreeness in both sides implies nonfreeness in the biconditional, deduction form. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.) |
⊢ (𝜑 → Ⅎ'𝑥𝜓) & ⊢ (𝜑 → Ⅎ'𝑥𝜒) ⇒ ⊢ (𝜑 → Ⅎ'𝑥(𝜓 ↔ 𝜒)) | ||
Theorem | bj-nnfv 33981* | A non-occurring variable is nonfree in a formula. (Contributed by BJ, 28-Jul-2023.) |
⊢ Ⅎ'𝑥𝜑 | ||
Theorem | bj-nnf-alrim 33982 | Proof of the closed form of alrimi 2204 from modalK (compare alrimiv 1919). See also bj-alrim 33925. Actually, most proofs between 19.3t 2192 and 2sbbid 2238 could be proved without ax-12 2167. (Contributed by BJ, 20-Aug-2023.) |
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))) | ||
Theorem | bj-nnf-exlim 33983 | Proof of the closed form of exlimi 2208 from modalK (compare exlimiv 1922). See also bj-sylget2 33853. (Contributed by BJ, 2-Dec-2023.) |
⊢ (Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜓))) | ||
Theorem | bj-dfnnf3 33984 | Alternate definition of nonfreeness when sp 2172 is available. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1776. (Proof modification is discouraged.) |
⊢ (Ⅎ'𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) | ||
Theorem | bj-nfnnfTEMP 33985 | New nonfreeness is equivalent to old nonfreeness on core FOL axioms plus sp 2172. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1776 except via df-nf 1776 directly. (Proof modification is discouraged.) |
⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ𝑥𝜑) | ||
Theorem | bj-nnfa1 33986 | See nfa1 2146. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
⊢ Ⅎ'𝑥∀𝑥𝜑 | ||
Theorem | bj-nnfe1 33987 | See nfe1 2145. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
⊢ Ⅎ'𝑥∃𝑥𝜑 | ||
Theorem | bj-19.12 33988 | See 19.12 2338. Could be labeled "exalimalex" for "'there exists for all' implies 'for all there exists'". This proof is from excom 2159 and modal (B) on top of modalK logic. (Contributed by BJ, 12-Aug-2023.) The proof should not rely on df-nf 1776 or df-bj-nnf 33954, directly or indirectly. (Proof modification is discouraged.) |
⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑) | ||
Theorem | bj-nnflemaa 33989 | One of four lemmas for nonfreeness: antecedent and consequent both expressed using universal quantifier. Note: this is bj-hbalt 33913. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)) | ||
Theorem | bj-nnflemee 33990 | One of four lemmas for nonfreeness: antecedent and consequent both expressed using existential quantifier. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
⊢ (∀𝑥(∃𝑦𝜑 → 𝜑) → (∃𝑦∃𝑥𝜑 → ∃𝑥𝜑)) | ||
Theorem | bj-nnflemae 33991 | One of four lemmas for nonfreeness: antecedent expressed with universal quantifier and consequent expressed with existential quantifier. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → (∃𝑥𝜑 → ∀𝑦∃𝑥𝜑)) | ||
Theorem | bj-nnflemea 33992 | One of four lemmas for nonfreeness: antecedent expressed with existential quantifier and consequent expressed with universal quantifier. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
⊢ (∀𝑥(∃𝑦𝜑 → 𝜑) → (∃𝑦∀𝑥𝜑 → ∀𝑥𝜑)) | ||
Theorem | bj-nnfalt 33993 | See nfal 2334 and bj-nfalt 33943. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
⊢ (∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦∀𝑥𝜑) | ||
Theorem | bj-nnfext 33994 | See nfex 2335 and bj-nfext 33944. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
⊢ (∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦∃𝑥𝜑) | ||
Theorem | bj-stdpc5t 33995 | Alias of bj-nnf-alrim 33982 for labeling consistency (a standard predicate calculus axiom). Closed form of stdpc5 2199 proved from modalK (obsoleting stdpc5v 1930). (Contributed by BJ, 2-Dec-2023.) Use bj-nnf-alrim 33982 instead. (New usaged is discouraged.) |
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))) | ||
Theorem | bj-19.21t 33996 | Statement 19.21t 2197 proved from modalK (obsoleting 19.21v 1931). (Contributed by BJ, 2-Dec-2023.) |
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))) | ||
Theorem | bj-19.23t 33997 | Statement 19.23t 2201 proved from modalK (obsoleting 19.23v 1934). (Contributed by BJ, 2-Dec-2023.) |
⊢ (Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))) | ||
Theorem | bj-19.36im 33998 | One direction of 19.36 2223 from the same axioms as 19.36imv 1937. (Contributed by BJ, 2-Dec-2023.) |
⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))) | ||
Theorem | bj-19.37im 33999 | One direction of 19.37 2225 from the same axioms as 19.37imv 1939. (Contributed by BJ, 2-Dec-2023.) |
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))) | ||
Theorem | bj-19.42t 34000 | Closed form of 19.42 2229 from the same axioms as 19.42v 1945. (Contributed by BJ, 2-Dec-2023.) |
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))) |
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